A030010 -id:A030010 - 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

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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

A030011 Inverse Euler transform of {1, primes}.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, -1, -2, -3, -2, 4, -1, 5, 3, -4, -5, -9, 3, -3, 15, 19, 0, 6, -39, -27, -22, 5, 57, 50, 107, -49, -96, -142, -213, 83, 138, 468, 365, 0, -327, -1215, -618, -526, 957, 2572, 1831, 1673, -2820, -4516, -6155, -3880, 5998, 9282, 18213, 7414

Offset: 1

N. J. A. Sloane

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			(1-x)^(-1) * (1-x^2)^(-1) * (1-x^3)^(-1) * (1-x^4)^(-1) * (1-x^5)^(-1) * (1-x^6)^(-1) * (1-x^7) * (1-x^8)^2 * ... = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 13*x^7 + 17*x^8 + ... .

Seiichi Manyama, Table of n, a(n) for n = 1..5000
N. J. A. Sloane, Transforms

Cf. A000040, A030010.

pp = Prepend[Prime[Range[n = 100]], 1]; s = {};
For[i = 1, i <= n + 1, i++, AppendTo[s, i*pp[[i]] - Sum[s[[d]]*pp[[i - d]], {d, i - 1}]]];
Table[Sum[If[Divisible[i, d], MoebiusMu[i/d], 0]*s[[d]], {d, 1, i}]/i, {i,
n + 1}] (* Jean-François Alcover, May 10 2019 *)

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

A072508 Decimal expansion of Backhouse constant.

Original entry on oeis.org

1, 4, 5, 6, 0, 7, 4, 9, 4, 8, 5, 8, 2, 6, 8, 9, 6, 7, 1, 3, 9, 9, 5, 9, 5, 3, 5, 1, 1, 1, 6, 5, 4, 3, 5, 5, 7, 6, 5, 3, 1, 7, 8, 3, 7, 4, 8, 4, 7, 1, 3, 1, 5, 4, 0, 2, 7, 0, 7, 0, 2, 4, 3, 7, 4, 1, 4, 0, 0, 1, 5, 0, 6, 2, 6, 5, 3, 8, 9, 8, 9, 5, 5, 9, 9, 6, 4, 5, 3, 1, 9, 4, 0, 1, 8, 6, 0, 3, 0, 9, 1, 0, 9, 9, 2

Offset: 1

Robert G. Wilson v, Aug 03 2002

The reciprocal (A088751) of Backhouse's constant is the real zero of a certain power series. - T. D. Noe, Oct 14 2003

			1.4560749485826896713995953...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.5, p. 294.

S. R. Finch, Backhouse's constant, 1995. [Cached copy, with permission]
Philippe Flajolet, in response to the previous document from S. R. Finch, Backhouse's constant, 1995.
S. R. Finch, Kalmar's Composition Constant, Section 5.5 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 292-295, 2003. [Cached copy, with permission]
S. R. Finch, Kalmar's composition constant, June 5, 2003. [A different version. Cached copy, with permission of the author]
Simon Plouffe, The Backhouse constant.
Eric Weisstein's World of Mathematics, Backhouse's Constant.

Continued fraction is in A074269.

Cf. A030010, A088751, A247818.

RealDigits[-1/x /. FindRoot[0 == 1 + Sum[x^n Prime[n], {n, 1000}], {x, {0, 1}}, WorkingPrecision -> 100]][[1]] (* T. D. Noe, corrected Apr 26 2013 *)

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

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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.

A088751 Decimal expansion of -x, the real root of the equation 0 = 1 + Sum_{k>=1} prime(k) x^k. The inverse of Backhouse's constant (A072508).

Original entry on oeis.org

6, 8, 6, 7, 7, 7, 8, 3, 4, 4, 6, 0, 6, 3, 4, 9, 5, 4, 4, 2, 6, 5, 4, 0, 2, 2, 3, 7, 0, 6, 7, 6, 9, 2, 6, 9, 2, 2, 7, 0, 0, 2, 6, 3, 7, 6, 2, 2, 5, 0, 4, 2, 0, 7, 3, 9, 3, 4, 2, 5, 8, 2, 9, 4, 0, 1, 1, 5, 3, 1, 0, 0, 8, 7, 7, 0, 0, 4, 3, 7, 3, 6, 6, 9, 6, 9, 5, 3, 0, 1, 0, 6, 7, 6, 8, 2, 5, 9, 0, 1

Offset: 0

T. D. Noe, Oct 14 2003

This constant is computed in Finch's article. This number is easier to compute than Backhouse's constant. Except for an additional term of 0, the continued fraction expansion is the same as that of Backhouse's constant.

			0.68677783446063...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.5, p. 294.

S. R. Finch, Backhouse's constant, 1995. [Cached copy, with permission]
Philippe Flajolet, in response to the previous document from S. R. Finch, Backhouse's constant, 1995.
S. R. Finch, Kalmar's Composition Constant, Section 5.5 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 292-295, 2003. [Cached copy, with permission]
S. R. Finch, Kalmar's composition constant, June 5, 2003. [A different version. Cached copy, with permission of the author]
Eric Weisstein's World of Mathematics, Backhouse's Constant.

Cf. A030010, A072508.

RealDigits[ -x/.FindRoot[0==1+Sum[x^n Prime[n], {n, 1000}], {x, {0, 1}}, WorkingPrecision->100]][[1]]

A305882 -1 + Product_{n>=1} 1/(1 + a(n)*x^n) = g.f. of A000040 (prime numbers).

Original entry on oeis.org

-2, 1, 1, 4, 4, 13, 16, 44, 52, 112, 182, 411, 620, 1318, 2142, 5148, 7676, 15228, 27530, 58660, 98372, 207392, 364464, 763263, 1341508, 2773990, 4923220, 10470948, 18510902, 37546152, 69269976, 148419094, 258284232, 534761242, 981480012, 2004302204

Offset: 1

Ilya Gutkovskiy, Jun 13 2018

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

Cf. A000040, A030010, A030018, A145519, A147541, A147557, A305871, A305881.

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

Product_{n>=1} (1 + a(n)*x^n) = Sum_{k>=0} A030018(k)*x^k.

A308298 Expansion of Sum_{k>=1} mu(k)log(1 + Sum_{j>=1} x^(prime(j)k))/k.

Original entry on oeis.org

0, 1, 1, -1, 0, -1, 1, 0, -1, -1, 2, 1, 0, -3, 0, 1, 3, -2, -1, 0, 4, -3, -1, -5, 6, 2, 2, -11, 4, 4, 13, -16, -5, -8, 30, -8, -7, -33, 42, 8, 16, -82, 27, 19, 95, -116, -21, -45, 223, -82, -40, -264, 326, 46, 135, -629, 242, 99, 752, -942, -105, -421, 1826, -717, -240

Offset: 1

Ilya Gutkovskiy, May 19 2019

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Inverse Euler transform of A010051.

Cf. A000607, A008683, A010051, A030010.

nmax = 65; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + Sum[x^(Prime[j] k), {j, 1, nmax}]]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

-1 + Product_{n>=1} 1/(1 - x^n)^a(n) = g.f. of A010051.

A348128 Product_{n>=1} (1 + a(n)*x^n) = 1 + Sum_{n>=1} x^prime(n).

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 1, 0, -1, -1, 2, 1, 0, -2, 0, 1, 3, -2, -1, 0, 4, 0, -1, -4, 6, 2, 2, -10, 4, 4, 13, -15, -7, -2, 30, -7, -7, -33, 42, 8, 16, -70, 27, 22, 95, -116, -21, -39, 223, -61, -48, -261, 326, 51, 129, -581, 242, 109, 752, -932, -105, -330, 1806, -612, -240, -2140, 2750, 227, 1245, -4865

Offset: 1

Ilya Gutkovskiy, Oct 01 2021

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Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000040, A002121, A010051, A030010, A147557, A305871, A305882, A308298, A328777, A348127.

A348127 Product_{n>=1} 1 / (1 - a(n)*x^n) = 1 + Sum_{n>=1} x^prime(n).

Original entry on oeis.org

0, 1, 1, -1, 0, -1, 1, -1, -1, -1, 2, 0, 0, -3, 0, 0, 3, -3, -1, -1, 4, -4, -1, -5, 6, 2, 2, -17, 4, 4, 13, -16, -7, -11, 30, -14, -7, -34, 42, 7, 16, -80, 27, 6, 95, -117, -21, -60, 223, -97, -48, -265, 326, 53, 129, -800, 242, 93, 752, -948, -105, -499, 1806, -853, -240, -2189, 2750, 124

Offset: 1

Ilya Gutkovskiy, Oct 01 2021

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Robert Israel, Table of n, a(n) for n = 1..2500

Cf. A000040, A002121, A010051, A030010, A147557, A305871, A305882, A308298, A328777, A348128.

N:= 20: # for a(1)..a(N)
P:= 1: a:= Vector(N):
for n from 1 to N do
  c:= coeff(P,x,n);
  if isprime(n) then a[n]:= 1-c  else a[n]:= -c fi;
  P:= series(P/(1-a[n]*x^n),x,N+1);
od:
convert(a,list); # Robert Israel, Mar 01 2022

A380498 Inverse Euler transform of primorial numbers.

Original entry on oeis.org

2, 3, 20, 150, 1860, 24950, 444060, 8583780, 202071920, 5992771854, 186947632200, 7001535703840, 288868991951760, 12455290280427090, 587972068547997856, 31327583556941402160, 1856116108295418943020, 113366872636395265380920, 7619343577986975410930880, 541957669076266398658079700

Offset: 1

Ilya Gutkovskiy, Jan 25 2025

nonn

Eric Weisstein's World of Mathematics, Primorial.

Cf. A002110, A030010, A030011, A112354, A380497.

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

primorial[n_] := Product[Prime[j], {j, 1, n}]; b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i] + j - 1, j] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := primorial[n] - b[n, n - 1]; a /@ Range[20]

Product_{n>=1} 1 / (1 - x^n)^a(n) = Sum_{n>=0} prime(n)# * x^n.

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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A030011 Inverse Euler transform of {1, primes}.

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A072508 Decimal expansion of Backhouse constant.

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

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A088751 Decimal expansion of -x, the real root of the equation 0 = 1 + Sum_{k>=1} prime(k) x^k. The inverse of Backhouse's constant (A072508).

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A305882 -1 + Product_{n>=1} 1/(1 + a(n)*x^n) = g.f. of A000040 (prime numbers).

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A308298 Expansion of Sum_{k>=1} mu(k)*log(1 + Sum_{j>=1} x^(prime(j)*k))/k.

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A348128 Product_{n>=1} (1 + a(n)*x^n) = 1 + Sum_{n>=1} x^prime(n).

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A348127 Product_{n>=1} 1 / (1 - a(n)*x^n) = 1 + Sum_{n>=1} x^prime(n).

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A308298 Expansion of Sum_{k>=1} mu(k)log(1 + Sum_{j>=1} x^(prime(j)k))/k.