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.

A088830 a(n) = Min{x : sigma(x) = n*phi(x), x is not a prime}, the least nonprime solutions to sigma(x) = n*phi(x); special balanced numbers.

Original entry on oeis.org

1, 35, 15, 14, 56, 6, 12, 42, 30, 168, 2580, 210, 630, 420, 840, 20790, 416640, 9240, 291060, 83160, 120120, 5165160, 1719277560, 43825320, 26860680, 277560360, 1304863560, 569729160, 587133466920, 16522145640, 33044291280, 563462139240, 1140028049160
Offset: 1

Views

Author

Labos Elemer, Nov 03 2003

Keywords

Comments

a(33) > 10^12. - Donovan Johnson, Sep 03 2013
a(34) <= 9015394227840, a(35) <= 1255683068640. - Giovanni Resta, May 08 2017

Links

Crossrefs

Compare A087979, which has a slightly different definition.
Cf. A074891, A000203, A000010, A087979, A020492.
Cf. A055234.

Programs

  • Mathematica
    ds[x_, de_] := DivisorSigma[1, x]-de*EulerPhi[x] a[n_] := Block[{m=1, s=ds[m, n]}, While[(s !=0||PrimeQ[m])&&!Greater[m, 100000], m++ ]; m]; Table[a[n], {n, 22}]

Formula

For n > 3, a(n) = A055234(n). - David Wasserman, Aug 18 2005

Extensions

More terms from David Wasserman, Aug 18 2005
a(32) from Donovan Johnson, Sep 03 2013
a(33) from Giovanni Resta, May 08 2017

A087979 a(n) = Min {x : sigma(x) = phi(n*x), x is not a prime}, least nonprime solutions to sigma(x) = phi(n*x).

Original entry on oeis.org

1, 1, 15, 14, 14, 6, 6, 42, 30, 42, 168, 210, 210, 420, 840, 20790, 20790, 9240, 9240, 83160, 120120, 3984120, 5165160, 43825320, 26860680, 43825320, 1304863560, 569729160, 569729160, 16522145640, 18176198040, 563462139240, 1140028049160, 3844800479520, 1255683068640, 65361608151840, 65361608151840, 65361608151840, 413956851628320, 1241870554884960, 1241870554884960
Offset: 1

Views

Author

Labos Elemer, Sep 29 2003

Keywords

Comments

If x is prime then by necessity we have x = 3 and n = 4. Hence, except for a(4), this sequence is the same as A256527. - Max Alekseyev, Sep 29 2023

Examples

			n=4: a(4)=14, sigma(14) = 24 = phi(4*14) = phi(56).
n=5: a(5)=14, sigma(14) = 24 = phi(5*14) = phi(70).

Crossrefs

Cf. A074891, A000203, A000010, A088830, A256527.

Programs

  • Mathematica
    ds[x_, de_] := DivisorSigma[1, x]-EulerPhi[de*x] a[n_] := Block[{m=1, s=ds[m, n]}, While[(s !=0||PrimeQ[m])&&!Greater[m, 4000000], m++ ]; m]; Table[a[n], {n, 22}]

Formula

For n >= 5, a(n) = A256527(n). - Conjectured by Manfred Scheucher, May 28 2015; proved by Max Alekseyev, Sep 29 2023

Extensions

More terms from David Wasserman, Jun 20 2005
a(26)-a(31) from Donovan Johnson, Feb 06 2010
a(32)-a(34) from Donovan Johnson confirmed, a(35) added by Giovanni Resta, May 24 2016
a(36)-a(41) from Max Alekseyev, Oct 10 2024

A068400 Numbers k such that sigma(k) = phi(k*bigomega(k)).

Original entry on oeis.org

248, 357, 1045, 3596, 3956, 4064, 5396, 8636, 20026, 20320, 23374, 24871, 25714, 29029, 33915, 35074, 39585, 41656, 50065, 55154, 56134, 56536, 58435, 61344, 64285, 74613, 79000, 87087, 87685, 137885, 140335, 142240, 353133, 383656, 393104, 423657, 474548, 479864
Offset: 1

Views

Author

Benoit Cloitre, Mar 02 2002

Keywords

Comments

For all the numbers listed, gcd(n!-1,2^n+1) = 2n+1. Is this always true? - Giovanni Resta, Nov 15 2006
This is plausible and true for n up to 45000. - Max Alekseyev, Nov 16 2006

Examples

			248 = 31*2^3 is here because sigma(248) = 480 = eulerphi(248*bigomega(2^3 * 31)) = eulerphi(248*4). - _David A. Corneth_, Jun 29 2025

Links

Crossrefs

Cf. A000010 (phi), A000203 (sigma), A001222 (bigomega), A074891.

Programs

  • PARI
    isok(k) = {my(f = factor(k)); sigma(f) == eulerphi(k * bigomega(f));} \\ Amiram Eldar, May 09 2025
Showing 1-3 of 3 results.

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