A067573 -id:A067573 - cp's OEIS frontend

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

Don Reble, Mar 19 2005

nonn

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

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.

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

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 *)

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

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

A293059 Numbers k such that sigma(phi(k))/k > sigma(phi(m))/m for all m < k, where sigma is the sum of divisors function (A000203) and phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 5, 7, 13, 31, 37, 61, 181, 241, 421, 899, 1321, 1333, 1763, 2161, 2521, 5183, 7561, 12601, 15121, 28187, 30241, 55441, 110881, 167137, 278263, 332641, 555911, 666917, 722473, 1443853, 2165407, 3607403, 4324321, 7212581, 8654539, 10817761, 21631147, 36768847

Offset: 1

Amiram Eldar, Oct 15 2017

nonn

Alaoglu and Erdős proved that lim sup sigma(phi(n))/n = oo, thus this sequence is infinite.

Amiram Eldar, Table of n, a(n) for n = 1..55
Leon Alaoglu and Paul Erdős, A conjecture in elementary number theory, Bulletin of the American Mathematical Society, Vol. 50, No. 12 (1944), pp. 881-882.
Florian Luca and Carl Pomerance, On some problems of Makowski-Schinzel and Erdős concerning the arithmetical functions phi and sigma, Colloquium Mathematicae, Vol. 92, No. 1 (2002), pp. 111-130.

Cf. A000010, A000203, A062402, A067573.

a={}; rm=0; Do[r = DivisorSigma[1, EulerPhi[n]]/n; If[r>rm, rm=r; AppendTo[a,n]],{n,1,100000}]; a

lista(nn) = {my(rmax = 0); for (n=1, nn, if ((r=sigma(eulerphi(n))/n) > rmax, rmax = r; print1(n, ", ")););} \\ Michel Marcus, Oct 18 2017

cp's OEIS Frontend

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

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