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

6, 8, 28, 81, 108, 496, 2500, 2700, 3375, 5292, 8128, 13068, 15625, 18252, 31212, 38988, 57132, 67228, 90828, 94500, 103788, 147852, 181548, 199692, 231525, 238572, 303372, 375948, 401868, 484812, 544428, 575532, 674028, 713097, 744012, 855468, 1016172, 1058841, 1101708, 1145772, 1236492, 1283148, 1379052, 1500625

Offset

1,1

Comments

Numbers k such that A345001(k)*A048250(k) is equal to A342001(k)*A344753(k).

Conjecture: Sequence is a disjoint union of A000396 and A301939.

Links

Table of n, a(n) for n=1..44.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Mathematica

A003415[n_] := If[n<2, 0, Module[{f = FactorInteger[n]}, If[PrimeQ[n], 1, Total[n*f[[All, 2]]/f[[All, 1]]]]]];
A003557[n_] := n * Times @@ (1/FactorInteger[n][[All, 1]]);
A048250[n_] := Select[Divisors[n], SquareFreeQ] // Total;
A344753[n_] := Sum[d + If[SquareFreeQ[n/d], d, 0], {d, Most[Divisors[n]]}];
A342001[n_] := A003415[n]/A003557[n];
A345001[n_] := DivisorSigma[1, n] + A003415[n] - 2 n;
A344998[n_] := A342001[n]*A344753[n];
A344999[n_] := A048250[n]*A345001[n];
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 *)

Prog

(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A048250(n) = factorback(apply(p -> p+1, factor(n)[, 1]));
A344753(n) = sumdiv(n, d, (d<n)*(d+(issquarefree(n/d) * d)));
A342001(n) = (A003415(n) / A003557(n));
A345001(n) = (sigma(n)+A003415(n)-(2*n));
isA345003(n) = (A345001(n)*A048250(n) == A342001(n)*A344753(n));

Crossrefs

Positions of zeros in A345043.

Cf. A001615, A003415, A003557, A048250, A342001, A344753, A344998, A344999, A345001.

Cf. A000396, A301939, A345004, A345005 (subsequences).

Cf. also A345051.

Sequence in context: A237290 A229335 A007829 * A000773 A258283 A039720

Adjacent sequences: A345000 A345001 A345002 * A345004 A345005 A345006

Keyword

nonn

Author

Antti Karttunen, Jun 05 2021

Status

approved