cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Michel ten Voorde

Keywords

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

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

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.

Original entry on oeis.org

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
Offset: 1

Views

Author

Antti Karttunen, Jun 30 2017

Keywords

Comments

a(1) = 1 by convention.

Examples

			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.

Links

Crossrefs

Cf. A000005, A000010, A286561, A286628, A289276.
Cf. A015733 (positions of zeros), A020491 (of nonzeros).

Programs

  • PARI
    A286627(n) = valuation(eulerphi(n), numdiv(n));

Formula

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

A303435 Numbers n such that uphi(n) (the unitary totient function A047994) is a power of the number of unitary divisors of n (A034444).

Original entry on oeis.org

1, 2, 3, 5, 9, 10, 17, 30, 34, 85, 170, 257, 514, 765, 1285, 1542, 4369, 8738, 39321, 65537, 131070, 131074, 327685, 655370, 1114129, 2949165, 3342387, 16843009, 33686018, 100271610, 151587081, 572662306, 2863311530
Offset: 1

Views

Author

Amiram Eldar, Apr 24 2018

Keywords

Comments

The unitary version of A289276.
Since A034444(n)=2^omega(n) is a power of 2, all the terms are products of 2 and the Fermat primes (A019434), each with multiplicity < 2, except for 3 that may be of multiplicity of 2 (since 3^2 = 2^3 + 1). If there is no 6th Fermat prime, then this sequence is finite with 33 terms.

Examples

			2863311530 = 2 * 5 * 17 * 257 * 65537 is in the sequence since it has 2^5 unitary divisors, and its uphi value is 2^30 = (2^5)^6.

Crossrefs

Cf. A019434, A034444, A047994, A092506, A289276.

Programs

  • Mathematica
    uphi[n_]:=If[n == 1,1,(Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@ FactorInteger [n]))[[1]]]; aQ[n_] := If[n == 1, True, IntegerQ[Log[2, uphi[n]]/PrimeNu[n]]]; v = Union[Times @@@ Rest[Subsets[{1, 2, 3, 5, 17, 257, 65537}]]]; w = Union[v, 3*v]; s = {}; Do[w1 = w[[k]]; If[aQ[w1], AppendTo[s, w1]], {k, 1, Length[w]}]; s
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.