A344754 -id:A344754 - cp's OEIS frontend

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

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

A344755 Numbers k such that A344753(k) is a multiple of A048250(k), and k is a multiple of A344753(k)/A048250(k).

Original entry on oeis.org

6, 28, 150, 496, 528, 1980, 4560, 8128, 8736, 11400, 19872, 20664, 75840, 82080, 253080, 254880, 741744, 1627290, 5130300, 5607360, 7529760, 19645440, 20718720, 33550336, 35092512, 45643392, 45995040, 56424960, 86944320, 169910136, 174013920, 180442080, 196378992, 242040960, 304577280, 314511360, 326611440, 451344960

Offset: 1

Antti Karttunen, May 29 2021

nonn

Numbers k for which A344753(k)/A048250(k) is a divisor of k.

Perfect numbers (A000396, including also any hypothetical odd terms) are included as only on them A001615 coincides with A344753, and because A001615(n) = A003557(n)*A048250(n), with A003557(n) being a divisor of n.

Cf. A000396 (subsequence), A001065, A001615, A003557, A306927, A048250, A344753.
Subsequence of A344754.
Cf. also A344700.

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A344753(n) = sumdiv(n,d,(dA344755(n) = { my(t=A344753(n),u=A048250(n)); ((0==(t%u))&&(0==(n%(t/u)))); };

A344700 Numbers k for which A306927(k) [= A001615(k)-k] is a multiple of A344705(k) [= A001615(k)-A001065(k)], and their quotient is nonnegative.

Original entry on oeis.org

1, 6, 24, 28, 168, 496, 864, 1080, 1520, 1836, 2016, 2088, 2112, 2520, 2912, 2976, 3000, 3024, 3240, 3800, 8128, 9000, 11088, 11232, 11448, 14160, 14688, 16920, 17028, 18360, 19872, 20520, 20880, 25280, 25488, 27552, 29376, 30800, 31200, 31320, 31968, 35400, 39240, 44064, 48768, 49896, 50760, 51480, 51660, 52200, 55680

Offset: 1

Antti Karttunen, May 28 2021

nonn

Numbers k for which A344704(k) = A344705(k), i.e., numbers k such that gcd(A001615(k)-k, A001615(k)-A001065(k)) = A001615(k) - A001065(k).
Note that A306927(k) is always nonnegative, but A344705(k) = A033879(k) + A306927(k) gets also negative values. Number k is perfect only when A033879(k) = A344705(k) - A306927(k) = 0, that is, when A344705(k) = A306927(k), which necessitates that A306927(k) should be a multiple of A344705(k), and their quotient should be nonnegative (actually = +1).
In the range 1 .. 2^31 there are 782 such numbers, of which only the initial 1 is odd.

Cf. A000203, A001065, A001615, A033879, A244963, A306927, A344704, A344705, A344752 (gives the quotient A306927(k)/A344705(k) computed for these terms), A344753.
Cf. A000396 (subsequence).
Cf. also A344754, A344755.

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
isA344700(n) = { my(t=A001615(n), s=sigma(n), u = (n+t)-s); (gcd(t-n,u)==u); };
\\ Alternatively as:
isA344700(n) = { my(t=A001615(n), s=sigma(n), u = (n+t)-s); ((u>0)&&(0==((t-n)%u))); };

A345004 Numbers k such that A345001(k)A048250(k) is equal to A342001(k)A344753(k) and A345001(n)/A342001(n) is an integer.

Original entry on oeis.org

6, 28, 496, 5292, 8128, 33550336

Offset: 1

Antti Karttunen, Jun 05 2021

Equally, we may require that A344753(k)/A048250(k) is an integer, thus this is a subsequence of A344754.

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

Cf. A000396 (subsequence), A003415, A003557, A048250, A342001, A344753, A345001.

Intersection of A344754 and A345003.

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));
isA345004(n) = { my(u=A345001(n),v=A342001(n)); (v>0&&1==denominator(u/v)&&(u*A048250(n) == v*A344753(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)))); };

cp's OEIS Frontend

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

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A344755 Numbers k such that A344753(k) is a multiple of A048250(k), and k is a multiple of A344753(k)/A048250(k).

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A344700 Numbers k for which A306927(k) [= A001615(k)-k] is a multiple of A344705(k) [= A001615(k)-A001065(k)], and their quotient is nonnegative.

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A345004 Numbers k such that A345001(k)*A048250(k) is equal to A342001(k)*A344753(k) and A345001(n)/A342001(n) is an integer.

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

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A345004 Numbers k such that A345001(k)A048250(k) is equal to A342001(k)A344753(k) and A345001(n)/A342001(n) is an integer.