A344998 -id:A344998 - 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));

A345043 a(n) = A344998(n) - A344999(n).

Original entry on oeis.org

1, 2, 6, 1, 20, 0, 42, 0, 10, 22, 110, -16, 156, 60, 96, -1, 272, -15, 342, -12, 220, 184, 506, -12, 76, 270, 14, 0, 812, -492, 930, -2, 612, 490, 792, -38, 1332, 624, 880, -10, 1640, -660, 1806, 48, 132, 940, 2162, 56, 246, 63, 1560, 84, 2756, -36, 2128, -8, 1972, 1534, 3422, -1248, 3660, 1764, 330, -3, 3036, -996, 4422

Offset: 1

Antti Karttunen, Jun 06 2021

sign

Each term a(n) seems to be a multiple of A033879(n). Factor (quotient) is negative at least for 16, 32, 48, 64, 72, 80, 96, 112, 128, 144, 160, 176, 192, 200, 208, 216, 224, 240, 243, 256, 272, 288, 304, 320, 324, 336, 352, 368, 384, 392, 400, 416, 432, 448, 464, 480, 486, ...

Cf. A033879, A344998, A344999, A345003 (positions of zeros).

a(n) = A344998(n) - A344999(n).

A344753 a(n) = sigma(n) + psi(n) - 2n = Sum_{d|n, d

Original entry on oeis.org

0, 2, 2, 5, 2, 12, 2, 11, 7, 16, 2, 28, 2, 20, 18, 23, 2, 39, 2, 38, 22, 28, 2, 60, 11, 32, 22, 48, 2, 84, 2, 47, 30, 40, 26, 91, 2, 44, 34, 82, 2, 108, 2, 68, 60, 52, 2, 124, 15, 83, 42, 78, 2, 120, 34, 104, 46, 64, 2, 192, 2, 68, 74, 95, 38, 156, 2, 98, 54, 148, 2, 195, 2, 80, 94, 108, 38, 180, 2, 170, 67, 88, 2

Offset: 1

Antti Karttunen, May 28 2021

Sigma is the sum of divisors (A000203), and psi is Dedekind psi-function (A001615). Coincides with the latter only on perfect numbers (A000396).

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001065, A001615, A008683, A008966, A033879, A173557, A306927, A344705, A342001, A344705, A344754, A344755, A344997, A344998.
Cf. A000396 [where a(n) = A001615(n)], A005100 [a(n) < A001615(n)], A005101 [a(n) > A001615(n)].
Cf. also A051709, A345001.
Cf. A013661, A082020.

a[n_] := Sum[d + If[SquareFreeQ[n/d], d, 0], {d, Most[Divisors[n]]}];
Array[a, 100] (* Jean-François Alcover, Jun 12 2021 *)

PARI
```
A344753(n) = sumdiv(n,d,(d
				
```

a(n) = Sum_{d|n, dA008966(n/d) * d).
a(n) = A001065(n) + A306927(n).
a(n) = A001615(n) - A033879(n).
a(n) = A344705(n) + 2*A001065(n) - n.
For squarefree n, a(n) = 2*A001065(n).
a(n) = A344997(n) / A173557(n) = A344998(n) / A342001(n). - Antti Karttunen, Jun 06 2021
Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = Pi^2/6 + 15/Pi^2 - 2 = 1.164751... . - Amiram Eldar, Dec 08 2023

New primary definition added by Antti Karttunen, Jun 06 2021

A344999 a(n) = A048250(n) * A345001(n).

Original entry on oeis.org

-1, 0, -4, 9, -18, 60, -40, 33, 4, 90, -108, 240, -154, 120, 48, 93, -270, 288, -340, 468, 0, 180, -504, 672, -54, 210, 52, 768, -810, 3096, -928, 237, -192, 270, -480, 948, -1330, 300, -336, 1404, -1638, 5088, -1804, 1584, 648, 360, -2160, 1680, -216, 684, -720, 2100, -2754, 1116, -1584, 2400, -960, 450, -3420, 10080

Offset: 1

Antti Karttunen, Jun 05 2021

sign

Cf. A000203, A003415, A048250, A345001.

Cf. A345003 [gives k for which a(k) = A344998(k)], A345004, A345005.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A345001(n) = (sigma(n)+A003415(n)-(2*n));
A344999(n) = (A048250(n) * A345001(n));

a(n) = A048250(n) * A345001(n).

a(n) = A344998(n) - A345043(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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A345043 a(n) = A344998(n) - A344999(n).

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A344753 a(n) = sigma(n) + psi(n) - 2n = Sum_{d|n, d

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A344999 a(n) = A048250(n) * A345001(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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