A165552 -id:A165552 - cp's OEIS frontend

A176891 Triangle T(n,k) = k if k

1, 1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 1, 2, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 4, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 0, 6, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gary W. Adamson and Mats Granvik, Apr 28 2010

A variant of A127093, which has T(n,n) = n. [The original definition said "Subsequence of A127093". Since all nonnegative integers are repeated infinitely often in both sequences, each one is a subsequence of the other, but there is no such relation on a row-by-row basis. - M. F. Hasler, Aug 08 2016]

Let A=A176891*A176891, B=A*A, C=B*B, D=C*C and so on, then the leftmost column in the last matrix (D) converges to A165552.

			Triangle begins:
1,
1,0,
1,0,0,
1,2,0,0,
1,0,0,0,0,
1,2,3,0,0,0,
1,0,0,0,0,0,0,
1,2,0,4,0,0,0,0,

Cf. A127093, A165552.

T(n,k) = if n=1 and k=1 then 1 elseif n=k then 0 elseif k divides n then k else 0.

Definition corrected by M. F. Hasler, Aug 08 2016

A330774 Number of n-color perfect compositions of n.

Original entry on oeis.org

1, 1, 1, 5, 1, 11, 1, 61, 7, 15, 1, 259, 1, 19, 17, 1901, 1, 383, 1, 511, 21, 27, 1, 14147, 11, 31, 187, 859, 1, 1403, 1, 147661, 29, 39, 25, 39307, 1, 43, 33, 42351, 1, 2303, 1, 1843, 947, 51, 1, 1815811, 15, 1255, 41, 2479, 1, 46697, 33, 97339, 45, 63, 1, 219347, 1

Offset: 0

Augustine O. Munagi, Dec 30 2019

nonn

An n-color perfect composition of v is a composition into j types of each part j whose sequence of parts contains one composition of every positive integer less than n.

			a(5)=11 because v=5 has eleven n-color perfect compositions: (1,1,1,1,1), (1,2,2),(2,2,1), (1,2',2'), (2',2',1), (1,1,3), (3,1,1), (1,1,3'), (3',1,1), (1,1,3''), (3'',1,1).

A. O. Munagi, Perfect Compositions of Numbers, J. Integer Seq. 23 (2020), art. 20.5.1.

Cf. A330773, A165552, A001906, A002033, A074206.

A342403 a(1) = 1; a(n) = -Sum_{d|n, d < n} d * a(d).

Original entry on oeis.org

1, -1, -1, 1, -1, 4, -1, -3, 2, 6, -1, -24, -1, 8, 7, 21, -1, -38, -1, -58, 9, 12, -1, 288, 4, 14, -16, -108, -1, -180, -1, -315, 13, 18, 11, 930, -1, 20, 15, 1126, -1, -314, -1, -256, -116, 24, -1, -6960, 6, -154, 19, -354, -1, 1078, 15, 2940, 21, 30, -1, 6664, -1, 32, -198, 9765, 17

Offset: 1

Ilya Gutkovskiy, Mar 10 2021

sign

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

Cf. A008683, A055615, A165552, A321384.

a[1] = 1; a[n_] := a[n] = -Sum[d a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 65}]

a(n) = if (n==1, 1, -sumdiv(n, d, if (dMichel Marcus, Mar 11 2021

up_to = 20000;
A342403list(n) = { my(v=vector(n)); v[1] = 1; for(n=2, #v, v[n] = -sumdiv(n, d, if(d==n, 0, v[d]*d))); (v); };
v342403 = A342403list(up_to);
A342403(n) = v342403[n]; \\ Antti Karttunen, Jan 22 2025

from sympy import divisors
def A342403(n): return 1 if n == 1 else -sum(d*A342403(d) for d in divisors(n) if d < n) # Chai Wah Wu, Mar 11 2021

A326824 a(1) = 1; a(n) = Sum_{d|n, d < n} phi(n/d) * d * a(d).

Original entry on oeis.org

1, 1, 2, 4, 4, 12, 6, 24, 18, 32, 10, 124, 12, 60, 72, 240, 16, 336, 18, 440, 132, 140, 22, 2088, 100, 192, 378, 1044, 28, 2096, 30, 4320, 300, 320, 312, 9636, 36, 396, 408, 10384, 40, 5040, 42, 3500, 3000, 572, 46, 61584, 294, 3920, 672, 5544, 52, 23148, 680

Offset: 1

Ilya Gutkovskiy, Feb 23 2020

nonn

Cf. A000010, A001710, A006579, A006874, A018804, A165552.

a[1] = 1; a[n_] := Sum[If[d < n, EulerPhi[n/d] d a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 55}]
a[1] = 1; a[n_] := Sum[GCD[n, k] a[GCD[n, k]], {k, 1, n - 1}]; Table[a[n], {n, 1, 55}]

a(1) = 1; a(n) = Sum_{k=1..n-1} gcd(n, k) * a(gcd(n, k)).

A348661 a(1) = 1; a(n) = Sum_{d|n, d < n} d * a(d)^(n/d).

Original entry on oeis.org

1, 1, 1, 3, 1, 6, 1, 39, 4, 8, 1, 330, 1, 10, 9, 12495, 1, 1446, 1, 1620, 11, 14, 1, 1792050, 6, 16, 580, 10158, 1, 53002, 1, 2516534175, 15, 20, 13, 469241466, 1, 22, 17, 774558756, 1, 1696170, 1, 712914, 20160, 26, 1, 108457624531554, 8, 328588, 21, 6383964

Offset: 1

Ilya Gutkovskiy, Oct 28 2021

nonn

Cf. A006241, A008578 (positions of 1's), A157313, A165552, A196545, A281145.

a[1] = 1; a[n_] := a[n] = Sum[If[d < n, d a[d]^(n/d), 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 52}]

For n > 1, a(n) is the coefficient of x^n/n in expansion of -log(Product_{k=1..n-1} (1 - a(k)*x^k)).

cp's OEIS Frontend

A176891 Triangle T(n,k) = k if k

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A330774 Number of n-color perfect compositions of n.

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A342403 a(1) = 1; a(n) = -Sum_{d|n, d < n} d * a(d).

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A326824 a(1) = 1; a(n) = Sum_{d|n, d < n} phi(n/d) * d * a(d).

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A348661 a(1) = 1; a(n) = Sum_{d|n, d < n} d * a(d)^(n/d).

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