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 A336664 Number of distinct divisors d of n with the property that d = (the number of nonnegative bases m < n such that m^k == m (mod n))/(the number of nonnegative bases m < n such that -m^k == m (mod n)) for some nonnegative k. 3
 1, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 2, 2, 4, 2, 2, 2, 3, 3, 3, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 4, 2, 2, 3, 2, 3, 3, 2, 2, 2, 3, 2, 3, 2, 2, 4, 3, 2, 2, 2, 3, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS For any k >= 0, the value of (the number of nonnegative bases m < n such that m^k == m (mod n))/(the number of nonnegative bases m < n such that -m^k == m (mod n)) is the value of some part of n, and is equal or unequal to the value of some divisor of n. Starting with k = 2, these infinite sequences of the parts of n are periodic with a period equal to A002322(n) (or A000010(n)). Also these sequences are different, but among n there are pairs of numbers (1 and 2, 3 and 6, ...) for which they are the same. For n >= 1, minimal d is equal to 1 and maximal d is equal to A026741(n). If n: 1, 2, 3, 5, 6, 10, 15, 17, 30, 34, 51, 85, 102, ..., then d = (the number of nonnegative bases m < n such that m^k = m)/(the number of nonnegative bases m < n such that -m^k = m) for all nonnegative k. LINKS Antti Karttunen, Table of n, a(n) for n = 1..3755 EXAMPLE For n = 1 the a(1) = 1 solution d is 1 (k = 0), n = 2 the a(2) = 1 solution d is 1 (k = 0), n = 3 the a(3) = 2 solutions d are 1 (k = 0) and 3 (k = 1), n = 4 the a(4) = 2 solutions d are 1 (k = 0) and 2 (k = 1), n = 5 the a(5) = 2 solutions d are 1 (k = 0) and 5 (k = 1), n = 6 the a(6) = 2 solutions d are 1 (k = 0) and 3 (k = 1), n = 7 the a(7) = 2 solutions d are 1 (k = 0) and 7 (k = 1), n = 8 the a(8) = 2 solutions d are 1 (k = 0) and 4 (k = 1), n = 9 the a(9) = 3 solutions d are 1 (k = 0), 3 (k = 3) and 9 (k = 1). PROG (PARI) T(n, k) = sum(m=0, n-1, Mod(m, n)^k == m)/sum(m=0, n-1, Mod(-m, n)^k == m); \\ A334006 vec(n) = vecsort(vector(n, k, T(n, k-1)), , 8); a(n) = { my(v=vec(n)); sumdiv(n, d, vecsearch(v, d) != 0); }; \\ Michel Marcus, Aug 27 2020, edited for speed by Antti Karttunen, Dec 13 2021 CROSSREFS Cf. A000010, A002322, A026741, A334006. Sequence in context: A351399 A024708 A096917 * A335108 A359238 A320011 Adjacent sequences: A336661 A336662 A336663 * A336665 A336666 A336667 KEYWORD nonn AUTHOR Juri-Stepan Gerasimov, Jul 29 2020 STATUS approved

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Last modified September 13 08:24 EDT 2024. Contains 375902 sequences. (Running on oeis4.)