A030009 -id:A030009 - cp's OEIS frontend

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

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).

A007441 1 + Sum_{n>=1} a_n x^n = Product_{n>=1} (1-x^n)^prime(n).

Original entry on oeis.org

1, -2, -2, 1, 3, 7, 5, 6, -10, -27, -50, -42, -30, 41, 148, 241, 345, 303, 167, -275, -858, -1685, -2342, -2813, -2316, -536, 2914, 8228, 14531, 20955, 24370, 22393, 10265, -13839, -53386, -104364, -161593, -209463, -228141, -188750, -62023, 177547, 541310, 1009998, 1527972, 1976120, 2189974

Offset: 0

N. J. A. Sloane

sign

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

Cf. A030009, A030010, A061150, A061151, A061152.

a(n) = (1/n)*Sum_{k=1..n} a(n-k)*b(k), k>0, a(0)=1, b(k)=-Sum_{d|k} d*prime(d), cf. A061150.

Better description from Vladeta Jovovic, Apr 16 2001

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).

A353065 Euler transform of odd primes.

Original entry on oeis.org

1, 3, 11, 32, 92, 239, 608, 1465, 3450, 7858, 17525, 38165, 81653, 171497, 354785, 723084, 1454642, 2889854, 5676607, 11031046, 21224439, 40453596, 76428636, 143192339, 266172016, 491072611, 899583306, 1636775949, 2958900040, 5316004485, 9494514599

Offset: 0

Ilya Gutkovskiy, Apr 21 2022

nonn

Cf. A005380, A030009, A030012, A065091.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*ithprime(d+1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 21 2022

nmax = 30; CoefficientList[Series[Product[1/(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, 30}]

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

A380497 Euler transform of primorial numbers.

Original entry on oeis.org

1, 2, 9, 46, 314, 3072, 37641, 603510, 11148030, 249327430, 7040987792, 216220333314, 7895699690498, 321315600822232, 13770543972819903, 644232544408157820, 33954066516677635554, 1994206929690480710244, 121461036181617491970561, 8111955386813996410196454, 574814471423312085719652432

Offset: 0

Ilya Gutkovskiy, Jan 25 2025

nonn

Eric Weisstein's World of Mathematics, Primorial.

Cf. A002110, A030009, A030012, A107895, A380498.

p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      add(d*p(d), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 25 2025

nmax = 20; CoefficientList[Series[Product[1/(1 - x^k)^Product[Prime[j], {j, k}], {k, 1, nmax}], {x, 0, nmax}], x]
primorial[n_] := Product[Prime[j], {j, 1, n}]; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d primorial[d], {d, Divisors[j]}] a[n - j], {j, 1, n}]/n]; Table[a[n], {n, 0, 20}]

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

A197132 Euler transform of composite numbers.

Original entry on oeis.org

1, 4, 16, 52, 157, 434, 1144, 2862, 6906, 16090, 36449, 80430, 173555, 366802, 761102, 1552569, 3118508, 6174461, 12064383, 23283027, 44419855, 83834278, 156626605, 289839251, 531534746, 966483534, 1743164649, 3119864511, 5543030861, 9779552117, 17139055493

Offset: 0

Franklin T. Adams-Watters, Oct 10 2011

nonn

Robert Israel, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Cf. A002808, A030009.

N:= 100: # to use composites <= N
comps:= remove(isprime,[$4..N]):
M:= nops(comps):
G:= mul((1-x^k)^(-comps[k]),k=1..M):
S:= series(G, x, M+1):
seq(coeff(S,x,j),j=0..M); # Robert Israel, Jan 30 2018

a[ns_Integer?NonNegative, nf_Integer?NonNegative] := CoefficientList[Series[Product[(1 - x^k)^-FixedPoint[k + PrimePi[#] + 1 &, k], {k, 1, nf}], {x, 0, nf}], x][[ns + 1 ;; nf + 1]]; a[0, 30] (* Robert P. P. McKone, Nov 08 2023 *)

G.f.: Product_{k>=1} (1-x^k)^-composite(k), where composite(k) = A002808(k) is the k-th composite number.

A301971 a(n) = [x^n] Product_{k>=1} 1/(1 - x^prime(k))^n.

Original entry on oeis.org

1, 0, 2, 3, 10, 30, 77, 252, 682, 2136, 6182, 18766, 56173, 169351, 512990, 1551828, 4720170, 14348289, 43751984, 133502873, 408029510, 1248460587, 3823949824, 11724787763, 35980251181, 110510334780, 339674840715, 1044812449722, 3215861978150, 9904301974294, 30521063942312, 94103983534015

Offset: 0

Ilya Gutkovskiy, Mar 29 2018

nonn

Number of partitions of n into prime parts of n kinds.

Index entries for sequences related to partitions

Cf. A000607, A008485, A030009, A117278, A219224, A259254, A291647, A298436, A299168.

Table[SeriesCoefficient[Product[1/(1 - x^Prime[k])^n, {k, 1, n}], {x, 0, n}], {n, 0, 31}]

A302238 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^prime(k).

Original entry on oeis.org

1, 4, 14, 46, 136, 382, 1022, 2626, 6530, 15784, 37218, 85842, 194146, 431358, 943038, 2031454, 4316884, 9058662, 18787730, 38542526, 78264298, 157403290, 313712482, 619919350, 1215125262, 2363570168, 4563951858, 8751621598, 16670498062, 31553539214, 59361428202

Offset: 0

Ilya Gutkovskiy, Apr 03 2018

nonn

Convolution of the sequences A030009 and A061152.

N. J. A. Sloane, Transforms

Cf. A000040, A015128, A030009, A061152, A156616, A206622, A206623, A206624, A260916, A261386, A261452, A261519, A261520, A301554, A301555.

nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x]

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

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

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

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A007441 1 + Sum_{n>=1} a_n x^n = 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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A353065 Euler transform of odd primes.

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Maple

Mathematica

Formula

A380497 Euler transform of primorial numbers.

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Maple

Mathematica

Formula

A197132 Euler transform of composite numbers.

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Maple

Mathematica

Formula

A301971 a(n) = [x^n] Product_{k>=1} 1/(1 - x^prime(k))^n.

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Mathematica

A302238 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^prime(k).

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