A341945 -id:A341945 - cp's OEIS frontend

A341947 Number of partitions of n into 4 primes (counting 1 as a prime).

1, 1, 2, 2, 4, 3, 5, 4, 6, 4, 7, 4, 9, 6, 10, 6, 12, 6, 14, 8, 15, 8, 18, 9, 21, 10, 20, 9, 23, 10, 26, 12, 27, 12, 31, 13, 34, 13, 33, 14, 39, 15, 42, 16, 43, 17, 48, 18, 53, 19, 52, 19, 58, 20, 61, 20, 61, 20, 68, 23, 73, 23, 73, 26, 82, 26, 84, 23, 84, 27, 92, 28, 98

Offset: 4

Ilya Gutkovskiy, Feb 24 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..10000

Cf. A008578, A034891, A259194, A341945, A341946, 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, 5)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 4):
seq(a(n), n=4..76);  # 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, 5}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 4];
Table[a[n], {n, 4, 76}] (* Jean-François Alcover, Feb 15 2022, after Alois P. Heinz *)

A341948 Number of partitions of n into 5 primes (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 5, 8, 6, 10, 7, 12, 9, 15, 10, 18, 12, 21, 14, 25, 15, 29, 18, 33, 21, 37, 20, 41, 23, 46, 26, 51, 27, 58, 31, 63, 34, 68, 33, 77, 39, 83, 42, 90, 43, 101, 48, 107, 53, 116, 52, 128, 58, 134, 61, 142, 61, 157, 68, 165, 73, 176, 73, 194, 82, 201, 84, 211

Offset: 5

Ilya Gutkovskiy, Feb 24 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..10000

Cf. A008578, A034891, A259195, A341945, A341946, A341947, 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, 6)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 5):
seq(a(n), n=5..73);  # 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, 6}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 5];
Table[a[n], {n, 5, 73}] (* Jean-François Alcover, Feb 15 2022, after Alois P. Heinz *)

A341949 Number of partitions of n into 6 primes (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 6, 9, 8, 12, 10, 16, 12, 19, 15, 24, 18, 29, 21, 35, 25, 41, 29, 49, 33, 56, 37, 63, 41, 72, 46, 82, 51, 91, 58, 105, 63, 115, 68, 128, 77, 143, 83, 158, 90, 174, 101, 193, 107, 211, 116, 231, 128, 250, 134, 273, 142, 294, 157, 321, 165, 347, 176, 374

Offset: 6

Ilya Gutkovskiy, Feb 24 2021

nonn

Cf. A008578, A034891, A259196, A341945, A341946, A341947, A341948, 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, 7)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 6):
seq(a(n), n=6..70);  # 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, 7}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 6];
Table[a[n], {n, 6, 70}] (* Jean-François Alcover, Feb 15 2022, after Alois P. Heinz *)

A341950 Number of partitions of n into 7 primes (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 7, 10, 9, 14, 12, 19, 16, 23, 19, 30, 24, 37, 29, 44, 35, 55, 41, 65, 49, 75, 56, 89, 63, 102, 72, 116, 82, 134, 91, 153, 105, 171, 115, 194, 128, 220, 143, 242, 158, 273, 174, 305, 193, 334, 211, 374, 231, 412, 250, 447, 273, 494, 294, 541, 321

Offset: 7

Ilya Gutkovskiy, Feb 24 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..10000

Cf. A008578, A034891, A259197, A341945, A341946, A341947, A341948, A341949, 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, 8)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 7):
seq(a(n), n=7..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, 8}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 7];
Table[a[n], {n, 7, 68}] (* Jean-François Alcover, Feb 15 2022, after Alois P. Heinz *)

A341946 Number of partitions of n into 3 primes (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 4, 2, 4, 2, 6, 3, 6, 2, 6, 3, 8, 3, 8, 3, 9, 4, 10, 3, 9, 2, 10, 4, 12, 3, 12, 4, 13, 4, 13, 3, 14, 3, 15, 5, 16, 4, 17, 4, 18, 6, 19, 4, 19, 3, 20, 6, 20, 3, 20, 4, 23, 7, 23, 4, 26, 5, 26, 6, 23, 3, 27, 5, 28, 7, 28, 6, 33, 5, 31, 7, 30, 5, 34

Offset: 3

Ilya Gutkovskiy, Feb 24 2021

Alois P. Heinz, Table of n, a(n) for n = 3..10000

Cf. A008578, A034891, A068307, A341945, 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, 4)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 3):
seq(a(n), n=3..83);  # 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, 4}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 3];
Table[a[n], {n, 3, 83}] (* Jean-François Alcover, Feb 14 2022, after Alois P. Heinz *)

A341951 Number of partitions of n into 8 primes (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 7, 11, 10, 15, 14, 21, 19, 27, 23, 35, 30, 44, 37, 54, 44, 67, 55, 81, 65, 96, 75, 115, 89, 133, 102, 155, 116, 180, 134, 206, 153, 236, 171, 271, 194, 305, 220, 346, 242, 391, 273, 438, 305, 489, 334, 551, 374, 608, 412, 674, 447, 750, 494, 823

Offset: 8

Ilya Gutkovskiy, Feb 24 2021

nonn

Cf. A008578, A034891, A259198, A341945, A341946, A341947, A341948, A341949, A341950.

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, 9)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 8):
seq(a(n), n=8..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, 9}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 8];
Table[a[n], {n, 8, 68}] (* Jean-François Alcover, Feb 15 2022, after Alois P. Heinz *)

A341973 Number of partitions of n into 2 distinct primes (counting 1 as a prime).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 0, 2, 1, 2, 1, 2, 0, 3, 1, 3, 1, 2, 0, 4, 1, 2, 0, 2, 0, 4, 1, 3, 1, 3, 0, 4, 0, 2, 1, 3, 0, 5, 1, 4, 1, 3, 0, 6, 1, 4, 0, 3, 0, 6, 1, 3, 0, 3, 0, 7, 1, 3, 1, 5, 0, 6, 0, 3, 1, 5, 0, 7, 1, 5, 1, 5, 0, 7, 0, 5, 1, 4, 0, 9, 1, 4, 0, 4, 0, 10, 1, 4, 0, 4, 0, 7

Offset: 3

Ilya Gutkovskiy, Feb 24 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..10000

Cf. A008578, A036497, A117929, A154804, A341945, A341974, A341975, A341976, A341977.

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, 3)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 2):
seq(a(n), n=3..96);  # Alois P. Heinz, Feb 24 2021

a[n_] := Select[IntegerPartitions[n, {2}, Join[{1},
    Prime[Range[PrimePi[n-1]]]]], #[[1]] != #[[2]]&] // Length;
a /@ Range[3, 100] (* Jean-François Alcover, Jul 13 2021 *)

a(n) = A117929(n) + A010051(n-1). - R. J. Mathar, Oct 01 2021

A341982 Number of ways to write n as an ordered sum of 2 primes (counting 1 as a prime).

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 4, 2, 3, 0, 4, 2, 5, 2, 4, 0, 6, 2, 6, 2, 5, 0, 8, 2, 5, 0, 4, 0, 8, 2, 6, 2, 7, 0, 8, 0, 5, 2, 6, 0, 10, 2, 8, 2, 7, 0, 12, 2, 8, 0, 6, 0, 12, 2, 6, 0, 7, 0, 14, 2, 7, 2, 10, 0, 12, 0, 6, 2, 10, 0, 14, 2, 11, 2, 10, 0, 14, 0, 10, 2, 9, 0, 18, 2, 9, 0, 8

Offset: 2

Ilya Gutkovskiy, Feb 24 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Cf. A008578, A073610, A096139, A280917, A283762, A341945, A341973, A341983, A341984.

b:= proc(n) option remember; series(`if`(n=0, 1, x*add(
      `if`(j=1 or isprime(j), b(n-j), 0), j=1..n)), x, 3)
    end:
a:= n-> coeff(b(n), x, 2):
seq(a(n), n=2..88);  # Alois P. Heinz, Feb 24 2021

nmax = 88; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^2, {x, 0, nmax}], x] // Drop[#, 2] &

G.f.: ( x + Sum_{k>=1} x^prime(k) )^2.

A341719 Number of partitions of n into 9 primes (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 7, 11, 11, 16, 15, 23, 21, 30, 27, 39, 35, 51, 44, 63, 54, 78, 67, 97, 81, 116, 96, 139, 115, 166, 133, 194, 155, 227, 180, 265, 206, 305, 236, 351, 271, 403, 305, 460, 346, 522, 391, 592, 438, 668, 489, 751, 551, 844, 608, 942, 674, 1050, 750

Offset: 9

Ilya Gutkovskiy, Feb 24 2021

nonn

Cf. A008578, A034891, A259200, A341945, A341946, A341947, A341948, A341949, A341950, A341951, A341972.

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, 10)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 9):
seq(a(n), n=9..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, 10}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 9];
Table[a[n], {n, 9, 68}] (* Jean-François Alcover, Feb 26 2022, after Alois P. Heinz *)

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 *)

cp's OEIS Frontend

A341947 Number of partitions of n into 4 primes (counting 1 as a prime).

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A341948 Number of partitions of n into 5 primes (counting 1 as a prime).

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A341949 Number of partitions of n into 6 primes (counting 1 as a prime).

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Maple

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A341950 Number of partitions of n into 7 primes (counting 1 as a prime).

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Maple

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A341946 Number of partitions of n into 3 primes (counting 1 as a prime).

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Maple

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A341951 Number of partitions of n into 8 primes (counting 1 as a prime).

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Maple

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A341973 Number of partitions of n into 2 distinct primes (counting 1 as a prime).

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Maple

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A341982 Number of ways to write n as an ordered sum of 2 primes (counting 1 as a prime).

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Maple

Mathematica

Formula

A341719 Number of partitions of n into 9 primes (counting 1 as a prime).

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Maple