A007446 -id:A007446 - cp's OEIS frontend

A303073 L.g.f.: log(1 + Sum_{k>=1} prime(k)x^k) = Sum_{n>=1} a(n)x^n/n.

2, 2, 5, 2, 12, -13, 16, -30, 41, -18, 46, -73, 132, -278, 315, -318, 580, -805, 1218, -1998, 2665, -3958, 5936, -7761, 11612, -17678, 25313, -38134, 54754, -76833, 114392, -166334, 240685, -356454, 515996, -748441, 1095572, -1581482, 2303163, -3375550, 4903684, -7149365, 10417010, -15111622

Offset: 1

Ilya Gutkovskiy, Apr 18 2018

sign

			L.g.f.: L(x) = 2*x + 2*x^2/2 + 5*x^3/3 + 2*x^4/4 + 12*x^5/5 - 13*x^6/6 + 16*x^7/7 - 30*x^8/8 + 41*x^9/9 - 18*x^10/10 + ...
exp(L(x)) = 1 + 2*x + 3*x^2 + 5*x^3 + 7*x^4 + 11*x^5 + 13*x^6 + 17*x^7 + 19*x^8 + 23*x^9 + 29*x^10 + ... + A000040(n)*x^n + ...

Cf. A000040, A007446, A007447, A008578, A030017, A030018, A302194.

nmax = 44; Rest[CoefficientList[Series[Log[1 + Sum[Prime[k] x^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]

A343623 E.g.f.: -log(1 - x - Sum_{k>=2} prime(k-1) * x^k / k!).

Original entry on oeis.org

1, 3, 11, 59, 416, 3683, 39093, 484220, 6854176, 109150227, 1931303809, 37589753206, 798135918850, 18358887315769, 454779141016707, 12070296596154136, 341715021307029876, 10278722402921420619, 327369178071821161755, 11005696560250745851048, 389469699942038630639524

Offset: 1

Ilya Gutkovskiy, Aug 04 2021

nonn

Cf. A000040, A007446, A007447, A008578, A300632, A336185, A343622.

nmax = 21; CoefficientList[Series[-Log[1 - x - Sum[Prime[k - 1] x^k/k!, {k, 2, nmax}]], {x, 0, nmax}], x] Range[0, nmax]! // Rest

a(n) = A008578(n) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * A008578(n-k) * k * a(k).

A352914 Expansion of e.g.f. exp(Sum_{k>=1} prime(k)*x^k).

Original entry on oeis.org

1, 2, 10, 74, 676, 7592, 97024, 1416200, 23015248, 412777952, 8090869984, 171435904928, 3908548404160, 95264270043776, 2470715015425024, 67913132377486208, 1971038886452490496, 60212661838223997440, 1930529043247940342272, 64801071784954698480128

Offset: 0

Seiichi Manyama, Apr 28 2022

nonn

Cf. A000040, A007446, A033286, A353162, A353166.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      ithprime(j)*j*binomial(n, j)*j!, j=1..n)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 28 2022

a[0] = 1; a[n_] := a[n] = (n-1)! Sum[k Prime[k] a[n-k]/(n-k)!, {k, 1, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 28 2022 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, prime(k)*x^k))))

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*prime(k)*a(n-k)/(n-k)!));

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A033286(k) * a(n-k)/(n-k)!.

A353079 Exponential transform of odd primes.

Original entry on oeis.org

1, 3, 14, 79, 521, 3876, 31935, 287225, 2791122, 29066589, 322292257, 3784650052, 46857941291, 609360372095, 8296220760974, 117914344818807, 1745211622467633, 26838798853062516, 428009369349905497, 7065576909286562195, 120545067517808693300, 2122393931891338237325, 38512344746420591905771

Offset: 0

Ilya Gutkovskiy, Apr 22 2022

nonn

Cf. A007446, A065091, A300632.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      ithprime(j+1)*binomial(n-1, j-1), j=1..n))
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 27 2022

nmax = 22; CoefficientList[Series[Exp[Sum[Prime[k + 1] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Prime[k + 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

E.g.f.: exp( Sum_{k>=1} prime(k+1) * x^k / k! ).

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * prime(k+1) * a(n-k).

A365106 Sum_{n>=0} a(n) * x^n / n!^2 = exp( Sum_{n>=1} prime(n) * x^n / n!^2 ).

Original entry on oeis.org

1, 2, 11, 107, 1577, 32201, 860460, 28921567, 1187475909, 58232016701, 3350187053856, 222857979706305, 16935374386652282, 1455271176236200143, 140181486948923188907, 15023106134895469195114, 1779460642743292348315607, 231607462899834684300774917, 32954119475274480307491604062, 5102159139278049158548905019487

Offset: 0

Ilya Gutkovskiy, Aug 21 2023

nonn

Cf. A007446, A023998.

nmax = 19; CoefficientList[Series[Exp[Sum[Prime[k] x^k/k!^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k Prime[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * prime(k) * a(n-k).

cp's OEIS Frontend

A303073 L.g.f.: log(1 + Sum_{k>=1} prime(k)*x^k) = Sum_{n>=1} a(n)*x^n/n.

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A343623 E.g.f.: -log(1 - x - Sum_{k>=2} prime(k-1) * x^k / k!).

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A352914 Expansion of e.g.f. exp(Sum_{k>=1} prime(k)*x^k).

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A353079 Exponential transform of odd primes.

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Maple

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A365106 Sum_{n>=0} a(n) * x^n / n!^2 = exp( Sum_{n>=1} prime(n) * x^n / n!^2 ).

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A303073 L.g.f.: log(1 + Sum_{k>=1} prime(k)x^k) = Sum_{n>=1} a(n)x^n/n.