cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A341977 Number of partitions of n into 6 distinct primes (counting 1 as a prime).

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 1, 5, 0, 4, 1, 5, 2, 8, 1, 7, 2, 8, 4, 12, 2, 12, 6, 14, 7, 17, 5, 18, 8, 20, 11, 26, 10, 27, 15, 30, 18, 36, 17, 36, 22, 41, 28, 48, 25, 49, 35, 56, 40, 61, 38, 64, 50, 73, 56, 77, 54, 82, 70, 93, 74, 96, 78, 106, 92, 114, 100
Offset: 29

Views

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

Links

Crossrefs

Cf. A008578, A036497, A219200, A341949, A341973, A341974, A341975, A341976.

Programs

  • Maple
    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, 7)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 6):
seq(a(n), n=29..100);  # Alois P. Heinz, Feb 24 2021
  • Mathematica
    m = 6;
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, m + 1}];
a[n_] :=  Coefficient[b[n, PrimePi[n]], x, m];
Table[a[n], {n, 29, 100}] (* Jean-François Alcover, Mar 06 2021, 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

Views

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

Links

Crossrefs

Cf. A008578, A036497, A125688, A341946, A341973, A341975, A341976, A341977.

Programs

  • Maple
    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
  • Mathematica
    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 *)

A341975 Number of partitions of n into 4 distinct primes (counting 1 as a prime).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 3, 2, 4, 2, 4, 3, 5, 4, 5, 3, 5, 6, 7, 6, 6, 7, 8, 9, 9, 10, 7, 10, 9, 12, 10, 12, 9, 15, 12, 16, 13, 18, 12, 20, 14, 22, 16, 23, 13, 27, 16, 29, 19, 30, 14, 33, 19, 36, 21, 35, 15, 43, 23, 43, 23, 43, 18, 52, 26, 51, 26, 52, 21, 64, 29, 58, 28, 64
Offset: 11

Views

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

Crossrefs

Cf. A008578, A036497, A219198, A341947, A341973, A341974, A341976, A341977.

Programs

  • Maple
    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, 5)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 4):
seq(a(n), n=11..88);  # Alois P. Heinz, Feb 24 2021
  • Mathematica
    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, 5}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 4];
Table[a[n], {n, 11, 100}] (* Jean-François Alcover, Jul 13 2021, after Alois P. Heinz *)

A341976 Number of partitions of n into 5 distinct primes (counting 1 as a prime).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 2, 1, 3, 1, 4, 0, 3, 2, 6, 2, 6, 2, 7, 5, 9, 4, 10, 5, 10, 8, 12, 7, 12, 8, 15, 12, 16, 12, 18, 14, 20, 17, 22, 18, 23, 20, 27, 26, 29, 27, 30, 30, 33, 36, 36, 36, 35, 41, 43, 48, 43, 49, 43, 56, 52, 61, 51, 64, 52, 73, 64, 77, 58, 82, 64, 93
Offset: 18

Views

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

Crossrefs

Cf. A008578, A036497, A219199, A341948, A341973, A341974, A341975, A341977.

Programs

  • Maple
    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, 6)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 5):
seq(a(n), n=18..91);  # Alois P. Heinz, Feb 24 2021
  • Mathematica
    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, 6}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 5];
Table[a[n], {n, 18, 100}] (* Jean-François Alcover, Jul 13 2021, after Alois P. Heinz *)

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

Views

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

Links

Crossrefs

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

Programs

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

Formula

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

A341978 Number of partitions of n into 7 distinct primes (counting 1 as a prime).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 2, 0, 4, 0, 2, 1, 6, 1, 7, 0, 5, 2, 8, 1, 11, 1, 10, 4, 15, 3, 18, 3, 17, 7, 22, 6, 28, 6, 25, 11, 35, 11, 40, 11, 38, 19, 50, 18, 56, 18, 54, 30, 70, 28, 74, 30, 78, 45, 92, 40, 100, 46, 104, 63, 123, 60, 133, 69, 140, 88, 157, 86, 173
Offset: 42

Views

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

Links

Crossrefs

Cf. A008578, A036497, A219201, A341950, A341973, A341974, A341975, A341976, A341977, A341979, A341980, A341981.

Programs

  • Maple
    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, 8)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 7):
seq(a(n), n=42..114);  # Alois P. Heinz, Feb 24 2021
  • Mathematica
    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, 8}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 7];
Table[a[n], {n, 42, 114}] (* Jean-François Alcover, Feb 24 2022, after Alois P. Heinz *)

A341979 Number of partitions of n into 8 distinct primes (counting 1 as a prime).

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 4, 0, 3, 0, 3, 1, 7, 0, 6, 1, 6, 1, 11, 0, 11, 2, 11, 3, 19, 1, 18, 3, 18, 5, 30, 4, 28, 6, 30, 10, 45, 6, 40, 11, 46, 16, 63, 11, 60, 19, 69, 25, 88, 18, 86, 32, 97, 36, 121, 32, 123, 47, 131, 55, 164, 49, 164, 69, 181, 80
Offset: 59

Views

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

Crossrefs

Cf. A008578, A036497, A219202, A341951, A341973, A341974, A341975, A341976, A341977, A341978, A341980, A341981.

Programs

  • Maple
    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, 9)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 8):
seq(a(n), n=59..130);  # Alois P. Heinz, Feb 24 2021
  • Mathematica
    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, 9}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 8];
Table[a[n], {n, 59, 130}] (* Jean-François Alcover, Feb 24 2022, after Alois P. Heinz *)

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

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)

A341981 Number of partitions of n into 10 distinct primes (counting 1 as a prime).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 5, 0, 4, 0, 2, 0, 9, 0, 7, 1, 7, 1, 14, 0, 10, 0, 12, 2, 22, 0, 19, 2, 22, 3, 34, 1, 31, 4, 32, 5, 54, 3, 48, 7, 50, 9, 78, 7, 70, 11, 76, 16, 113, 9, 100, 19, 114, 26, 155, 17, 147, 32, 164, 37, 212, 26
Offset: 101

Views

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

Crossrefs

Cf. A008578, A036497, A219204, A341972, A341973, A341974, A341975, A341976, A341977, A341978, A341979, A341980.

Programs

  • Maple
    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, 11)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 10):
seq(a(n), n=101..174);  # Alois P. Heinz, Feb 24 2021
  • Mathematica
    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, 11}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 10];
Table[a[n], {n, 101, 174}] (* Jean-François Alcover, Feb 24 2022, after Alois P. Heinz *)
Showing 1-9 of 9 results.

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