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.

A038843 Unitary superperfect numbers: numbers n such that usigma(usigma(n)) = 2*n, where usigma(n) is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

2, 9, 165, 238, 1640, 4320, 10250, 10824, 13500, 23760, 58500, 66912, 425880, 520128, 873180, 931392, 1899744, 2129400, 2253888, 3276000, 4580064, 4668300, 13722800, 15459840, 40360320, 201801600, 439021440, 3809332800, 15359485680, 794436968640, 1407035080704
Offset: 1

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Author

Yasutoshi Kohmoto

Keywords

Comments

May be called (2,2)-unitary perfect numbers, analogous to (k,l)-perfect numbers.
Sitaramaiah and Subbarao found the first 22 terms. Also in the sequence is 12189313382400. - Amiram Eldar, Feb 27 2019

Links

Crossrefs

Cf. A034448, A019279.
Cf. A064012 (usigma(usigma(n)) = 3n).

Programs

  • Mathematica
    usigma[n_] := Times @@ (Apply[ Power, FactorInteger[n], {1}] + 1); n = 1; A038843 = {}; While[n < 10^7, If[ usigma[ usigma[n] ] == 2n, Print[n]; AppendTo[ A038843, n] ]; n++]; A038843 (* Jean-François Alcover, Dec 07 2011 *)
  • PARI
    {usigma(n,s=1,fac,i)= fac=factor(n); for(i=1,matsize(fac)[1], s=s*(1+fac[i,1]^fac[i,2]) ); return(s);}
for(n=1,10^7, if(usigma(usigma(n))==2*n, print1(n, ", ")))

Extensions

Corrected by Jason Earls, Aug 25 2001
More terms from Jud McCranie, Oct 28 2001
Offset corrected and a(28) from Donovan Johnson, Jul 23 2012
Name edited and a(29) from Amiram Eldar, Feb 27 2019
a(30)-a(31) from Giovanni Resta, Mar 08 2019

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

A328132 Exponential (2,3)-perfect numbers: numbers m such that esigma(esigma(m)) = 3m, where esigma(m) is the sum of exponential divisors of m (A051377).

Original entry on oeis.org

300, 2100, 3300, 3900, 5100, 5700, 6900, 8700, 9300, 11100, 12100, 12300, 12900, 14100, 15900, 17700, 18300, 20100, 21300, 21900, 23100, 23700, 23760, 24900, 26700, 27300, 29100, 30300, 30900, 32100, 32700, 33900, 35700, 38100, 39300, 39900, 41100, 41700, 42900
Offset: 1

Views

Author

Amiram Eldar, Oct 04 2019

Keywords

References

  • József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 53.

Links

Crossrefs

The exponential version of A019281.
Cf. A051377, A064012.

Programs

  • Mathematica
    f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; espQ[n_] := esigma[esigma[n]] == 3n; Select[Range[50000], espQ]
  • PARI
    esigma(n) = {my(f = factor(n)); prod(k = 1, #f~, sumdiv(f[k, 2], d, f[k, 1]^d));}
isok(k) = esigma(esigma(k)) == 3*k; \\ Amiram Eldar, Jan 09 2025

Formula

300 is in the sequence since esigma(300) = 540, and esigma(540) = 900 = 3*300.
Showing 1-3 of 3 results.

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

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