A341946 -id:A341946 - cp's OEIS frontend

A341945 Number of partitions of n into 2 primes (counting 1 as a prime).

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

Offset: 2

Ilya Gutkovskiy, Feb 24 2021

Number of partitions of n into 2 noncomposite numbers, A008578. - Antti Karttunen, Dec 13 2021

Antti Karttunen, Table of n, a(n) for n = 2..20000

Cf. A001031, A008578, A034891, A061358, 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, 3)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 2):
seq(a(n), n=2..90);  # Alois P. Heinz, Feb 24 2021

a[n_] := If[2 == n, 1, Module[{s = 0}, For[p = 2, True, p = NextPrime[p], If[p > n-p, Return[s + Boole[PrimeQ[n-1]]], s += Boole[PrimeQ[n-p]]]]]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Jan 03 2022, after Antti Karttunen *)

A341945(n) = if(2==n,1,my(s=0); forprime(p=2,,if(p>(n-p), return(s+isprime(n-1)), s += isprime(n-p)))); \\ Antti Karttunen, Dec 13 2021

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

Original entry on oeis.org

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 *)

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 *)

A341974 Number of partitions of n into 3 distinct primes (counting 1 as a prime).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 3, 4, 2, 5, 4, 5, 2, 5, 2, 7, 4, 6, 3, 8, 3, 9, 4, 7, 2, 9, 3, 10, 5, 9, 4, 12, 3, 13, 6, 12, 4, 14, 3, 16, 6, 13, 3, 16, 3, 19, 7, 14, 3, 19, 5, 21, 6, 15, 3, 23, 5, 23, 7, 18, 5, 26, 5, 26, 7, 21, 5, 29, 4, 28, 9, 25, 4, 30, 4

Offset: 6

Ilya Gutkovskiy, Feb 24 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..10000

Cf. A008578, A036497, A125688, A341946, A341973, 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, 4)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 3):
seq(a(n), n=6..90);  # 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, 4}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 3];
Table[a[n], {n, 6, 1000}] (* Jean-François Alcover, Jul 13 2021, after Alois P. Heinz *)

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

A341945 Number of partitions of n into 2 primes (counting 1 as a prime).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

A341974 Number of partitions of n into 3 distinct primes (counting 1 as a prime).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica