A345051 -id:A345051 - 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));

A345048 a(n) = A342001(n) * A051709(n).

Original entry on oeis.org

0, 0, 0, 2, 0, 10, 0, 9, 2, 14, 0, 64, 0, 18, 16, 28, 0, 63, 0, 120, 20, 26, 0, 220, 2, 30, 12, 192, 0, 620, 0, 75, 28, 38, 24, 310, 0, 42, 32, 442, 0, 984, 0, 384, 156, 50, 0, 616, 2, 117, 40, 504, 0, 270, 32, 736, 44, 62, 0, 2944, 0, 66, 238, 186, 36, 1952, 0, 792, 52, 1652, 0, 975, 0, 78, 154, 960, 36, 2556, 0, 1276, 52

Offset: 1

Antti Karttunen, Jun 06 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A345049, A345050, A345051.

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]));
A342001(n) = (A003415(n) / A003557(n));
A051709(n) = ((sigma(n) + eulerphi(n)) - (2*n));
A345048(n) = (A342001(n) * A051709(n));

a(n) = A342001(n) * A051709(n).

a(n) = A345049(n) + A345050(n).

A345049 a(n) = A173557(n) * A345001(n).

Original entry on oeis.org

-1, 0, -2, 3, -12, 10, -30, 11, 2, 20, -90, 40, -132, 30, 16, 31, -240, 48, -306, 104, 0, 50, -462, 112, -36, 60, 26, 192, -756, 344, -870, 79, -80, 80, -240, 158, -1260, 90, -144, 312, -1560, 636, -1722, 440, 216, 110, -2070, 280, -162, 152, -320, 600, -2652, 186, -880, 600, -432, 140, -3306, 1120, -3540, 150, 348, 191

Offset: 1

Antti Karttunen, Jun 06 2021

sign

Cf. A173557, A345001, A345048, A345050, A345051.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A173557(n) = factorback(apply(p -> p-1,factor(n)[,1]));
A345001(n) = (sigma(n)+A003415(n)-(2*n));
A345049(n) = (A173557(n) * A345001(n));

a(n) = A173557(n) * A345001(n).

a(n) = A345048(n) - A345050(n).

A345050 a(n) = A345048(n) - A345049(n).

Original entry on oeis.org

1, 0, 2, -1, 12, 0, 30, -2, 0, -6, 90, 24, 132, -12, 0, -3, 240, 15, 306, 16, 20, -24, 462, 108, 38, -30, -14, 0, 756, 276, 870, -4, 108, -42, 264, 152, 1260, -48, 176, 130, 1560, 348, 1722, -56, -60, -60, 2070, 336, 164, -35, 360, -96, 2652, 84, 912, 136, 476, -78, 3306, 1824, 3540, -84, -110, -5, 1380, 492, 4290

Offset: 1

Antti Karttunen, Jun 06 2021

sign

Cf. A345048, A345049, A345051 (positions of zeros).

PARI
```
A345050(n) = (A345048(n) - A345049(n));
```

a(n) = A345048(n) - A345049(n).

A344994 Numbers k such that A173557(k) divides nonzero A051709(k).

Original entry on oeis.org

4, 6, 8, 12, 16, 24, 27, 28, 32, 42, 48, 54, 60, 64, 96, 108, 112, 120, 126, 128, 150, 168, 176, 192, 204, 216, 240, 243, 250, 256, 294, 312, 378, 384, 396, 432, 440, 448, 456, 460, 480, 486, 496, 500, 504, 512, 540, 588, 672, 700, 768, 774, 828, 840, 864, 888, 924, 960, 972, 1000, 1014, 1024, 1080, 1134, 1176, 1216

Offset: 1

Antti Karttunen, Jun 05 2021

nonn

Cf. A051709, A173557.
Cf. A344995, A345054 (subsequences).
Cf. also A344754, A345051.

A051709(n) = ((sigma(n) + eulerphi(n)) - (2*n));
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
isA344994(n) = { my(u=A051709(n)); ((u>0)&&(0==(u%A173557(n)))); };

A345054 Odd numbers k such that A173557(k) divides nonzero A051709(k).

Original entry on oeis.org

27, 243, 1377, 2187, 3125, 19683, 28125, 55233, 68445, 177147, 195625, 203125, 239805, 253125, 453125, 823543, 907137, 1323297, 1378125, 1464561, 1594323, 1953125, 2278125, 3341637, 3572829, 5255361, 5877117, 9034497, 9819837, 11701053, 14348907, 17453125, 19460393, 20503125, 22209633, 26010621, 30074733, 44910045

Offset: 1

Antti Karttunen, Jun 06 2021

nonn

Question: Are there any common terms with A345051?

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

Odd terms in A344994.

Cf. A051709, A173557, A345051.

A051709(n) = ((sigma(n) + eulerphi(n)) - (2*n));
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
isA345054(n) = if(!(n%2),0,my(u=A051709(n)); ((u>0)&&(0==(u%A173557(n)))));

cp's OEIS Frontend

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

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A345048 a(n) = A342001(n) * A051709(n).

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A345049 a(n) = A173557(n) * A345001(n).

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A345050 a(n) = A345048(n) - A345049(n).

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A344994 Numbers k such that A173557(k) divides nonzero A051709(k).

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A345054 Odd numbers k such that A173557(k) divides nonzero A051709(k).

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