A345001 -id:A345001 - cp's OEIS frontend

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

-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).

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).

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))); };

A345002 Numbers k such that A345001(k)/A342001(k) is a positive natural number and a divisor of k.

Original entry on oeis.org

6, 28, 171, 496, 8128, 478800, 32317272, 33550336

Offset: 1

Antti Karttunen, Jun 05 2021

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

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

Cf. also A344755.

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]));
isA345002(n) = if(1==n,0,my(d=A003415(n), s=sigma(n)+d-(n+n), r=s/(d/A003557(n))); (r>0&&1==denominator(r)&&!(n%r)));

A051709 a(n) = sigma(n) + phi(n) - 2n.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 1, 2, 0, 8, 0, 2, 2, 7, 0, 9, 0, 10, 2, 2, 0, 20, 1, 2, 4, 12, 0, 20, 0, 15, 2, 2, 2, 31, 0, 2, 2, 26, 0, 24, 0, 16, 12, 2, 0, 44, 1, 13, 2, 18, 0, 30, 2, 32, 2, 2, 0, 64, 0, 2, 14, 31, 2, 32, 0, 22, 2, 28, 0, 75, 0, 2, 14, 24, 2, 36, 0

Offset: 1

Jud McCranie and Carlos Rivera

nonn

Sigma is the sum of divisors (A000203), and phi is the Euler totient function (A000010). - Michael B. Porter, Jul 05 2013
Because sigma and phi are multiplicative functions, it is easy to show that (1) if a(n)=0, then n is prime or 1 and (2) if a(n)=2, then n is the product of two distinct prime numbers. Note that a(n) is the n-th term of the Dirichlet series whose generating function is given below. Using the generating function, it is theoretically possible to compute a(n). Hence a(n)=0 could be used as a primality test and a(n)=2 could be used as a test for membership in P2 (A006881). - T. D. Noe, Aug 01 2002
It appears that a(n) - A002033(n) = zeta(s-1) * (zeta(s) - 2 + 1/zeta(s)) + 1/(zeta(s)-2). - Eric Desbiaux, Jul 04 2013
a(n) = 1 if and only if n = prime(k)^2 (n is in A001248). It seems that a(n) = k has only finitely many solutions for k >= 3. - Jianing Song, Jun 27 2021

			a(5) = sigma(5) + phi(5) - 2*5 = 6 + 4 - 10 = 0.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (First 1000 terms from T. D. Noe.)
Carlos Rivera, Puzzle 76. z(n)=sigma(n) + phi(n) - 2n, The Prime Puzzles and Problems Connection.

Cf. A000010, A000203, A001065, A001248, A005843, A006881, A051612, A051953, A065387, A072780, A228498 (= a(n^2)), A297159, A324048, A344994, A344995, A344996, A345048, A345054.
Cf. A278373 (range of this sequence), A056996 (numbers not present).
Cf. also A344753, A345001 (analogous sequences).

Table[DivisorSigma[1,n]+EulerPhi[n]-2n,{n,80}] (* Harvey P. Dale, Apr 08 2015 *)

a(n)=sigma(n)+eulerphi(n)-2*n \\ Charles R Greathouse IV, Jul 05 2013

A051709(n) = -sumdiv(n,d,(dAntti Karttunen, Mar 02 2018

Dirichlet g.f.: zeta(s-1) * (zeta(s) - 2 + 1/zeta(s)). - T. D. Noe, Aug 01 2002
From Antti Karttunen, Mar 02 2018: (Start)
a(n) = A001065(n) - A051953(n).  [Difference between the sum of proper divisors of n and their Moebius-transform.]
a(n) = -Sum_{d|n, dA008683(n/d)*A001065(d). (End)
Sum_{k=1..n} a(k) = (3/(Pi^2) + Pi^2/12 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 03 2023

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

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

Original entry on oeis.org

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));

A345051 Numbers k such that A345048(k) is equal to A345049(k).

Original entry on oeis.org

2, 6, 9, 15, 28, 496, 625, 1225, 3993, 8128, 117649, 218491, 857375, 3788435, 4259571, 33550336, 69302975, 136410197, 200533921, 313742585, 603439225, 1516358753, 2563893625, 3326174929, 5655792025, 8589869056, 10214476341

Offset: 1

Antti Karttunen, Jun 06 2021

Numbers k for which A342001(n) * A051709(n) = A173557(n) * A345001(n).

Conjecture: Sequence is a disjoint union of A000396 and A166374, i.e., there are no terms of any other kind.

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

Cf. A051709, A173557, A342001, A345001, A345048, A345049.
Positions of zeros in A345050.
Cf. A000396, A166374 (subsequences).
Cf. also A345003.

PARI
```
isA345051(n) = (0==A345050(n));
```

a(21)-a(27) from Amiram Eldar, Dec 08 2023

cp's OEIS Frontend

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

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

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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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A345002 Numbers k such that A345001(k)/A342001(k) is a positive natural number and a divisor of k.

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A051709 a(n) = sigma(n) + phi(n) - 2n.

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Formula

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

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

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A345051 Numbers k such that A345048(k) is equal to A345049(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.