A305236 -id:A305236 - cp's OEIS frontend

A307527 Odd terms in A305236.

63, 513, 2107, 12625, 26533, 39609, 355023, 3190833, 4457713, 19854847, 38463283, 242138449, 370634743, 516465451, 574336561, 701607583, 1380336193, 2324581983, 4882890625, 6174434113, 12859758577, 14793096853, 20578440583, 43522669657, 85504120021

Offset: 1

Jianing Song, Apr 12 2019

nonn

These are numbers of the form p^e*((p-1)*p^(e-1) + 1) where p is an odd prime and (p-1)*p^(e-1) + 1 is prime.
{A002322(a(n))} = {6, 18, 42, 100, 156, 162, ...} is a permutation of A114874 without the terms that are powers of 2 (but they don't have the same order: 6563187324027001 and 6575415997816513 are both terms but A002322(6563187324027001) = 81009000 while A002322(6575415997816513) = 80995248).
A305236 = {a(n)} U {2*a(n)} U {8, 12}.

			See A305236 for examples.

Jianing Song, Table of n, a(n) for n = 1..156 (all terms below 10^16)

Cf. A305236, A114874, A002322.

the_first_entries(nn) = my(u=[]); for(n=2, sqrt(nn), my(v=factor(n), d=#v[, 1], p=v[d, 1], e=v[d, 2]); if(isprime(n+1) && p!=2 && n==(p-1)*p^e, u=concat(u, [(n+1)*p^(e+1)]))); u=vecsort(select(i->(i

A316089 Number of ordered (k_1, k_2, ..., k_r) such that (Z/nZ)* is isomorphic to C_{k_1} X C_{k_2} X ... X C_{k_r}. Here (Z/nZ)* is the multiplicative group of integers modulo n, r = rank((Z/nZ)*).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 4, 2, 1, 1, 4, 3, 1, 2, 1, 2, 4, 1, 1, 3, 1, 1, 2, 4, 1, 1, 4, 3, 2, 1, 1, 3, 1, 1, 1, 2, 2, 2, 1, 2, 2, 4, 1, 3, 1, 1, 4, 2, 4, 4, 1, 3, 1, 1, 1, 3, 2, 1, 4

Offset: 1

Jianing Song, Jun 27 2018

nonn

The rank of a finitely generated group rank(G) is defined as the size of the minimal generating sets (the generating sets with the least elements) of G. By convention a(1) = a(2) = 1, since the multiplicative group of integers modulo 1 or 2 has rank 0, and the empty direct product yields the trivial group.
This sequence is related to the number of minimal generating sets {x_1, x_2, ..., x_r} of (Z/nZ)* such that Product_{i=1..r} x_i^e_i == 1 (mod n) implies that x_i^e_i == 1 (mod n) for 1 <= i <= r. See A302257.
For n >= 3, also the number of ordered (k_1, k_2, ..., k_r) such that (Z/nZ)* is isomorphic to C_{k_1} X C_{k_2} X ... X C_{k_r} and gcd(k_1, k_2, ..., k_r) > 1.
a(n) = 1 iff there exists an integer m such that (Z/nZ)* = (C_m)^r, that is, n is a term in A033948 or A305236 or n = 24.
First occurrence of k, or -1 if k never occurs: 1, 15, 40, 35, -1, 160, -1, 143, 104, -1, -1, 480, -1, -1, -1, ... It's easy to show that prime number >= 5 never occurs.
In general, let (Z/nZ)* = C_(Product_{j=1..k} p_j^e_1j) X C_(Product_{j=1..k} p_j^e_2j) X ... X C_(Product_{j=1..k} p_j^e_rj) where p_1, p_2, ..., p_k are different primes, 0 <= e_1j <= e_2j <= ... <= e_rj for 1 <= j <= k (for n = 1 or 2, (Z/nZ)* is the trivial group, thus r = k = 0), then a(n) is the number of matrices C = (c_ij) such that (c_1j, c_2j, ..., c_rj) is a permutation of (e_1j, e_2j, ..., e_rj) for 1 <= j <= k. - Jianing Song, Feb 04 2019

			For n = 35, rank((Z/35Z)*) = 2, and (Z/35Z)* = C_2 X C_12 = C_12 X C_2 = C_4 X C_6 = C_6 X C_4, so a(35) = 4. Also, (Z/35Z)* = C_(2^1*3^0) X C_(2^2*3^1), e_11 = 1, e_12 = 0, e_21 = 2, e_22 = 1, so b_11 = b_12 = b_21 = b_22 = 1, then a(35) = (2!)^2 = 4.
For n = 105, rank((Z/105Z)*) = 3, and (Z/105Z)* = C_2 X C_2 X C_12 (along with its two permutations) = C_2 X C_4 X C_6 (along with its five permutations), so a(105) = 9. Also, (Z/35Z)* = C_(2^1*3^0) X C_(2^1*3^0) X C_(2^2*3^1), e_11 = 1, e_12 = 0, e_21 = 1, e_22 = 0, e_31 = 2, e_32 = 1, so b_11 = b_12 = b_21 = b_22 = 2, b_31 = b_32 = 1, then a(105) = (3!)^2/(2!*2!) = 9.

Andrew Howroyd, Table of n, a(n) for n = 1..10000
Wikipedia, Multiplicative group of integers modulo n

Cf. A033948, A302257, A305236.

zp(g)={sum(i=1, #g, my(f=factor(g[i])); sum(j=1, #f~, x^f[j,1]*(y^f[j,2]-1)))}
a(n)={my(g=znstar(n).cyc); my(p=zp(g), r=#g); prod(i=1, poldegree(p), my(u=Vec(r + polcoeff(p,i))); r!/prod(j=1, #u, u[j]!))} \\ Andrew Howroyd, Jun 30 2018

Let (Z/nZ)* = C_(Product_{j=1..k} p_j^e_1j) X C_(Product_{j=1..k} p_j^e_2j) X ... X C_(Product_{j=1..k} p_j^e_rj) where p_1, p_2, ..., p_k are different primes, 0 <= e_1j <= e_2j <= ... <= e_rj for 1 <= j <= k. Define b_ij = max{1 <= t <= r | e_tj = e_ij} - min{1 <= t <= r | e_tj = e_ij} + 1, then a(n) = (r!)^k/Product_{i=1..r, j=1..k} (b_ij)!^(1/b_ij).

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A307527 Odd terms in A305236.

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