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.

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A063885 z(sigma(n)) = 2n, where z(n) = A048146.

Original entry on oeis.org

24, 1536, 1631, 47360, 82458
Offset: 1

Views

Author

Jason Earls, Aug 28 2001

Keywords

Crossrefs

Cf. A000203, A034448, A048146, A019284.

Programs

  • PARI
    u(n) = sumdiv(n,d, if(gcd(d,n/d)==1,d));
z(n) = sigma(n)-u(n);
for(n=1,10^7, if(z(sigma(n))==2*n,print1(n, ", ")))

A063891 Numbers k such that nusigma(usigma(k)) = 2k, where usigma(k) is the sum of unitary divisors of k (A034448) and nusigma(k) is the sum of non-unitary divisors of k (A048146).

Original entry on oeis.org

1631, 2016, 8928, 11808, 36576, 45360, 1486080, 2359008, 3093552, 37748448, 101350656, 150994656, 2885670144
Offset: 1

Views

Author

Jason Earls, Aug 28 2001

Keywords

Comments

a(14) > 2*10^11. All the numbers of the form 2^5 * 3^2 * p where p>3 is a Mersenne prime (A000668) are in the sequence, so a(14) <= 618475290336. - Giovanni Resta, Apr 10 2019

Crossrefs

Cf. A000203, A034448, A048146, A019284, A063885.

Programs

  • Mathematica
    usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); nusigma[n_] := DivisorSigma[1,n] - usigma[n]; Select[Range[12000], nusigma[usigma[#]] == 2# &] (* Amiram Eldar, Apr 10 2019 *)
  • PARI
    u(n) = sumdiv(n,d, if(gcd(d,n/d)==1,d));
z(n)=sigma(n)-u(n) ;
for(n=1,10^8, if(z(u(n))==2*n,print1(n, ", ")))

Extensions

More terms from Thomas Baruchel, Oct 22 2003
a(11)-a(13) from Amiram Eldar, Apr 10 2019

A173431 Count of consecutive coprime iterations of sum-of-divisors function.

Original entry on oeis.org

1, 6, 5, 4, 2, 1, 3, 2, 3, 1, 2, 1, 2, 1, 1, 5, 2, 1, 2, 1, 4, 1, 2, 1, 5, 1, 2, 1, 2, 1, 4, 3, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 4, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 4, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 4, 3, 1, 5, 2, 1, 2, 1, 1
Offset: 1

Views

Author

Walter Nissen, Feb 18 2010

Keywords

Comments

The last of these iterates is the value in A173430.

Examples

			Calculating sum-of-divisors ( ... sum-of-divisors ( sum-of-divisors ( 7 ) ) ... ) the iterates are 7, 8, 15, 24, ... .
The initial, consecutive, pairwise, coprime iterates are 7, 8, 15, and there are 3 of these, so a(7) = 3.
Here sigma ( 7 ) = 8, sigma ( sigma ( 7 ) ) = sigma ( 8 ) = 15, etc.

References

  • Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
  • Oystein Ore, Number Theory and Its History, 1988, Dover Publications, ISBN 0486656209, pp. 88-96.

Links

  • Leonard Eugene Dickson, History of the Theory of Numbers, Volume I, Divisibility and Primality, Carnegie Institution of Washington, 1919, Chapters II and X

Crossrefs

Cf. A173430, A129246 and the references there, A019294, A019295, A000203, A051027, A019284, A019277.

Programs

  • PARI
    a(n)=my(t,s);if(n==1,1,while(1,s++;t=sigma(n);if(gcd(t,n)==1,n=t,return(s)))) \\ Charles R Greathouse IV, Feb 06 2012
Previous Showing 11-13 of 13 results.

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

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