A057660 -id:A057660 - cp's OEIS frontend

A350900 Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(i,n) / gcd(gcd(i,k),n) for 1 <= k <= n.

1, 3, 2, 5, 5, 3, 8, 5, 8, 4, 9, 9, 9, 9, 5, 15, 10, 9, 10, 15, 6, 13, 13, 13, 13, 13, 13, 7, 20, 12, 20, 9, 20, 12, 20, 8, 21, 21, 11, 21, 21, 11, 21, 21, 9, 27, 18, 27, 18, 15, 18, 27, 18, 27, 10, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 11, 40, 25, 24, 20, 40, 15, 40, 20, 24, 25, 40, 12

Offset: 1

Werner Schulte, Jan 21 2022

Subtriangle (triangle without main diagonal) is symmetrical.

Conjecture: Let f be an arbitrary arithmetic function. Define for n > 0 the sequence a(f; n) = Sum_{i=1..n, k=1..n} f(gcd(i,n)/gcd(gcd(i,k),n)); a(f; n) equals Dirichlet convolution of f(n)*A000010(n) and A057660(n); if f is multiplicative, then a(f; n) is multiplicative; row sums of this triangle use f(n) = n (see formula section).

			The triangle T(n, k) for 1 <= k <= n starts:
n \k :   1   2   3   4   5   6   7   8   9  10  11  12
======================================================
   1 :   1
   2 :   3   2
   3 :   5   5   3
   4 :   8   5   8   4
   5 :   9   9   9   9   5
   6 :  15  10   9  10  15   6
   7 :  13  13  13  13  13  13   7
   8 :  20  12  20   9  20  12  20   8
   9 :  21  21  11  21  21  11  21  21   9
  10 :  27  18  27  18  15  18  27  18  27  10
  11 :  21  21  21  21  21  21  21  21  21  21  11
  12 :  40  25  24  20  40  15  40  20  24  25  40  12
  etc.

Row sums gives A373059.

Cf. A000010, A002618, A018804, A057660, A050873.

T(n, k) = sum(i=1, n, gcd(i,n) / gcd(gcd(i,k),n));
row(n) = vector(n, k, T(n,k)); \\ Michel Marcus, Jan 22 2022

T(n, 1) = A018804(n); T(n, n) = n.
T(n, k) = T(n, n-k) for 1 <= k < n.
Conjecture: Row sums equal Dirichlet convolution of A002618 and A057660.

A365076 Number of length-n binary words x such that the infinite word xxxx... is balanced.

Original entry on oeis.org

2, 4, 8, 12, 22, 22, 44, 44, 62, 64, 112, 78, 158, 130, 148, 172, 274, 184, 344, 232, 302, 334, 508, 302, 522, 472, 548, 474, 814, 442, 932, 684, 778, 820, 904, 672, 1334, 1030, 1100, 904, 1642, 904, 1808, 1222, 1282, 1522, 2164, 1198, 2102, 1564, 1912, 1728

Offset: 1

Jeffrey Shallit, Aug 20 2023

nonn

A binary word w is "balanced" if for all lengths and all blocks b of the same length appearing in it, the number of 1's in b can take only two different values. For example, 00111 is not balanced because 00 has no 1's, 01 has one, and 11 has two.

			For n = 4, the 12 such words are 0000, 0001, 0010, 0100, 0101, 0111 and their bitwise binary complements.

Laurent Vuillon, Balanced words, Bull. Belg. Math. Soc. Simon Stevin 10(5): 787-805 (December 2003).

Cf. A057660, A057661.

from math import gcd
def A365076(n): return sum(n//gcd(n,k) for k in range(1,n+1))+1 # Chai Wah Wu, Aug 24 2023

from math import prod
from sympy import factorint
def A365076(n): return 1+prod((p**((e<<1)+1)+1)//(p+1) for p,e in factorint(n).items()) # Chai Wah Wu, Aug 05 2024

a(n) = 2*A057661(n).

a(n) = A057660(n) + 1.

A292302 Expansion of (1 - x)Sum_{k>=1} kphi(k)*x^k/(1 - x^k), where phi() is the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 4, 4, 10, 0, 22, 0, 18, 2, 48, -34, 80, -28, 18, 24, 102, -90, 160, -112, 70, 32, 174, -206, 220, -50, 76, -74, 340, -372, 490, -248, 94, 42, 84, -232, 662, -304, 70, -196, 738, -738, 904, -586, 60, 240, 642, -966, 904, -538, 348, -184, 1030, -1116, 690, -482, 552, 38, 984, -1806

Offset: 1

Ilya Gutkovskiy, Sep 14 2017

sign

First differences of A057660.

Cf. A057660, A174405.

nmax = 60; Rest[CoefficientList[Series[(1 - x) Sum[EulerPhi[k] k x^k /(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]

G.f.: (1 - x)*Sum_{k>=1} k*phi(k)*x^k/(1 - x^k).

A307705 Expansion of Product_{k>=1} 1/(1 - x^k)^(k-phi(k)), where phi() is the Euler totient function (A000010).

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 8, 5, 16, 15, 34, 30, 75, 66, 144, 150, 285, 292, 566, 585, 1062, 1170, 1988, 2205, 3729, 4159, 6755, 7785, 12214, 14147, 21957, 25560, 38709, 45839, 67884, 80747, 118332, 141244, 203614, 245330, 348396, 420971, 592439, 717659, 998248, 1215439, 1672544, 2040210, 2786687

Offset: 0

Ilya Gutkovskiy, Apr 22 2019

nonn

Euler transform of A051953.

Cf. A000010, A001157, A051953, A053650, A057660, A061255, A065764, A065827.

nmax = 48; CoefficientList[Series[Product[1/(1 - x^k)^(k - EulerPhi[k]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[(DivisorSigma[2, k] - DivisorSigma[2, k^2]/DivisorSigma[1, k^2]) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 - EulerPhi[d^2], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 48}]

G.f.: exp(Sum_{k>=1} (sigma_2(k) - sigma_2(k^2)/sigma_1(k^2)) * x^k/k).
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} cototient(d^2) ) * x^k/k).
a(n) ~ exp(3*((Pi^2 - 6)*Zeta(3))^(1/3) * n^(2/3) / (2*Pi)^(2/3) + 1/4) * ((Pi^2 - 6)*Zeta(3))^(1/4) / (A^3 * 2^(1/12) * 3^(1/2) * Pi^(5/6) * n^(3/4)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 06 2019

A317480 Numbers m such that Sum_{k=1..m} 1/gcd(k,m) is an integer.

Original entry on oeis.org

1, 614341, 618233, 1854699, 11746427, 26584019, 35239281, 79752057, 85393399, 118082503, 345592247, 354247509, 505096361, 802597537, 1036776741, 1062742527, 1515289083, 2149579159, 2243567557, 3695178641, 5077547629, 5250772527, 6566252693, 6730702671

Offset: 1

Amiram Eldar, Jul 29 2018

nonn

Also numbers n such that n | A057660(n).

a(1)-a(21) are taken from De Koninck's book.

Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, page 265.

Cf. A057660.

fun[p_, e_]:=(p^(2e+1)+1)/(p+1); aQ[n_] := Divisible[Times @@ (fun @@@ FactorInteger[n]), n]; Select[Range[10000000], aQ]

xi(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = (p^(2*e+1)+1)/(p+1); f[i, 2] = 1; ); factorback(f); } for(n=1, 1e7, if(xi(n) % n == 0, print(n)))

cp's OEIS Frontend

A350900 Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(i,n) / gcd(gcd(i,k),n) for 1 <= k <= n.

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A365076 Number of length-n binary words x such that the infinite word xxxx... is balanced.

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A292302 Expansion of (1 - x)*Sum_{k>=1} k*phi(k)*x^k/(1 - x^k), where phi() is the Euler totient function (A000010).

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A307705 Expansion of Product_{k>=1} 1/(1 - x^k)^(k-phi(k)), where phi() is the Euler totient function (A000010).

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A317480 Numbers m such that Sum_{k=1..m} 1/gcd(k,m) is an integer.

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A292302 Expansion of (1 - x)Sum_{k>=1} kphi(k)*x^k/(1 - x^k), where phi() is the Euler totient function (A000010).