The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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
 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 (list; graph; refs; listen; history; text; internal format)
 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 19 18:01 EDT 2024. Contains 376014 sequences. (Running on oeis4.)