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 1-4 of 4 results.

A045795 Unitary-sigma sigma multiply perfect numbers: numbers k such that A061765(k) = m*k for some integer m.

Original entry on oeis.org

1, 2, 4, 8, 10, 16, 24, 27, 30, 54, 63, 64, 108, 126, 165, 238, 252, 360, 432, 504, 512, 660, 864, 952, 1008, 1536, 1728, 2016, 2464, 2640, 4032, 4096, 5544, 10560, 13824, 16728, 17640, 23040, 32256, 45500, 47360, 60928, 65536, 110592, 152064, 153600
Offset: 1

Views

Author

Yasutoshi Kohmoto

Keywords

Examples

			Sigma(10) = 18 = 2*3^2, usigma(18) = (2+1)*(9+1) = 30, divisible by 10, so 10 is in the sequence.
Sigma(24) = 60 = 2^2*3*5, usigma(60) = 5*4*6 = 120, divisible by 24, so 24 is in the sequence.

Links

Crossrefs

Cf. A000203, A034448, A045796, A061765.

Programs

  • Mathematica
    q[n_] := Divisible[Times @@ (1 + Power @@@ FactorInteger[DivisorSigma[1, n]]), n]; Select[Range[200000], q] (* Amiram Eldar, Aug 26 2022 *)
  • PARI
    for(n=1, 10^9, s=sigma(n); om=omega(s); f=factorint(s); pr=1; for(j=1, om, pr=pr*(f[j,1]^f[j,2]+1)); if(pr%n==0, print(n))) \\ Donovan Johnson, Mar 12 2013

Extensions

Corrected and extended by Jud McCranie, Oct 28 2001
Missing first term added and offset corrected by Donovan Johnson, Mar 12 2013

A055033 a(n) = usigma(usigma(n)), where usigma(n) is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

1, 4, 5, 6, 12, 20, 9, 10, 18, 30, 20, 30, 24, 36, 36, 18, 30, 72, 30, 72, 33, 50, 36, 50, 42, 96, 40, 54, 72, 90, 33, 48, 68, 84, 68, 78, 60, 120, 72, 84, 96, 132, 60, 120, 120, 90, 68, 90, 78, 168, 90, 144, 84, 160, 90, 90, 102, 180, 120, 216, 96
Offset: 1

Views

Author

N. J. A. Sloane, Oct 30 2001

Keywords

Links

Crossrefs

Cf. A034448, A051027, A061765, A064012.

Programs

  • Mathematica
    usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; a[n_] := usigma[usigma[n]]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)
  • PARI
    usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
a(n) = usigma(usigma(n)); \\ Amiram Eldar, Jul 24 2024

Formula

a(n) = A034448(A034448(n)). - Amiram Eldar, Jul 24 2024

A045796 Numbers m = usigma(sigma(k))/k such that usigma(sigma(k)) is divisible by k.

Original entry on oeis.org

1, 2, 2, 3, 3, 2, 5, 2, 3, 4, 2, 2, 4, 4, 2, 2, 4, 7, 4, 6, 3, 4, 5, 3, 4, 5, 4, 5, 3, 4, 4, 2, 5, 4, 6, 4, 8, 7, 6, 4, 5, 3, 2, 4, 5, 7, 7, 4, 4, 2, 9, 5, 5, 4, 8, 4, 4, 4, 8, 7, 4, 4, 4, 5, 6, 4, 8, 5, 8, 8, 6, 4, 6, 4, 5, 6, 4, 4, 4, 8, 5, 4, 6, 5, 8, 7, 5
Offset: 1

Views

Author

Yasutoshi Kohmoto

Keywords

Comments

a(n) = m values in A045795. - Donovan Johnson, Mar 12 2013

Links

Crossrefs

Cf. A000203, A034448, A045795, A061765.

Programs

  • Maple
    A034448 := proc(n) local ans, i: ans := 1: for i from 1 to nops(ifactors(n)[ 2 ]) do ans := ans*(1+ifactors(n)[2][i][1]^ifactors(n)[2][i][2]): od: RETURN(ans) end: isA045795 := proc(n) if A034448(numtheory[sigma](n)) mod n = 0 then A034448(numtheory[sigma](n))/n ; else -1 ; fi ; end: A045796 := proc() local n,a : n := 2: while true do a := isA045795(n) ; if a>=0 then printf("%d, ",a) ; fi ; n := n+1: od : end: A045796() ; # R. J. Mathar, Jun 26 2007
  • Mathematica
    s[n_] := Times @@ (1 + Power @@@ FactorInteger[DivisorSigma[1, n]])/n; s[1] = 1; Select[s /@ Range[10^6], IntegerQ] (* Amiram Eldar, Aug 26 2022 *)

Formula

a(n) = usigma(sigma(A045795(n)))/A045795(n).

Extensions

Corrected and extended by R. J. Mathar, Jun 26 2007
Missing first term added and offset corrected by Donovan Johnson, Mar 12 2013

A063836 Numbers k such that usigma(sigma(k)) is prime.

Original entry on oeis.org

3, 217, 57337
Offset: 1

Views

Author

Robert G. Wilson v, Aug 21 2001

Keywords

Comments

Also, numbers k such that sigma(k) + 1 is a Fermat prime (A019434). Equivalently, numbers k that are a product of distinct Mersenne primes (A000668), k = (2^p_1 - 1) * (2^p_2 - 1) * ... * (2^p_m - 1), p_i are in A000043 and m >= 1, such that p_1 + p_2 + ... + p_m = 2^s and 2^(2^s) + 1 is prime. - Amiram Eldar, Jan 25 2025

Crossrefs

Cf. A000043, A000203, A000668, A019434, A034448, A061765, A063103.
Subsequence of A046528.

Programs

  • Mathematica
    us[n_Integer] := (d = Divisors[n]; l = Length[d]; k = 1; s = n; While[k < l, If[ GCD[ d[[k]], n/d[[k]] ] == 1, s = s + d[[k]]]; k++ ]; s); Do[m = n; If[ PrimeQ[ us[ DivisorSigma[1, n]]], Print[n]], {n, 1, 10^7/4} ]
  • PARI
    usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
isok(k) = isprime(usigma(sigma(k))); \\ Amiram Eldar, Jan 25 2025
Showing 1-4 of 4 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.