A061152 -id:A061152 - cp's OEIS frontend

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

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

A305871 -1 + Product_{n>=1} (1 + x^n)^a(n) = g.f. of A000040 (prime numbers).

Original entry on oeis.org

2, 2, 1, 2, 2, -2, 2, -2, 4, -1, 4, -7, 10, -19, 20, -20, 34, -42, 64, -100, 126, -178, 258, -326, 464, -675, 936, -1371, 1888, -2550, 3690, -5208, 7292, -10467, 14742, -20808, 29610, -41586, 59052, -84438, 119602, -170153, 242256, -343534, 489550, -697815

Offset: 1

Ilya Gutkovskiy, Jun 12 2018

sign

Inverse weigh transform of A000040.

			(1 + x)^2 * (1 + x^2)^2 * (1 + x^3) * (1 + x^4)^2 * (1 + x^5)^2 * (1 + x^6)^(-2) * ... * (1 + x^n)^a(n) * ... = 1 + 2*x + 3*x^2 + 5*x^3 + 7*x^4 + 11*x^5 + 13*x^6 + ... + A000040(k)*x^k + ...

Alois P. Heinz, Table of n, a(n) for n = 1..3000
N. J. A. Sloane, Transforms

Cf. A000040, A030010, A030011, A061152.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; ithprime(n)-b(n, n-1) end:
seq(a(n), n=1..50);  # Alois P. Heinz, Jun 13 2018

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
     Sum[Binomial[a[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := a[n] = Prime[n] - b[n, n - 1];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 18 2022, after Alois P. Heinz *)

Product_{n>=1} (1 + x^n)^a(n) = 1 + Sum_{k>=1} prime(k)*x^k.

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).

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).

A318367 a(n) = Sum_{d|n} (-1)^(n/d+1)dprime(d).

Original entry on oeis.org

2, 4, 17, 20, 57, 67, 121, 116, 224, 239, 343, 371, 535, 487, 777, 660, 1005, 958, 1275, 1095, 1669, 1401, 1911, 1715, 2482, 2097, 3005, 2295, 3163, 2987, 3939, 3156, 4879, 3727, 5391, 4502, 5811, 4925, 7063, 5271, 7341, 6619, 8215, 6433, 9849, 7249, 9919, 8691, 11244, 9264

Offset: 1

Ilya Gutkovskiy, Aug 24 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A033286, A061150, A061151, A061152.

f:= proc(n) local d; add((-1)^(n/d+1)*d*ithprime(d), d = numtheory:-divisors(n)); end proc:map(f, [$1..100]); # Robert Israel, Aug 01 2023

Table[Sum[(-1)^(n/d + 1) d Prime[d], {d, Divisors[n]}], {n, 50}]
nmax = 50; Rest[CoefficientList[Series[Sum[k Prime[k] x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 50; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^Prime[k], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]

a(n) = sumdiv(n, d, (-1)^(n/d+1)*d*prime(d)); \\ Michel Marcus, Aug 25 2018

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

L.g.f.: log(Product_{k>=1} (1 + x^k)^prime(k)) = Sum_{n>=1} a(n)*x^n/n.

A353169 Expansion of Product_{k>=1} (1 + x^k)^prime(k+1).

Original entry on oeis.org

1, 3, 8, 23, 57, 137, 317, 705, 1524, 3224, 6667, 13521, 26980, 52985, 102624, 196248, 370849, 693159, 1282537, 2350584, 4269912, 7692044, 13748080, 24390170, 42966637, 75187515, 130737631, 225957706, 388279308, 663533206, 1127936772, 1907676978, 3210783522, 5378798428

Offset: 0

Ilya Gutkovskiy, Apr 28 2022

nonn

Weigh transform of odd primes.

Cf. A061152, A065091, A353065, A353160, A353170.

nmax = 33; 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[(-1)^(k/d + 1) d Prime[d + 1], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]

A380613 Expansion of Product_{k>=1} (1 + x^k)^prime(k)#.

Original entry on oeis.org

1, 2, 7, 42, 291, 2970, 36950, 597100, 11070875, 248103940, 7018494836, 215718595582, 7881561212732, 320881902092122, 13754717161317416, 643588827524430916, 33926485821837232397, 1992916854095359256932, 121393059052727838936847, 8107963745977267426512386, 574571379331620422000295082

Offset: 0

Ilya Gutkovskiy, Jan 28 2025

nonn

Weigh transform of primorial numbers.

Eric Weisstein's World of Mathematics, Primorial.

Cf. A002110, A061152, A261052, A380497, A380614.

p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(p(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 28 2025

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

A371308 Expansion of e.g.f. Product_{k>=1} (1 + x^k/k!)^prime(k).

Original entry on oeis.org

1, 2, 5, 23, 101, 511, 3300, 20499, 147249, 1158047, 9284124, 82250155, 762408746, 7406758725, 75928931645, 815389826454, 9127145085135, 106002459387831, 1287304713397098, 16132127163478581, 209381715443456410, 2814011969429674997, 38957100435462040565

Offset: 0

Ilya Gutkovskiy, Mar 24 2024

nonn

"EGJ" (unordered, element, labeled) transform of primes.

C. G. Bower, Transforms (2)

Cf. A000040, A007837, A032315, A032316, A061152.

nmax = 22; CoefficientList[Series[Product[(1 + x^k/k!)^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

cp's OEIS Frontend

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

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A305871 -1 + Product_{n>=1} (1 + x^n)^a(n) = g.f. of A000040 (prime numbers).

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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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Mathematica

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

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A318367 a(n) = Sum_{d|n} (-1)^(n/d+1)*d*prime(d).

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Maple

Mathematica

PARI

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A353169 Expansion of Product_{k>=1} (1 + x^k)^prime(k+1).

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A318367 a(n) = Sum_{d|n} (-1)^(n/d+1)dprime(d).