A007441 -id:A007441 - cp's OEIS frontend

A030009 Euler transform of primes.

1, 2, 6, 15, 37, 85, 192, 414, 879, 1816, 3694, 7362, 14480, 28037, 53644, 101379, 189587, 350874, 643431, 1169388, 2108045, 3770430, 6694894, 11804968, 20679720, 35999794, 62298755, 107198541, 183462856, 312357002, 529173060, 892216829, 1497454396, 2502190992

Offset: 0

N. J. A. Sloane

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Cf. A007441.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*ithprime(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  #  Alois P. Heinz, Sep 06 2008

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*Prime[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 16 2014, after Alois P. Heinz *)

a(n)=if(n<0,0,polcoeff(prod(i=1,n,(1-x^i)^-prime(i),1+x*O(x^n)),n))

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: nth_prime(n))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

G.f.: Product_{n>=1} (1-x^n)^(-prime(n)).

A061152 Expansion of Product_{n>=1} (1+x^n)^prime(n).

Original entry on oeis.org

1, 2, 4, 11, 23, 51, 107, 216, 430, 839, 1614, 3046, 5684, 10465, 19046, 34321, 61225, 108245, 189779, 330093, 569916, 977139, 1664304, 2817039, 4740000, 7930740, 13198108, 21851556, 36001483, 59035979, 96373100, 156644241, 253550911

Offset: 0

Vladeta Jovovic, Apr 16 2001

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A007441, A007445, A030009, A061150, A061151.

a(n) = (1/n) * Sum_{k=1..n} a(n-k)*b(k), k>0, a(0)=1, b(k)=Sum_{d|k} (-1)^(k/d+1)*d*prime(d).

A061150 a(n) = Sum_{d|n} d*prime(d).

Original entry on oeis.org

2, 8, 17, 36, 57, 101, 121, 188, 224, 353, 343, 573, 535, 729, 777, 1036, 1005, 1406, 1275, 1801, 1669, 2087, 1911, 2861, 2482, 3167, 3005, 3753, 3163, 4541, 3939, 5228, 4879, 5737, 5391, 7314, 5811, 7475, 7063, 8873, 7341, 9957, 8215, 10607, 9849

Offset: 1

Vladeta Jovovic, Apr 16 2001

			a(4)=36 because the divisors of 4 are 1,2,4 and 1*p(1) + 2*p(2) + 4*p(4) = 1*2 + 2*3 + 4*7 = 36.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A007441, A007445, A030009, A061151-A061152.

Cf. A127093.

with(numtheory): a:=proc(n) local div: div:=divisors(n): sum(div[j]*ithprime(div[j]),j=1..tau(n)) end: seq(a(n),n=1..55); # Emeric Deutsch, Jan 20 2007

a(n) = sumdiv(n, d, d*prime(d)); \\ Michel Marcus, Jun 24 2018

Equals M * V, where M = A127093 as an infinite lower triangular matrix and V = A000040, the sequence of primes as a vector. E.g., a(4) = 36 = 1*2 + 2*3 + 4*7, where (1, 2, 0, 4) = row 4 of A127093 and 2, 3 and 7 are p(1), p(2), p(4). - Gary W. Adamson, Jan 11 2007

L.g.f.: log(Product_{k>=1} 1/(1 - x^k)^prime(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 10 2017

Edited by N. J. A. Sloane, May 04 2007

A061151 1 + Sum_{n>=1} a_n x^n = 1/Product_{n>=1} (1+x^n)^prime(n).

Original entry on oeis.org

1, -2, 0, -3, 5, -3, 14, -4, 25, -32, 16, -88, 18, -155, 108, -153, 393, -88, 855, -160, 1255, -974, 1122, -3172, 370, -6794, 383, -10017, 5004, -9460, 19380, -2635, 45790, 5008, 76263, -7353, 87597, -77967, 48886, -244397, -45016, -500016, -115318, -734277, 56213, -710603, 810177, -161662, 2432173, 910752, 4767086

Offset: 0

Vladeta Jovovic, Apr 16 2001

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A007441, A007445, A030009, A061150, A061152.

a(n) = (1/n) * Sum_{k=1..n} a(n-k)*b(k), k>0, a(0)=1, b(k)=Sum_{d|k} (-1)^(k/d)*d*prime(d).

A291647 Expansion of Product_{k>=1} (1 + x^prime(k))^prime(k).

Original entry on oeis.org

1, 0, 2, 3, 1, 11, 3, 20, 21, 20, 64, 35, 112, 117, 160, 269, 284, 477, 598, 819, 1116, 1495, 1899, 2718, 3389, 4596, 6121, 7627, 10460, 13128, 17350, 22506, 28696, 37063, 47779, 60249, 78642, 98783, 126058, 160758, 200795, 257750, 321768, 407930, 511526, 640636, 802816, 1005618, 1252820, 1567454, 1946162

Offset: 0

Ilya Gutkovskiy, Aug 28 2017

nonn

Number of partitions of n into distinct prime parts, where prime(k) different parts of size prime(k) are available (2a, 2b, 3a, 3b, 3c, ...).

			a(6) = 3 because we have [3a, 3b], [3a, 3c] and [3b, 3c].

Index entries for related partition-counting sequences

Cf. A000040, A000586, A007441, A026007, A061151, A061152, A219224, A262736.

nmax = 50; CoefficientList[Series[Product[(1 + x^Prime[k])^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A000040(k))^A000040(k).

A304791 Expansion of Product_{k>=1} (1 - prime(k)*x^k).

Original entry on oeis.org

1, -2, -3, 1, 3, 18, 0, 35, -27, -85, -91, -109, -366, 118, 942, -957, 2791, 2091, 4855, -1157, -6903, 3341, 3162, -37034, -46480, -89890, 581, 131275, -296935, 167543, 108671, 801491, 616017, 2441581, -307733, -1864550, 4495872, 1158228, -2589768, -767646, -21062537

Offset: 0

Ilya Gutkovskiy, May 18 2018

sign

Convolution inverse of A145519.

Index entries for sequences related to partitions

Cf. A000040, A007441, A145519, A147655, A298159.

nmax = 40; CoefficientList[Series[Product[(1 - Prime[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[-Sum[d Prime[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 40}]

G.f.: Product_{k>=1} (1 - A000040(k)*x^k).

A353170 Expansion of Product_{k>=1} (1 - x^k)^prime(k+1).

Original entry on oeis.org

1, -3, -2, 7, 5, 9, -13, -27, -36, -36, 67, 117, 184, 171, -38, -356, -731, -883, -733, -90, 1194, 2828, 4202, 5008, 3993, 201, -6649, -15984, -26148, -32864, -30316, -13192, 22406, 75700, 139948, 196508, 222252, 184914, 53773, -192233, -547296, -968438, -1361207

Offset: 0

Ilya Gutkovskiy, Apr 28 2022

sign

Convolution inverse of A353065.

Cf. A007441, A065091, A353065, A353169.

nmax = 42; CoefficientList[Series[Product[(1 - x^k)^Prime[k + 1], {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Sum[d Prime[d + 1], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 42}]

A300521 Expansion of Product_{k>=1} (1 - x^prime(k))^prime(k).

Original entry on oeis.org

1, 0, -2, -3, 1, 1, 3, 0, 9, 8, 4, -31, -12, -13, 20, -13, 48, -17, 74, -87, 8, -143, 175, -174, 349, -164, 369, -651, 520, -1004, 1142, -1218, 1652, -1739, 3291, -3933, 3546, -5743, 6170, -8022, 11435, -13230, 17196, -18706, 22958, -31884, 38420, -49802, 58916

Offset: 0

Ilya Gutkovskiy, Mar 08 2018

sign

Cf. A000040, A000607, A005063, A007441, A030009, A046675, A073592, A219224, A291647, A291696.

nmax = 48; CoefficientList[Series[Product[(1 - x^Prime[k])^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[-Sum[DivisorSum[k, Boole[PrimeQ[#]] #^2 &] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 - x^A000040(k))^A000040(k).

G.f.: exp(-Sum_{k>=1} A005063(k)*x^k/k).

cp's OEIS Frontend

A030009 Euler transform of primes.

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A061152 Expansion of Product_{n>=1} (1+x^n)^prime(n).

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A061150 a(n) = Sum_{d|n} d*prime(d).

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A061151 1 + Sum_{n>=1} a_n x^n = 1/Product_{n>=1} (1+x^n)^prime(n).

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A291647 Expansion of Product_{k>=1} (1 + x^prime(k))^prime(k).

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A304791 Expansion of Product_{k>=1} (1 - prime(k)*x^k).

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A353170 Expansion of Product_{k>=1} (1 - x^k)^prime(k+1).

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A300521 Expansion of Product_{k>=1} (1 - x^prime(k))^prime(k).

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