A341719 -id:A341719 - cp's OEIS frontend

A341980 Number of partitions of n into 9 distinct primes (counting 1 as a prime).

1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 3, 0, 1, 0, 3, 0, 5, 0, 4, 1, 6, 0, 10, 0, 6, 1, 11, 1, 16, 1, 11, 2, 19, 2, 25, 1, 18, 5, 32, 4, 36, 2, 32, 9, 47, 7, 55, 7, 49, 14, 69, 10, 80, 12, 74, 22, 98, 19, 117, 22, 106, 34, 140, 31, 158, 32, 149, 54, 194, 48, 215, 50

Offset: 78

Ilya Gutkovskiy, Feb 24 2021

nonn

Cf. A008578, A036497, A219203, A341719, A341973, A341974, A341975, A341976, A341977, A341978, A341979, A341981.

b:= proc(n, i) option remember; series(`if`(n=0, 1,
     `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i-1)))(
     `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 10)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 9):
seq(a(n), n=78..151);  # Alois P. Heinz, Feb 24 2021

b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
     If[i < 0, 0, Function[p, If[p > n, 0, x*b[n - p, i - 1]]][
     If[i == 0, 1, Prime[i]]] + b[n, i - 1]]], {x, 0, 10}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 9];
Table[a[n], {n, 78, 151}] (* Jean-François Alcover, Feb 24 2022, after Alois P. Heinz *)

A341988 Number of ways to write n as an ordered sum of 9 primes (counting 1 as a prime).

Original entry on oeis.org

1, 9, 45, 156, 423, 954, 1887, 3384, 5661, 8935, 13446, 19332, 26838, 36126, 47691, 61668, 78696, 98631, 122665, 150516, 184230, 222438, 268146, 318564, 379383, 445572, 525942, 610344, 712872, 817290, 947166, 1075680, 1238148, 1391475, 1591236, 1773684, 2022241

Offset: 9

Ilya Gutkovskiy, Feb 24 2021

nonn

Cf. A008578, A280917, A283762, A340965, A341719, A341980, A341982, A341983, A341984, A341985, A341986, A341987, A341989.

nmax = 45; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^9, {x, 0, nmax}], x] // Drop[#, 9] &

G.f.: ( x + Sum_{k>=1} x^prime(k) )^9.

A341972 Number of partitions of n into 10 primes (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 7, 11, 11, 17, 16, 24, 23, 32, 30, 43, 39, 56, 51, 71, 63, 89, 78, 111, 97, 134, 116, 164, 139, 197, 166, 232, 194, 275, 227, 324, 265, 374, 305, 438, 351, 505, 403, 578, 460, 665, 522, 760, 592, 859, 668, 978, 751, 1105, 844, 1239, 942, 1394

Offset: 10

Ilya Gutkovskiy, Feb 24 2021

nonn

Cf. A008578, A034891, A259201, A341719, A341945, A341946, A341947, A341948, A341949, A341950, A341951.

b:= proc(n, i) option remember; series(`if`(n=0, 1,
     `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i)))(
     `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 11)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 10):
seq(a(n), n=10..68);  # Alois P. Heinz, Feb 24 2021

b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
     If[i < 0, 0, Function[p, If[p > n, 0, x*b[n - p, i]]][
     If[i == 0, 1, Prime[i]]] + b[n, i - 1]]], {x, 0, 11}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 10];
Table[a[n], {n, 10, 68}] (* Jean-François Alcover, Feb 26 2022, after Alois P. Heinz *)

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A341980 Number of partitions of n into 9 distinct primes (counting 1 as a prime).

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A341988 Number of ways to write n as an ordered sum of 9 primes (counting 1 as a prime).

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A341972 Number of partitions of n into 10 primes (counting 1 as a prime).

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Author

Keywords

Crossrefs

Programs

Maple

Mathematica