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A345003 Numbers k for which A344998(k) = A344999(k).

%I A345003 #16 Jun 12 2021 06:06:02
%S A345003 6,8,28,81,108,496,2500,2700,3375,5292,8128,13068,15625,18252,31212,
%T A345003 38988,57132,67228,90828,94500,103788,147852,181548,199692,231525,
%U A345003 238572,303372,375948,401868,484812,544428,575532,674028,713097,744012,855468,1016172,1058841,1101708,1145772,1236492,1283148,1379052,1500625
%N A345003 Numbers k for which A344998(k) = A344999(k).
%C A345003 Numbers k such that A345001(k)*A048250(k) is equal to A342001(k)*A344753(k).
%C A345003 Conjecture: Sequence is a disjoint union of A000396 and A301939.
%H A345003 <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%t A345003 A003415[n_] := If[n<2, 0, Module[{f = FactorInteger[n]}, If[PrimeQ[n], 1, Total[n*f[[All, 2]]/f[[All, 1]]]]]];
%t A345003 A003557[n_] := n * Times @@ (1/FactorInteger[n][[All, 1]]);
%t A345003 A048250[n_] := Select[Divisors[n], SquareFreeQ] // Total;
%t A345003 A344753[n_] := Sum[d + If[SquareFreeQ[n/d], d, 0], {d, Most[Divisors[n]]}];
%t A345003 A342001[n_] := A003415[n]/A003557[n];
%t A345003 A345001[n_] := DivisorSigma[1, n] + A003415[n] - 2 n;
%t A345003 A344998[n_] := A342001[n]*A344753[n];
%t A345003 A344999[n_] := A048250[n]*A345001[n];
%t A345003 Reap[For[k = 1, k <= 2*10^6, k++, If[A344998[k] == A344999[k], Print[k]; Sow[k]]]][[2, 1]] (* _Jean-François Alcover_, Jun 12 2021 *)
%o A345003 (PARI)
%o A345003 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A345003 A003557(n) = (n/factorback(factorint(n)[, 1]));
%o A345003 A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
%o A345003 A344753(n) = sumdiv(n,d,(d<n)*(d+(issquarefree(n/d) * d)));
%o A345003 A342001(n) = (A003415(n) / A003557(n));
%o A345003 A345001(n) = (sigma(n)+A003415(n)-(2*n));
%o A345003 isA345003(n) = (A345001(n)*A048250(n) == A342001(n)*A344753(n));
%Y A345003 Positions of zeros in A345043.
%Y A345003 Cf. A001615, A003415, A003557, A048250, A342001, A344753, A344998, A344999, A345001.
%Y A345003 Cf. A000396, A301939, A345004, A345005 (subsequences).
%Y A345003 Cf. also A345051.
%K A345003 nonn
%O A345003 1,1
%A A345003 _Antti Karttunen_, Jun 05 2021