A301939 -id:A301939 - cp's OEIS frontend

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

Antti Karttunen, Jun 05 2021

nonn

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.

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

Positions of zeros in A345043.
Cf. A001615, A003415, A003557, A048250, A342001, A344753, A344998, A344999, A345001.
Cf. A000396, A301939, A345004, A345005 (subsequences).
Cf. also A345051.

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 *)

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,(dA342001(n) = (A003415(n) / A003557(n));
A345001(n) = (sigma(n)+A003415(n)-(2*n));
isA345003(n) = (A345001(n)*A048250(n) == A342001(n)*A344753(n));

A342458 a(n) = gcd(A001615(n), A003415(n)), where A001615 is Dedekind psi, and A003415 is the arithmetic derivative of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 12, 6, 1, 1, 8, 1, 3, 8, 8, 1, 3, 1, 12, 2, 1, 1, 4, 10, 3, 9, 16, 1, 1, 1, 16, 2, 1, 12, 12, 1, 3, 8, 4, 1, 1, 1, 24, 3, 1, 1, 16, 14, 45, 4, 28, 1, 27, 8, 4, 2, 1, 1, 4, 1, 3, 3, 96, 6, 1, 1, 36, 2, 1, 1, 12, 1, 3, 5, 40, 6, 1, 1, 16, 108, 1, 1, 4, 2, 3, 8, 4, 1, 3, 4, 48, 2, 1, 24, 16, 1, 7, 3, 20

Offset: 1

Antti Karttunen, Mar 28 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A001615, A003415, A003557, A342459, A342919.
Cf. A301939 (gives the positions at which a(n) = A001615(n) = A003415(n)).
Cf. also A175732, A342413, A342915.

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342458(n) = gcd(A001615(n), A003415(n));

a(n) = gcd(A001615(n), A003415(n)).
a(n) = A003557(n) * A342459(n).
a(n) = A003415(n) / A342919(n).

A345005 Odd numbers whose arithmetic derivative (A003415) is equal to Dedekind psi (A001615) applied to the same number.

Original entry on oeis.org

81, 3375, 15625, 231525, 713097, 1058841, 1500625, 4348377, 5764801, 16891497, 163555875, 209548647, 239010993, 239160735, 254205875, 267651475, 405189675, 451699875, 958403475, 1050284375, 1213014231, 1501534375, 1695809375, 1809323971, 1942143291

Offset: 1

Antti Karttunen, Jun 05 2021

Conjectured to be also the odd numbers k for which A344998(k) = A344999(k).

Odd terms in A301939, but see also A345003.

Cf. A001615, A003415, A344998, A344999.

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA345005(n) = ((n%2)&&(A003415(n)==A001615(n)));

cp's OEIS Frontend

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

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A342458 a(n) = gcd(A001615(n), A003415(n)), where A001615 is Dedekind psi, and A003415 is the arithmetic derivative of n.

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A345005 Odd numbers whose arithmetic derivative (A003415) is equal to Dedekind psi (A001615) applied to the same number.

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