A051281 Sum of divisors of n, sigma(n) (A000203), is a power of number of divisors of n, d(n) (A000005).

1, 3, 7, 31, 127, 217, 889, 2667, 3937, 8191, 27559, 57337, 131071, 172011, 253921, 524287, 917497, 1040257, 1777447, 3670009, 4063201, 11010027, 12189603, 16252897, 16646017, 66584449, 113770279, 116522119, 225735769, 677207307, 1073602561, 2147483647, 3612185689, 4294434817, 7515217927

Offset: 1

Author

Michel ten Voorde

Keywords

• nonn
• nice

Comments

All Mersenne primes (A000668) are terms.

Subsequence of A046528 (product of distinct Mersenne primes). - Michel Marcus, Feb 15 2020

Examples

			d(217) = 4; sigma(217) = 256 = 4^4.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A000005, A000203, A046528, A286628, A289276.

Programs

Mathematica

spdQ[n_]:=Module[{sd=DivisorSigma[1,n],nd=DivisorSigma[0,n]},sd == nd^IntegerExponent[sd,nd]]; Join[{1},Select[Range[2,226000000],spdQ]] (* Harvey P. Dale, May 02 2012 *)

PARI

is(n)=my(t,e=ispower(sigma(n),,&t)); if(!e,return(n==1),nd); nd=numdiv(n); fordiv(e,d,if(t^d==nd,return(1)));0 \\ Charles R Greathouse IV, Feb 19 2013

PARI

isA051281(n) = { if(n==1, return(1)); my(sig = sigma(n), ndiv = numdiv(n), v = valuation(sig, ndiv)); (ndiv^v == sig); } \\ Antti Karttunen, Jun 30 2017

Extensions

More terms from Jud McCranie

a(30)-a(32) from Donovan Johnson, Oct 03 2012

a(33)-a(35) from Michel Marcus, Feb 14 2020