A103199 Primes p such that p-1 has more divisors than any smaller prime-1.

2, 3, 5, 7, 13, 31, 37, 61, 181, 241, 421, 1009, 1321, 1801, 2161, 2521, 6301, 7561, 12601, 15121, 20161, 30241, 45361, 55441, 100801, 110881, 196561, 332641, 498961, 786241, 982801, 1108801, 1580041, 1940401, 1995841, 2402401, 3880801, 4324321, 11476081, 11531521

Offset

1,1

Comments

There are infinitely many primes p such that d(p-1) > exp(c*log(p)/log(log(p))), where d(k) is the number of divisors of k, and c > 0 is a constant (Prachar, 1955). Therefore, this sequence is infinite. - Amiram Eldar, Apr 16 2024

Links

David A. Corneth, Table of n, a(n) for n = 1..130 (first 75 terms from Amiram Eldar)

Karl Prachar, Über die Anzahl der Teiler einer natürlichen Zahl, welche die Form p-1 haben, Monatshefte für Mathematik, Vol. 59 (1955), pp. 91-97.

Mathematica

seq[pmax_] := Module[{d, dm = 0, s = {}, p = 1}, While[p < pmax, p = NextPrime[p]; d = DivisorSigma[0, p-1]; If[d > dm, dm = d; AppendTo[s, p]]]; s]; seq[10^6] (* Amiram Eldar, Apr 16 2024 *)

Prog

(PARI) lista(pmax) = {my(dm = 0, d); forprime(p = 1, pmax, d = numdiv(p-1); if(d > dm, dm = d; print1(p, ", "))); } \\ Amiram Eldar, Apr 16 2024

Crossrefs

Cf. A000005, A008328, A066814, A067573, A072826.

Sequence in context: A306317 A359483 A067573 * A054217 A355521 A048414

Adjacent sequences: A103196 A103197 A103198 * A103200 A103201 A103202

Keyword

nonn,changed

Author

Don Reble, Mar 19 2005

Extensions

a(38)-a(40) added and name clarified by Amiram Eldar, Apr 16 2024

Status

approved