A030018 -id:A030018 - cp's OEIS frontend

A340991 Triangle T(n,k) whose k-th column is the k-fold self-convolution of the primes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 2, 0, 3, 4, 0, 5, 12, 8, 0, 7, 29, 36, 16, 0, 11, 58, 114, 96, 32, 0, 13, 111, 291, 376, 240, 64, 0, 17, 188, 669, 1160, 1120, 576, 128, 0, 19, 305, 1386, 3121, 4040, 3120, 1344, 256, 0, 23, 462, 2678, 7532, 12450, 12864, 8288, 3072, 512, 0, 29, 679, 4851, 16754, 34123, 44652, 38416, 21248, 6912, 1024

Offset: 0

Alois P. Heinz, Feb 01 2021

			Triangle T(n,k) begins:
  1;
  0,  2;
  0,  3,   4;
  0,  5,  12,    8;
  0,  7,  29,   36,   16;
  0, 11,  58,  114,   96,    32;
  0, 13, 111,  291,  376,   240,    64;
  0, 17, 188,  669, 1160,  1120,   576,  128;
  0, 19, 305, 1386, 3121,  4040,  3120, 1344,  256;
  0, 23, 462, 2678, 7532, 12450, 12864, 8288, 3072, 512;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-4 give (offsets may differ): A000007, A000040, A014342, A014343, A014344.
Main diagonal gives A000079.
Row sums give A030017(n+1).
T(2n,n) gives A340990.
Cf. A030018, A030281.

T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
      `if`(k=1, `if`(n=0, 0, ithprime(n)), (q->
       add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);
# Uses function PMatrix from A357368.
PMatrix(10, ithprime); # Peter Luschny, Oct 09 2022

T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
     If[k == 1, If[n == 0, 0, Prime[n]], With[{q = Quotient[k, 2]},
     Sum[T[j, q] T[n - j, k - q], {j, 0, n}]]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 10 2021, after Alois P. Heinz *)

T(n,k) = [x^n] (Sum_{j>=1} prime(j)*x^j)^k.
Sum_{k=0..n} k * T(n,k) = A030281(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A030018(n).
Conjecture: row polynomials are x*R(n,1) for n > 0 where R(n,k) = R(n-1,k+1) + x*R(n-1,1)*R(1,k) for n > 1, k > 0 with R(1,k) = prime(k) for k > 0. The same recursion seems to work for self-convolution of any other sequence. - Mikhail Kurkov, Apr 05 2025

A300662 Expansion of 1/(1 - x - Sum_{k>=2} prime(k-1)*x^k).

Original entry on oeis.org

1, 1, 3, 8, 22, 59, 160, 429, 1155, 3105, 8354, 22474, 60457, 162636, 437509, 1176941, 3166097, 8517138, 22912002, 61635707, 165806564, 446037175, 1199887133, 3227823181, 8683185454, 23358686444, 62837334885, 169039070970, 454732963567, 1223279724439, 3290751724917

Offset: 0

Ilya Gutkovskiy, Mar 10 2018

nonn

Invert transform of A008578.

Alois P. Heinz, Table of n, a(n) for n = 0..2327
N. J. A. Sloane, Transforms

Cf. A000040, A008578, A023626, A030011, A030012, A030013, A030014, A030015, A030016, A030017, A030018, A292744, A300632.

a:= proc(n) option remember; `if`(n=0, 1, add(
     `if`(j=1, 1, ithprime(j-1))*a(n-j), j=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Mar 10 2018

nmax = 30; CoefficientList[Series[1/(1 - x - Sum[Prime[k - 1] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]
p[1] = 1; p[n_] := p[n] = Prime[n - 1]; a[n_] := a[n] = Sum[p[k] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 30}]

G.f.: 1/(1 - Sum_{k>=1} A008578(k)*x^k).

A225127 Convolutory inverse of the nonprimes.

Original entry on oeis.org

1, -4, 10, -24, 59, -146, 360, -886, 2182, -5376, 13244, -32624, 80364, -197968, 487672, -1201319, 2959297, -7289859, 17957662, -44236464, 108971015, -268436517, 661259918, -1628931424, 4012669610, -9884711639, 24349755585, -59982589144, 147759635098

Offset: 1

Clark Kimberling, Apr 29 2013

Coefficients in 1/(1+g(x)), where g is the generating functions of the sequence of nonprimes: (1,4,6,8,9,...). For the convolutory inverse of the primes, see A030018. Conjecture: a(n+1)/a(n) has a limit, -2.4633754095588889..., analogous to the Backhouse constant.

The sequences with nonzero first term form a group under convolution. The identity is (1,0,0,0,...), and the inverse of a sequence r(1), r(2), r(3), ... is s(1), s(2), s(3),... given by s(1) = 1/r(1) and s(n) = -(r(2)*s(n-1) + ... + r(n)*s(1))/r(1). Thus, s(i) are the coefficients of the power series for 1/(r(1) + r(2)*x + r(3)*x^2 + ... ).

			(1,4,6,8,9,...)**(1,-4,10,-24,59,...) = (1,0,0,0,0,...), where ** denotes convolution.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A030018, A077607.

z = 1000; c = Complement[Range[z], Prime[Range[PrimePi[z]]]]; r[n_] := r[n] = c[[n]]; k[n_] := k[n] = 0; k[1] = 1; a[n_] := a[n] = (k[n] - Sum[r[i]*a[n - i + 1], {i, 2, n}])/r[1]; t = Table[a[n], {n, 1, 40}]   (* A225127 *)

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

sign

			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.

A225128 Numerators of the convolutory inverse of the primes of the form 4m+3.

Original entry on oeis.org

1, -7, 16, -52, 412, -2068, 6964, -19960, 81880, -396844, 1448908, -3853348, 9668860, -45544768, 238303744, -764868256, 1962327904, -9820441204, 62744531956, -306405293056, 1228176071080, -5276516025688, 26307346186816, -126143746044604, 534479888324932

Offset: 1

Clark Kimberling, Apr 29 2013

Coefficients in 1/(1+g(x)), where g is the generating functions of the sequence of primes (3,7,11,19,23,31,...) of primes congruent to 3 mod 4. For the convolutory inverse of the primes, see A030018. Conjecture: a(n+1)/a(n) -> -1.370819405....

			(3,7,11,19,23,...)**(1/3, -7/9, 16/27, -52/81, 412/243,...) = (1,0,0,0,0,...), where ** denotes convolution.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A030018, A225127, A225129.

q = {}; Do[If[PrimeQ[p = 4*n + 3], AppendTo[q, p]], {n, 0, 15000}]; r[n_] := q[[n]]; k[n_] := k[n] = 0; k[1] = 1; s[n_] := s[n] = (k[n] - Sum[r[k]*s[n - k + 1], {k, 2, n}])/r[1]; t = Table[s[n], {n, 1, 40}]; Numerator[t]

A225129 Numerators of the convolutory inverse of the primes of the form 4m+1.

Original entry on oeis.org

1, -13, 84, -712, 6916, -55788, 432584, -3555212, 28927916, -229458788, 1847086584, -14858027212, 118242773916, -945499611788, 7556178053084, -60048635124212, 477995366994916, -3810212526827288, 30296614848644584, -240796293647346212, 1916211884628153416

Offset: 1

Clark Kimberling, Apr 29 2013

Coefficients in 1/(1+g(x)), where g is the generating functions of the sequence of primes (5,13,17,29,37,...) of primes congruent to 1 mod 4. For the convolutory inverse of the primes, see A030018. Conjecture: a(n+1)/a(n) -> -1.59045463062282....

			(5,13,17,29,37,...)**(1/5, -13/25, 84/125, -712/625, 6916/3125,...) = (1,0,0,0,0,...), where ** denotes convolution.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A030018, A225127, A225128.

q = {}; Do[If[PrimeQ[p = 4*n + 1], AppendTo[q, p]], {n, 0, 15000}]; r[n_] := q[[n]]; k[n_] := k[n] = 0; k[1] = 1; s[n_] := s[n] = (k[n] - Sum[r[k]*s[n - k + 1], {k, 2, n}])/r[1]; t = Table[s[n], {n, 1, 40}]; Numerator[t]

A225130 Numerators of the convolutory inverse of the primes of the form 6m-1.

Original entry on oeis.org

1, -11, 36, -36, 36, -3786, 63786, -405036, 1215036, -4368786, 45022536, -380988786, 2242736286, -7681046286, 26949825036, -435049072536, 4543990507536, -25626723348786, 80068989783786, -100028016375036, 1579550678122536, -31186023693776286, 252408733196148786

Offset: 1

Clark Kimberling, Apr 29 2013

Coefficients in 1/(1+g(x)), where g is the generating functions of the sequence of primes (5,11,17,23,29,...) of primes congruent to -1 mod 6. For the convolutory inverse of the primes, see A030018. Conjecture: a(n+1)/a(n) -> -1.24066....

			(5,11,17,23,29,...)**(1/5, -11/25, 36/125, -36/625, 36/3125,...) = (1,0,0,0,0,...), where ** denotes convolution.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A030018, A225127, A225131.

q = {}; Do[If[PrimeQ[p = 6*n - 1], AppendTo[q, p]], {n, 0, 15000}]; r[n_] := q[[n]]; k[n_] := k[n] = 0; k[1] = 1; s[n_] := s[n] = (k[n] - Sum[r[k]*s[n - k + 1], {k, 2, n}])/r[1]; t = Table[s[n], {n, 1, 40}]; Numerator[t]

A225131 Numerators of the convolutory inverse of the primes of the form 6m+1.

Original entry on oeis.org

1, -13, 36, -258, 5622, -31716, -83460, 1766388, -2952900, 59171652, -2614259136, 25907667528, -87008484996, 410147565360, -10353918172170, 73320103253412, 409638469731702, -7210516315882284, 18236866211886120, -161388385633551558, 6594430509454957926

Offset: 1

Clark Kimberling, Apr 29 2013

Coefficients in 1/(1+g(x)), where g is the generating functions of the sequence of primes (7,13,19,31,37,...) of primes congruent to 1 mod 6. For the convolutory inverse of the primes, see A030018. Conjecture: a(n+1)/a(n) diverges.

			(7,13,19,31,37,...)**(1/7, -13/49, 36/343, -258/2401, 5622/16807,...) = (1,0,0,0,0,...), where ** denotes convolution.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A030018, A225127, A225130.

q = {}; Do[If[PrimeQ[p = 6*n - 1], AppendTo[q, p]], {n, 0, 15000}]; r[n_] := q[[n]]; k[n_] := k[n] = 0; k[1] = 1; s[n_] := s[n] = (k[n] - Sum[r[k]*s[n - k + 1], {k, 2, n}])/r[1]; t = Table[s[n], {n, 1, 40}]; Numerator[t]

A307898 Expansion of 1/(1 - x * Sum_{k>=1} prime(k)*x^k).

Original entry on oeis.org

1, 0, 2, 3, 9, 19, 48, 107, 258, 594, 1405, 3277, 7693, 18004, 42203, 98834, 231592, 542497, 1271003, 2977529, 6975674, 16342011, 38285178, 89691782, 210124363, 492265243, 1153247379, 2701752062, 6329489153, 14828313076, 34738805240, 81383803849, 190660665579, 446667359857, 1046423138962

Offset: 0

Ilya Gutkovskiy, May 04 2019

nonn

Antidiagonal sums of square array, in which row m equals the m-fold convolution of primes with themselves.

Cf. A000040, A030017, A030018, A300662, A307899.

nmax = 34; CoefficientList[Series[1/(1 - x Sum[Prime[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Prime[k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 34}]

Recurrence: a(n+1) = Sum_{k=1..n} prime(k)*a(n-k).

A307899 Expansion of 1/(1 + x * Sum_{k>=1} prime(k)*x^k).

Original entry on oeis.org

1, 0, -2, -3, -1, 5, 10, 9, -4, -26, -43, -33, 35, 148, 219, 98, -316, -857, -983, 23, 2296, 4501, 3712, -2906, -14257, -21771, -10811, 28282, 81209, 97292, 7960, -207185, -431595, -386033, 219344, 1322141, 2134126, 1226554, -2443765, -7684081, -9726127, -1791806, 18712361, 41428590, 39753658

Offset: 0

Ilya Gutkovskiy, May 04 2019

sign

Alternating antidiagonal sums of square array, in which row m equals the m-fold convolution of primes with themselves.

Cf. A000040, A030017, A030018, A307898.

nmax = 44; CoefficientList[Series[1/(1 + x Sum[Prime[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[Prime[k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 44}]

Recurrence: a(n+1) = -Sum_{k=1..n} prime(k)*a(n-k).

cp's OEIS Frontend

A340991 Triangle T(n,k) whose k-th column is the k-fold self-convolution of the primes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A300662 Expansion of 1/(1 - x - Sum_{k>=2} prime(k-1)*x^k).

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A225127 Convolutory inverse of the nonprimes.

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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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A225128 Numerators of the convolutory inverse of the primes of the form 4m+3.

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A225129 Numerators of the convolutory inverse of the primes of the form 4m+1.

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A225130 Numerators of the convolutory inverse of the primes of the form 6m-1.

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A225131 Numerators of the convolutory inverse of the primes of the form 6m+1.

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