A134382 a(n) is the smallest number k larger than a(n-1) such that n*d(k)*sopf(k)=sigma(k), where d is the number of divisors (A000005) and sopf the sum of prime factors without repetition (A008472).

20, 140, 464, 660, 1276, 1365, 2204, 2508, 2805, 2907, 5590, 5698, 5742, 6006, 7395, 8680, 14645, 15052, 18875, 19170, 19740, 23871, 34579, 34804, 35164, 35244, 35934, 38121, 106805, 114953, 261536, 503082

Offset: 1

Author

Enoch Haga, Oct 23 2007

Keywords

• nonn
• full
• fini

Comments

Sequence suggested by Puzzle 419 in Carlos Rivera's The Prime Puzzles & Problems Connection.

For n=33, the search for terms k that satisfy 33*d(k)*sopf(k)=sigma(k), without being greater than a(32), gives 21070, 25585, 30702, 36120, 41710, 49256, 52269, 68906, 74692, 92785, 95702, 111342, 117626, 383086 with no other terms up to 10^9. So this sequence might well be complete. - Michel Marcus, Oct 02 2019

I confirm that the solutions for n=33 listed above are complete, thus the sequence stops at n=32. - Max Alekseyev, Sep 18 2024

Links

• Carlos Rivera, Puzzle 419. Four SOPF questions, Prime Puzzles.

Crossrefs

Subsequence of A070222. - R. J. Mathar, Feb 05 2010

Cf. A134383, A134384, A134385, A134386.

Programs

Maple

A008472 := proc(n) local divs,i ; if n = 1 then 0; else divs := ifactors(n)[2] ; add( op(1,i),i=divs) ; fi ; end: A134382 := proc(n) option remember ; local k,kmin ; if n = 1 then kmin := 1 ; else kmin := procname(n-1)+1 ; fi ; for k from kmin do if numtheory[sigma](k) = n* numtheory[tau](k)*A008472(k) then RETURN(k) ; fi ; od: end: for n from 1 to 30 do print( A134382(n)) ; od: # R. J. Mathar, Nov 16 2007, Jun 24 2009

Mathematica

sopf[1] = 0; sopf[n_] := Total[FactorInteger[n][[All, 1]]]; a[n_] := a[n] = For[k = If[n == 1, 1, a[n-1] + 1], True, k++, If[DivisorSigma[1, k] == n*DivisorSigma[0, k]*sopf[k], Return[k]]]; Table[Print[a[n]]; a[n], {n, 1, 32}] (* Jean-François Alcover, Sep 12 2013 *)

PARI

lista(nn) = {lasta = 2; for (n=1, nn, k = lasta; while ((f = factor(k)) && (n*numdiv(k)*sum(j=1,#f~,f[j,1]) != sigma(k)), k++); print1(k, ", "); lasta = k;);} \\ Michel Marcus, Feb 25 2016

Formula

a(n) > a(n-1): n*A000005(a(n))*A008472(a(n)) = A000203(a(n)). - R. J. Mathar, Nov 16 2007, Jun 24 2009

Extensions

Edited by R. J. Mathar, Nov 16 2007

A-number in formula and Maple program corrected by R. J. Mathar, Jun 24 2009

a(32) from R. J. Mathar, Feb 05 2010

full,fini keywords added by Max Alekseyev, Sep 18 2024