A286628 -id:A286628

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A051281

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

+10
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

All Mersenne primes (A000668) are terms.

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

LINKS

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

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

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

KEYWORD

nonn,nice

AUTHOR

Michel ten Voorde

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

STATUS

approved

A286627

a(n) = exponent of the highest power of A000005(n) (number of divisors of n) dividing A000010(n) (totient function phi), a(1) = 1.

+10
3

1, 0, 1, 0, 2, 0, 1, 1, 1, 1, 1, 0, 2, 0, 1, 0, 4, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 2, 1, 1, 0, 1, 2, 1, 0, 2, 0, 1, 1, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 1, 2, 0, 1, 1, 1, 1, 1, 0, 2, 0, 2, 0, 2, 0, 1, 0, 1, 1, 1, 1, 3, 1, 0, 2, 1, 1, 1, 0, 0, 1, 1, 1, 3, 0, 1, 1, 3, 1, 1, 0, 1, 0, 1, 0, 5, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 4, 0, 1, 0, 2, 0, 2, 1, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

a(1) = 1 by convention.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = A286561(A000010(n), A000005(n)).

EXAMPLE

A000005(5) = 2, A000010(5) = 4, 2^2 is the highest power of 2 which divides 4, thus a(5) = 2.

A000005(6) = 4, A000010(6) = 2, 4^0 = 1 is the highest power of 4 which divides 2, thus a(6) = 0.