A176345 -id:A176345 - cp's OEIS frontend

A143869 An integer k is called regular (mod n) if there is an integer x such that k^2 x == k (mod n). Then these numbers are the sum of regular integers k (mod n) such that 1 <= k <= n for n=1,2,... .

1, 3, 6, 8, 15, 21, 28, 24, 36, 55, 66, 60, 91, 105, 120, 80, 153, 135, 190, 160, 231, 253, 276, 192, 275, 351, 270, 308, 435, 465, 496, 288, 561, 595, 630, 396, 703, 741, 780, 520, 861, 903, 946, 748, 810, 1081, 1128, 672, 1078, 1075, 1326, 1040, 1431, 1053

Offset: 1

Laszlo Toth, Sep 04 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Osama Alkam and Emad Abu Osba, On the regular elements in Zn, Turk J Math, 32 (2008), 31-39.
B. Apostol and L. Petrescu, Extremal Orders of Certain Functions Associated with Regular Integers (mod n), Journal of Integer Sequences, 2013, # 13.7.5.
L. Tóth, Regular integers modulo n, Annales Univ. Sci. Budapest., Sect. Comp., 29 (2008), 263-275.

Cf. A055653, A176345.

isregu(k, n) = {g = gcd(k, n); if ((n % g == 0) && (gcd(g, n/g) == 1), return(k), return(0));}
a(n) = sum(k=1, n, isregu(k, n)) \\ Michel Marcus, May 25 2013

a(n) = n*(A055653(n)+1)/2.

Extended by R. J. Mathar, Sep 05 2008

A335032 Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 + p^(1 - s) - p^(-s)).

Original entry on oeis.org

1, 4, 6, 10, 10, 24, 14, 22, 21, 40, 22, 60, 26, 56, 60, 46, 34, 84, 38, 100, 84, 88, 46, 132, 55, 104, 66, 140, 58, 240, 62, 94, 132, 136, 140, 210, 74, 152, 156, 220, 82, 336, 86, 220, 210, 184, 94, 276, 105, 220, 204, 260, 106, 264, 220, 308, 228, 232, 118

Offset: 1

Vaclav Kotesovec, Jun 20 2020

Dirichlet convolution of A000203 with abs(A097945).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms).

Cf. A000203, A001620, A097945, A176345, A326306.

Table[Sum[DivisorSigma[1, n/d] * Abs[MoebiusMu[d]] * EulerPhi[d], {d, Divisors[n]}], {n, 1, 100}]

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X)/(1 - p*X))[n], ", "))

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X/(1 - X))/(1 - p*X))[n], ", "))

Dirichlet g.f.: zeta(s) * zeta(s-1)^2 / zeta(2*s-2) * Product_{primes p} (1 - 1/(p^s + p)).
Dirichlet g.f.: zeta(s) * zeta(s-1)^2 * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)).
Let f(s) = Product_{primes p} (1 - 1/(p^s + p)), then Sum_{k=1..n} a(k) ~ n^2 * ((log(n)/2 + gamma - 3*zeta'(2)/Pi^2 - 1/4)*f(2) + f'(2)/2), where f(2) = A065463 = Product_{primes p} (1 - 1/(p*(p+1))) = 0.7044422009991655927366033503266372..., f'(2) = f(2) * Sum_{primes p} p*log(p) / ((p+1)*(p^2+p-1)) = 0.23219454323726621271960146689644280341444084188447499043209938838191022838..., for zeta'(2) see A073002 and gamma is the Euler-Mascheroni constant A001620.
a(n) = Sum_{d|n} A176345(d). - Ridouane Oudra, Jan 14 2022
Multiplicative with a(p^e) = sigma(p^e) + p^e - 1. - Amiram Eldar, Dec 25 2022

A349694 Dirichlet convolution of the squarefree kernel function (A007947) with itself.

Original entry on oeis.org

1, 4, 6, 8, 10, 24, 14, 12, 15, 40, 22, 48, 26, 56, 60, 16, 34, 60, 38, 80, 84, 88, 46, 72, 35, 104, 24, 112, 58, 240, 62, 20, 132, 136, 140, 120, 74, 152, 156, 120, 82, 336, 86, 176, 150, 184, 94, 96, 63, 140, 204, 208, 106, 96, 220, 168, 228, 232, 118, 480

Offset: 1

Ilya Gutkovskiy, Nov 25 2021

Cf. A007947, A097945, A176345, A191750.

Table[Sum[Last[Select[Divisors[d], SquareFreeQ]] Last[Select[Divisors[n/d], SquareFreeQ]], {d, Divisors[n]}], {n, 1, 60}]
f[p_, e_] := (e - 1)*p^2 + 2*p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Nov 25 2021 *)

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A349694(n) = sumdiv(n,d,A007947(n/d)*A007947(d)); \\ Antti Karttunen, Nov 25 2021

Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + p^(1-s) - p^(-s))^2.
a(n) = Sum_{d|n} A007947(d) * A007947(n/d).
a(n) = Sum_{d|n} abs(A097945(d)) * A191750(n/d).
Multiplicative with a(p^e) = (e-1)*p^2 + 2*p. - Amiram Eldar, Nov 25 2021
From Vaclav Kotesovec, Nov 26 2021: (Start)
Dirichlet g.f.: zeta(s-1)^2 * zeta(s)^2 * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s))^2.
Let f(s) = Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)), then
Sum_{k=1..n} a(k) ~ Pi^2 * f(2)^2 * n^2 / 144 * (Pi^2 * (2*log(n) + 4*gamma - 1 + 4*f'(2)/f(2)) + 24*zeta'(2)), where f(2) = Product_{primes p} (1 - 2/p^2 + 1/p^3) = A065464 = 0.428249505677094440218765707581823546121298513355936144..., f'(2) = f(2) * Sum_{primes p} log(p) * (3*p - 2) / (p^3 - 2*p + 1) = 0.6293283828324697510445630056425352981207558777167836747744750359407300858..., zeta'(2) = -A073002 and gamma is the Euler-Mascheroni constant A001620. (End)

A374432 Row sums of A374433.

Original entry on oeis.org

1, 2, 4, 6, 7, 10, 16, 14, 13, 16, 28, 22, 31, 26, 40, 46, 25, 34, 46, 38, 55, 66, 64, 46, 61, 46, 76, 46, 79, 58, 136, 62, 49, 106, 100, 118, 91, 74, 112, 126, 109, 82, 196, 86, 127, 136, 136, 94, 121, 92, 136, 166, 151, 106, 136, 190, 157, 186, 172, 118, 271

Offset: 0

Peter Luschny, Jul 10 2024

nonn

Cf. A374433, A176345 (a(n + 1) - 1).

seq(add(A374433(n, k), k = 0..n), n=0..60);

nn = 120; Do[Set[s[i], FactorInteger[i][[All, 1]]], {i, 0, nn}]; s[0] = {1};
Table[Sum[Times @@ Intersection[s[k], s[n]], {k, 0, n}], {n, 0, nn}] (* Michael De Vlieger, Jul 11 2024 *)

print([sum([A374433(n, k) for k in range(n + 1)]) for n in range(61)])

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A143869 An integer k is called regular (mod n) if there is an integer x such that k^2 x == k (mod n). Then these numbers are the sum of regular integers k (mod n) such that 1 <= k <= n for n=1,2,... .

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A335032 Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 + p^(1 - s) - p^(-s)).

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A349694 Dirichlet convolution of the squarefree kernel function (A007947) with itself.

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A374432 Row sums of A374433.

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