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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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

A241762 a(n) is the least number k > 0 such that sigma(k/n) = phi(k).

Original entry on oeis.org

1, 2, 45, 12, 70, 36, 42, 336, 270, 420, 1848, 2520, 2730, 5880, 12600, 332640, 353430, 166320, 175560, 1663200, 2522520, 87650640, 118798680, 1051807680, 671517000, 1139458320, 35231316120, 15952416480, 16522145640, 495664369200, 563462139240, 18030788455680, 37620925622280, 130723216303680, 43948907402400
Offset: 1

Views

Author

Paolo P. Lava, Apr 28 2014

Keywords

Examples

			For n=11, the least number is 1848. In fact, sigma(1848/11) = phi(1848) = 480.

Crossrefs

Cf. A000010, A000203, A087979.

Programs

  • Maple
    with(numtheory): P:=proc(q) local k,n;
for k from 1 to q do for n from k by k to q do
if sigma(n/k)=phi(n) then print(n); break; fi;
od; od; end: P(10^5);
  • PARI
    for(k=1,29,n=0;for(i=1,2^64,if(sigma(i)==eulerphi(i*k),n=i*k;break)); print(k,"  ",n)) \\ Dana Jacobsen, May 02 2014
  • Perl
    use Math::Prime::Util qw/:all/; for $k (1..29) { $i=1; $i++ while divisor_sum($i) != euler_phi($i*$k); say "$k  ",$i*$k; } # Dana Jacobsen, May 02 2014

Formula

a(n) = n * A256527(n). - Max Alekseyev, Sep 29 2023

Extensions

a(22)-a(26) from Giovanni Resta, Apr 29 2014
a(27)-a(29) from Dana Jacobsen, May 02 2014
a(30)-a(35) from Max Alekseyev, Sep 29 2023
Showing 1-2 of 2 results.

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