A340281 a(n) is the smallest prime p such that the number of distinct values of the ratio (number of nonnegative m < p such that m^k == m (mod p))/(number of nonnegative m < p such that -m^k == m (mod p)) is equal to n for some nonnegative k.

2, 3, 7, 19, 31, 163, 127, 1459, 211, 883, 811, 472393, 631, 8503057, 32077, 4051, 2311, 86093443, 4951, 6347497291777, 10531, 36451, 1299079, 251048476873, 8191, 388963, 5314411, 22051, 51031, 596046447753906250001, 28351, 411782264189299, 24571, 5904901

Offset

1,1

Comments

a(n) is the least prime p such that there are n distinct terms in the p-th row of A334006.

Conjecture: a(n) is the smallest prime p such that the number of distinct values of the ratio T(p, k) = (number of nonnegative m < p such that m^k == m (mod p))/(number of nonnegative m < p such that -m^k == m (mod p)) is equal to n for some 0 <= k <= floor((p + 2)/3).

Proof: for k > 1, iff t is a k-th power residue mod p, the number of nonnegative m < p such that m^k == t (mod p) is gcd(k, p - 1). Thus, the ratio T(p, 1+x) = T(p, 1+gcd(x, p-1)) and T(p, 2*t) = T(p, (p+1)/2) = 1. For odd prime p and 0 <= k < p - 1, notice that if k is an odd number of the form 1 + gcd(x, p-1) and x != (p - 1)/2, then k <= floor((p + 2)/3). - Jinyuan Wang, Jan 23 2021

For n >= 2, a(n) is the least prime p such that p - 1 has n - 1 odd divisors. - Jinyuan Wang, Jan 23 2021

Links

Seth A. Troisi, Table of n, a(n) for n = 1..200

Example

A334006 triangle begins:
   1 | 1;
   2 | 1, 1;   : 1 distinct value
   3 | 1, 3, 1;   : 2 distinct values
   4 | 1, 2, 1, 3;
   5 | 1, 5, 1, 1, 1;   : 2 distinct values
   6 | 1, 3, 1, 3, 1, 3;
   7 | 1, 7, 1, 3, 1, 3, 1;   : 3 distinct values

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
a(n) = my(p=2); while (#Set(vector(p, k, T(p, k))) != n, p = nextprime(p+1)); p; \\ Michel Marcus, Jan 21 2021
(PARI) lista(nn, show=50) = my(c, v=vector(show)); v[1]=2; forprime(p=3, nn, c=1+numdiv(p\2^valuation(p-1, 2)); if(c<=show && !v[c], v[c]=p)); v; \\ Jinyuan Wang, Jan 23 2021

Crossrefs

Cf. A001227, A019434, A058500, A074781, A334006.

Sequence in context: A092064 A152609 A298943 * A172461 A253971 A291339

Adjacent sequences: A340278 A340279 A340280 * A340282 A340283 A340284

Keyword

nonn

Author

Juri-Stepan Gerasimov, Jan 02 2021

Extensions

More terms from Jinyuan Wang, Jan 23 2021

Typo in a(34) corrected by Seth A. Troisi, May 22 2022

Status

approved