{"id":3916,"date":"2018-12-02T13:18:13","date_gmt":"2018-12-02T10:18:13","guid":{"rendered":"http:\/\/java.mazurok.com\/?p=3916"},"modified":"2018-12-19T00:30:05","modified_gmt":"2018-12-18T21:30:05","slug":"e-olymp-1128-%d0%bf%d1%80%d0%be%d0%b1%d0%bb%d0%b5%d0%bc%d0%b0-%d0%bb%d0%be%d0%bd%d0%b3%d0%b8","status":"publish","type":"post","link":"https:\/\/java.mazurok.com\/?p=3916","title":{"rendered":"e-olymp 1128. \u041f\u0440\u043e\u0431\u043b\u0435\u043c\u0430 \u041b\u043e\u043d\u0433\u0438"},"content":{"rendered":"

\u0417\u0430\u0434\u0430\u0447\u0430<\/h1>\n

\u041b\u043e\u043d\u0433\u0438 \u0445\u043e\u0440\u043e\u0448\u043e \u0440\u0430\u0437\u0431\u0438\u0440\u0430\u0435\u0442\u0441\u044f \u0432 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0435, \u043e\u043d \u043b\u044e\u0431\u0438\u0442 \u0437\u0430\u0434\u0443\u043c\u044b\u0432\u0430\u0442\u044c\u0441\u044f \u043d\u0430\u0434 \u0442\u0440\u0443\u0434\u043d\u044b\u043c\u0438 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u043c\u0438 \u0437\u0430\u0434\u0430\u0447\u0430\u043c\u0438, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043c\u043e\u0433\u0443\u0442 \u0431\u044b\u0442\u044c \u0440\u0435\u0448\u0435\u043d\u044b \u043f\u0440\u0438 \u043f\u043e\u043c\u043e\u0449\u0438 \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u0438\u0437\u044f\u0449\u043d\u044b\u0445 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u043e\u0432. \u0418 \u0432\u043e\u0442 \u0442\u0430\u043a\u0430\u044f \u0437\u0430\u0434\u0430\u0447\u043a\u0430 \u0432\u043e\u0437\u043d\u0438\u043a\u043b\u0430:
\n\u0414\u0430\u043d\u043e \u0446\u0435\u043b\u043e\u0435 \u0447\u0438\u0441\u043b\u043e [latex]n[\/latex] [latex](1 < n < 231)[\/latex], \u0412\u044b \u0434\u043e\u043b\u0436\u043d\u044b \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c [latex]\\sum\\limits_{i=1}^n gcd [\/latex] \u0434\u043b\u044f \u0432\u0441\u0435\u0445 [latex] 1 \u2264 i \u2264 n[\/latex].
\n"\u041e, \u044f \u0437\u043d\u0430\u044e, \u044f \u0437\u043d\u0430\u044e!" — \u0432\u043e\u0441\u043a\u043b\u0438\u043a\u043d\u0443\u043b \u041b\u043e\u043d\u0433\u0438! \u0410 \u0437\u043d\u0430\u0435\u0442\u0435 \u043b\u0438 \u0412\u044b? \u041f\u043e\u0436\u0430\u043b\u0443\u0439\u0441\u0442\u0430, \u0440\u0435\u0448\u0438\u0442\u0435 \u0435\u0451.<\/p>\n

\u0412\u0445\u043e\u0434\u043d\u044b\u0435 \u0434\u0430\u043d\u043d\u044b\u0435<\/h1>\n

\u041a\u0430\u0436\u0434\u0430\u044f \u0441\u0442\u0440\u043e\u043a\u0430 \u0441\u043e\u0434\u0435\u0440\u0436\u0438\u0442 \u043e\u0434\u043d\u043e \u0447\u0438\u0441\u043b\u043e [latex]n[\/latex].<\/p>\n

\u0412\u044b\u0445\u043e\u0434\u043d\u044b\u0435 \u0434\u0430\u043d\u043d\u044b\u0435<\/h1>\n

\u0414\u043b\u044f \u043a\u0430\u0436\u0434\u043e\u0433\u043e \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f [latex]n[\/latex] \u0441\u043b\u0435\u0434\u0443\u0435\u0442 \u0432\u044b\u0432\u0435\u0441\u0442\u0438 \u0432 \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u043e\u0439 \u0441\u0442\u0440\u043e\u043a\u0435 \u0441\u0443\u043c\u043c\u0443 [latex]\\sum\\limits_{i=1}^n gcd [\/latex] \u0434\u043b\u044f \u0432\u0441\u0435\u0445 [latex] 1 \u2264 i \u2264 n[\/latex].<\/p>\n

\u0422\u0435\u0441\u0442\u044b<\/h1>\n\n\n\n\n\n\n\n
\u0412\u0445\u043e\u0434\u043d\u044b\u0435 \u0434\u0430\u043d\u043d\u044b\u0435<\/td>\n\u0412\u044b\u0445\u043e\u0434\u043d\u044b\u0435 \u0434\u0430\u043d\u043d\u044b\u0435<\/td>\n<\/tr>\n
[latex]2[\/latex] [latex]6[\/latex]<\/td>\n$3$
\n$15$<\/td>\n<\/tr>\n
[latex]1[\/latex] [latex]50[\/latex] [latex]100[\/latex]<\/td>\n$1$
\n$195$
\n$520$
\n<\/tr>\n
[latex]7[\/latex] [latex]4791[\/latex] [latex]12345678[\/latex] [latex]478900[\/latex]<\/td>\n$13$
\n$15965$
\n$170994915$
\n$4980040$
\n<\/tr>\n
[latex]123[\/latex] [latex]7777[\/latex] [latex]157423949[\/latex] [latex]904573[\/latex] $2147483648$<\/td>\n$405$
\n$54873$
\n$613124817$
\n$1809145$
\n$35433480192$
\n<\/tr>\n<\/tbody>\n<\/table>\n

\u041a\u043e\u0434 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u043c\u044b<\/h1>\n
import java.util.*;\r\nimport java.lang.*;\r\nimport java.io.*;\r\n\r\nclass Main\r\n{\r\n\tpublic static void main (String[] args) throws java.lang.Exception\r\n\t{\r\n\t\tlong n, gcd, primeNumInPow, power;\r\n\t\tScanner in = new Scanner(System.in);\r\n    \twhile(in.hasNextLong()){\r\n    \t\tn = in.nextLong();\r\n        \tgcd = 1;\r\n        \tfor(long i = 2; i * i <= n; i++) {\r\n            \tif(n % i == 0) {\r\n                \tpower = 0; primeNumInPow = 1;\r\n                \twhile(n % i == 0) {\r\n                    \tpower++;\r\n                    \tprimeNumInPow *= i;\r\n                    \tn \/= i;\r\n                \t}\r\n                \tgcd *= (power + 1)*primeNumInPow - power*primeNumInPow\/i;\r\n            \t}\r\n        \t}       \r\n        \tif (n > 1) gcd *= (2*n - 1);\r\n        \tSystem.out.println(gcd);\r\n    \t}\r\n\t}\r\n}<\/pre>\n
\u0421\u043e\u0433\u043b\u0430\u0441\u043d\u043e \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0443 \u041d\u041e\u0414, \u0435\u0441\u043b\u0438 \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u0447\u0438\u0441\u043b\u0430 [latex]a_1[\/latex] \u0438 [latex]a_2[\/latex] \u0432\u0437\u0430\u0438\u043c\u043d\u043e \u043f\u0440\u043e\u0441\u0442\u044b, \u0442\u043e [latex]\\gcd \\left(a_1 \\cdot a_2, c\\right) = \\gcd \\left(a_1, c\\right) \\cdot \\gcd \\left(a_2, c\\right)[\/latex], \u0433\u0434\u0435 [latex]c[\/latex] — \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u0430\u044f \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u0430. \u0415\u0441\u043b\u0438 \u0436\u0435 \u0432\u043c\u0435\u0441\u0442\u043e [latex]c[\/latex] \u0432\u0437\u044f\u0442\u044c [latex]i[\/latex] ([latex] 1 \u2264 i \u2264 a_1 \\cdot a_2[\/latex]) \u0438 \u043f\u0440\u043e\u0441\u0443\u043c\u043c\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u043f\u043e [latex]i[\/latex] \u043e\u0431\u0435 \u0447\u0430\u0441\u0442\u0438 \u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u0430, \u043f\u043e\u043b\u0443\u0447\u0438\u043c:
\n[latex]\\sum\\limits_{i=1}^{a_1 \\cdot a_2} \\gcd \\left(a_1 \\cdot a_2, i\\right) = \\sum\\limits_{i=1}^{a_1 \\cdot a_2} \\left(\\gcd \\left(a_1, i\\right) \\cdot \\gcd \\left(a_2, i\\right)\\right) = \\sum\\limits_{i=1}^{a_1} \\gcd \\left(a_1, i\\right) \\cdot \\sum\\limits_{i=1}^{a_2} \\gcd \\left(a_2, i\\right)[\/latex].
\n\u0417\u043d\u0430\u0447\u0438\u0442 \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0434\u0430\u043d\u043d\u043e\u0435 \u0447\u0438\u0441\u043b\u043e \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0438\u0442\u044c \u043a\u0430\u043a \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0441\u0442\u044b\u0445 \u0432 \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u0441\u0442\u0435\u043f\u0435\u043d\u044f\u0445. \u042d\u0442\u0438 \u0447\u0438\u0441\u043b\u0430, \u043e\u0447\u0435\u0432\u0438\u0434\u043d\u043e, \u0431\u0443\u0434\u0443\u0442 \u0432\u0437\u0430\u0438\u043c\u043d\u043e \u043f\u0440\u043e\u0441\u0442\u044b\u043c\u0438, \u0438\u0437 \u0447\u0435\u0433\u043e \u0441\u043b\u0435\u0434\u0443\u0435\u0442 \u0432\u043e\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442\u044c \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u044f \u0434\u0430\u043d\u043d\u043e\u0433\u043e \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0438 \u043f\u043e\u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0433\u043e \u0441\u0443\u043c\u043c\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u043f\u043e [latex]i[\/latex].
\n\u0422\u0435\u043f\u0435\u0440\u044c \u0434\u043e\u043a\u0430\u0436\u0435\u043c, \u0447\u0442\u043e \u0434\u043b\u044f \u043b\u044e\u0431\u043e\u0433\u043e \u043f\u0440\u043e\u0441\u0442\u043e\u0433\u043e \u0447\u0438\u0441\u043b\u0430 [latex]p[\/latex] \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u0438 [latex]a\\geqslant 1[\/latex] \u0432\u0435\u0440\u043d\u043e \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0435 \u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u043e:
\n[latex]\\sum\\limits_{i=1}^{p^a} \\gcd\\left(p^a, i\\right) = \\left(a + 1\\right)\\cdot p^a — a \\cdot p^{a-1} [\/latex].
\n\u041e\u0431\u043e\u0437\u043d\u0430\u0447\u0438\u043c $\\sum\\limits_{i=1}^{r} \\gcd\\left(r, i\\right)$ \u043a\u0430\u043a $g\\left(r\\right)$.
\n\u0411\u0430\u0437\u0430 \u0438\u043d\u0434\u0443\u043a\u0446\u0438\u0438:
\n[latex]a = 1[\/latex]:
\n$$g\\left(p\\right) = \\gcd\\left(p, 1\\right) + \\gcd\\left(p, 2\\right) + \\ldots + \\gcd\\left(p, p\\right) = \\left(p — 1 \\right) + p = 2 \\cdot p — 1.$$
\n\u0415\u0441\u043b\u0438 [latex]a = 2[\/latex]:
\n$$g\\left(p^{2}\\right) = \\gcd\\left(p^{2}, 1\\right) + \\gcd\\left(p^{2}, 2\\right) + \\ldots + \\gcd\\left(p^{2}, p\\right) + \\gcd\\left(p^{2}, p + 1\\right) + \\ldots +  \\ + \\gcd\\left(p^{2}, 2 \\cdot p\\right) + \\ldots + \\gcd\\left(p^{2}, p^{2}\\right) = 1 + 1 + \\ldots + p + 1 + \\ldots + p + \\ldots + p^{2} = \\ = \\left( p^{2} — p \\right) + p \\cdot \\left( p — 1 \\right) + p^{2} = 3 \\cdot p^{2} — 2\\cdot p.$$
\n\u0414\u043b\u044f \u043b\u044e\u0431\u044b\u0445 $a \\geqslant 2$:
\n$$g\\left(p^{a}\\right) = \\sum\\limits_{j=1}^{p^{a-1}} \\gcd\\left(p^a, j\\right) + \\sum\\limits_{j=p^{a — 1} + 1}^{p^{a} — 1} \\gcd\\left(p^a, j\\right) + p^{a} =g\\left(p^{a — 1}\\right) + p^{a} + \\ + \\sum\\limits_{j=p^{a — 1} + 1}^{p^{a} — 1} \\gcd\\left(p^a — 1, j\\right).$$
\n\u041f\u0440\u0438\u0447\u0435\u043c:
\n$$\\sum\\limits_{j=p^{a — 1} + 1}^{p^{a} — 1} \\gcd\\left(p^a — 1, j\\right) = \\sum\\limits_{j=1}^{p^{a} — p^{a-1} — 1} \\gcd\\left(p^{a — 1}, j\\right) = \\ = \\sum\\limits_{j=1}^{p^{a} — p^{a-1}} \\gcd\\left(p^{a — 1}, j\\right) — p^{a — 1} = \\left( p — 1\\right)\\cdot g\\left(p^{a-1}\\right) — p^{a-1}.$$
\n\u041e\u0442\u043a\u0443\u0434\u0430 \u0441\u043b\u0435\u0434\u0443\u0435\u0442:
\n$$g\\left(p^{a}\\right) = p^{a} — p^{a-1} + p\\cdot g\\left(p^{a-1}\\right).$$
\n\u041f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0438\u043d\u0434\u0443\u043a\u0446\u0438\u0438:
\n\u041f\u0443\u0441\u0442\u044c [latex]a = b[\/latex]:
\n$$g\\left(p^{b}\\right) = \\left(b + 1\\right) \\cdot p^b — b \\cdot p^{b-1}.$$
\n\u0428\u0430\u0433 \u0438\u043d\u0434\u0443\u043a\u0446\u0438\u0438:
\n\u041f\u0443\u0441\u0442\u044c [latex]a = b + 1[\/latex]:
\n$$g\\left(p^{b + 1}\\right) = p^{b + 1} — p^{b} + p\\cdot g\\left(p^{b}\\right) = p^{b + 1} — p^{b} + p\\cdot \\left[\\left(b+1\\right) \\cdot p^{b} + b\\cdot p^{b-1}\\right] = \\ = \\left(b + 2\\right)\\cdot p^{b+1} — \\left(b + 1\\right)\\cdot p^{b}.$$<\/p>\n
\u0421\u0441\u044b\u043b\u043a\u0438<\/h1>\n
\u0423\u0441\u043b\u043e\u0432\u0438\u0435 \u0437\u0430\u0434\u0430\u0447\u0438 \u043d\u0430 e-olymp<\/a>
\n\u041a\u043e\u0434 \u0440\u0435\u0448\u0435\u043d\u0438\u044f<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"
\u0417\u0430\u0434\u0430\u0447\u0430 \u041b\u043e\u043d\u0433\u0438 \u0445\u043e\u0440\u043e\u0448\u043e \u0440\u0430\u0437\u0431\u0438\u0440\u0430\u0435\u0442\u0441\u044f \u0432 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0435, \u043e\u043d \u043b\u044e\u0431\u0438\u0442 \u0437\u0430\u0434\u0443\u043c\u044b\u0432\u0430\u0442\u044c\u0441\u044f \u043d\u0430\u0434 \u0442\u0440\u0443\u0434\u043d\u044b\u043c\u0438 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u043c\u0438 \u0437\u0430\u0434\u0430\u0447\u0430\u043c\u0438, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043c\u043e\u0433\u0443\u0442 \u0431\u044b\u0442\u044c \u0440\u0435\u0448\u0435\u043d\u044b \u043f\u0440\u0438 \u043f\u043e\u043c\u043e\u0449\u0438 \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u0438\u0437\u044f\u0449\u043d\u044b\u0445 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u043e\u0432. \u0418 \u0432\u043e\u0442 \u0442\u0430\u043a\u0430\u044f \u0437\u0430\u0434\u0430\u0447\u043a\u0430 \u0432\u043e\u0437\u043d\u0438\u043a\u043b\u0430: \u0414\u0430\u043d\u043e \u0446\u0435\u043b\u043e\u0435 \u0447\u0438\u0441\u043b\u043e [latex]n[\/latex] [latex](1 < n < 231)[\/latex], \u0412\u044b \u0434\u043e\u043b\u0436\u043d\u044b \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c [latex]\\sum\\limits_{i=1}^n gcd [\/latex] \u0434\u043b\u044f \u0432\u0441\u0435\u0445 [latex] 1 \u2264 i \u2264 n[\/latex]. "\u041e, \u044f \u0437\u043d\u0430\u044e, \u044f \u0437\u043d\u0430\u044e!" … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":131,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13],"tags":[481,363,483,147,482],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/java.mazurok.com\/index.php?rest_route=\/wp\/v2\/posts\/3916"}],"collection":[{"href":"https:\/\/java.mazurok.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/java.mazurok.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/java.mazurok.com\/index.php?rest_route=\/wp\/v2\/users\/131"}],"replies":[{"embeddable":true,"href":"https:\/\/java.mazurok.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3916"}],"version-history":[{"count":1,"href":"https:\/\/java.mazurok.com\/index.php?rest_route=\/wp\/v2\/posts\/3916\/revisions"}],"predecessor-version":[{"id":3917,"href":"https:\/\/java.mazurok.com\/index.php?rest_route=\/wp\/v2\/posts\/3916\/revisions\/3917"}],"wp:attachment":[{"href":"https:\/\/java.mazurok.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3916"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/java.mazurok.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3916"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/java.mazurok.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3916"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}