A323278 Numbers of the form p^2-1 that have a record-breaking number of divisors, where p is prime.

3, 8, 24, 48, 120, 288, 360, 840, 1680, 5040, 11880, 32760, 143640, 201600, 491400, 776160, 2042040, 3500640, 7447440, 9480240, 17297280, 34234200, 143256960, 514337040, 555120720, 569729160, 1656408600, 4283571600, 8148853440, 10951831800, 35415099720, 51437786400

Offset: 1

Author

G. L. Honaker, Jr., Jan 11 2019

Keywords

• nonn

Comments

a(11)-a(26) from Chuck Gaydos.

Examples

			a(7) = 360 because 360 has a record-breaking 24 divisors and 360 = p^2-1, where p = 19 is prime.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..63
• Chris Caldwell and G. L. Honaker, Jr., Prime Curio for 23869

Crossrefs

Cf. A000005, A084920.

Programs

Mathematica

Block[{s = Prime[Range[10^5]]^2 - 1, t}, t = DivisorSigma[0, s]; Map[s[[FirstPosition[t, #][[1]] ]] &, Union@ FoldList[Max, t]]] (* Michael De Vlieger, Jan 19 2019 *)

PARI

lista(nn) = {my(m = 0, p = 2, np); for (n=1, nn, np = p^2-1; if (((nd = numdiv(np)) > m), print1(np, ", "); m = nd); p = nextprime(p+1););} \\ Michel Marcus, Jan 12 2019

Python

from sympy import divisor_count, nextprime
A323278_list, p, nmax = [], 2 , -1
while len(A323278_list) < 100:
    n = divisor_count(p**2-1)
    if n > nmax:
        nmax = n
        A323278_list.append(p**2-1)
    p = nextprime(p) # Chai Wah Wu, Feb 09 2019

Extensions

a(27)-a(32) from Daniel Suteu, Jan 12 2019