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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).
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%I #33 Sep 19 2024 03:31:50

%S 20,140,464,660,1276,1365,2204,2508,2805,2907,5590,5698,5742,6006,

%T 7395,8680,14645,15052,18875,19170,19740,23871,34579,34804,35164,

%U 35244,35934,38121,106805,114953,261536,503082

%N 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).

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

%C 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

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

%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_419.htm">Puzzle 419. Four SOPF questions</a>, Prime Puzzles.

%F a(n) > a(n-1): n*A000005(a(n))*A008472(a(n)) = A000203(a(n)). - _R. J. Mathar_, Nov 16 2007, Jun 24 2009

%p 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

%t 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 *)

%o (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

%Y Subsequence of A070222. - _R. J. Mathar_, Feb 05 2010

%Y Cf. A134383, A134384, A134385, A134386.

%K nonn,full,fini

%O 1,1

%A _Enoch Haga_, Oct 23 2007

%E Edited by _R. J. Mathar_, Nov 16 2007

%E A-number in formula and Maple program corrected by _R. J. Mathar_, Jun 24 2009

%E a(32) from _R. J. Mathar_, Feb 05 2010

%E full,fini keywords added by _Max Alekseyev_, Sep 18 2024