A256527 a(n) is the least number k > 0 such that sigma(k) = phi(n*k).
1, 1, 15, 3, 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
Keywords
Examples
sigma(1) = phi(1*1) = 1; sigma(1) = phi(2*1) = 1; sigma(15) = phi(3*15) = 24; sigma(3) = phi(4*3) = 4; sigma(14) = phi(5*14) = 24; sigma(6) = phi(6*6) = 12; sigma(6) = phi(7*6) = 12; sigma(42) = phi(8*42) = 96; sigma(30) = phi(9*30) = 72; etc.
Programs
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Maple
with(numtheory): P:=proc(q) local k, n; for n from 1 to q do for k from 1 to q do if sigma(k)=phi(k*n) then lprint(n,k); break; fi; od; od; end: P(10^5);
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Mathematica
Table[k = 1; While[DivisorSigma[1, k] != EulerPhi[n k], k++]; k, {n, 20}] (* Michael De Vlieger, May 28 2015 *) -
PARI
a(n) = {k=1; while(sigma(k) != eulerphi(n*k), k++); k;} \\ Michel Marcus, Apr 01 2015
Formula
For n >= 5, a(n) = A087979(n). - Conjectured by Manfred Scheucher, May 28 2015; proved by Max Alekseyev, Sep 29 2023
Extensions
a(21)-a(23) from Michel Marcus, Apr 01 2015
a(24)-a(26) from Jon E. Schoenfield, Jun 28 2015
a(27)-a(35) from Giovanni Resta, May 24 2016
a(36)-a(41) from Max Alekseyev, Oct 10 2024