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 1-3 of 3 results.

A140952 Expansion of 1/((1-x^2)*(1-x^3)*(1-x^5)*(1-x^7)*(1-x^11)).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 22, 25, 28, 31, 35, 39, 43, 48, 53, 58, 64, 70, 77, 84, 91, 100, 108, 117, 127, 137, 148, 159, 172, 184, 198, 212, 227, 243, 259, 277, 295, 314, 334, 355, 377, 400, 424, 449, 475, 502, 531, 560
Offset: 0

Views

Author

Alois P. Heinz, Jul 25 2008

Keywords

Comments

Number of partitions of n into the first 5 primes.

Links

Crossrefs

Cf. A000040, A025795, A029144, A103221, A140953, A335106.

Programs

  • Maple
    M := Matrix(28, (i,j)-> if (i=j-1) or (j=1 and member(i, [2, 3, 11, 15, 19, 20, 28])) then 1 elif j=1 and member(i, [8, 9, 13, 17, 25, 26]) then -1 else 0 fi):
a:= n-> (M^(n))[1,1]:
seq(a(n), n=0..50);
  • Mathematica
    CoefficientList[Series[1/Times@@(1-x^Prime[Range[5]]),{x,0,70}],x] (* or *) LinearRecurrence[{0,1,1,0,0,0,0,-1,-1,0,1,0,-1,0,1,0,-1,0,1,1,0,0,0,0,-1,-1,0,1},{1,0,1,1,1,2,2,3,3,4,5,6,7,8,10,11,13,15,17,19,22,25,28,31,35,39,43,48},70] (* Harvey P. Dale, Jun 18 2021 *)

A140953 Expansion of 1/((1-x^2)*(1-x^3)*(1-x^5)*(1-x^7)*(1-x^11)*(1-x^13)).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 16, 19, 21, 25, 28, 32, 36, 41, 46, 52, 58, 65, 72, 80, 89, 98, 109, 119, 132, 144, 158, 173, 189, 206, 224, 244, 264, 287, 310, 336, 362, 391, 421, 453, 487, 523, 561, 601, 644, 688, 736, 785, 838, 893
Offset: 0

Views

Author

Alois P. Heinz, Jul 25 2008

Keywords

Comments

Number of partitions of n into the first 6 primes. [Corrected by Harvey P. Dale, Dec 05 2022]

Links

Crossrefs

Cf. A000040, A025795, A029144, A103221, A140952, A335106.

Programs

  • Maple
    M := Matrix(41, (i,j)-> if (i=j-1) or (j=1 and member(i, [2, 3, 11, 19, 20, 21, 22, 30, 38, 39])) then 1 elif j=1 and member(i, [8, 9, 16, 17, 24, 25, 32, 33, 41]) then -1 else 0 fi):
a:= n -> (M^(n))[1,1]:
seq(a(n), n=0..50);
  • Mathematica
    CoefficientList[Series[1/Times@@Table[1-x^p,{p,Prime[Range[6]]}],{x,0,60}],x] (* or *) LinearRecurrence[{0,1,1,0,0,0,0,-1,-1,0,1,0,0,0,0,-1,-1,0,1,1,1,1,0,-1,-1,0,0,0,0,1,0,-1,-1,0,0,0,0,1,1,0,-1},{1,0,1,1,1,2,2,3,3,4,5,6,7,9,10,12,14,16,19,21,25,28,32,36,41,46,52,58,65,72,80,89,98,109,119,132,144,158,173,189,206},70] (* Harvey P. Dale, Dec 05 2022 *)

A335285 a(n) is the greatest possible greatest part of any partition of n into prime parts.

Original entry on oeis.org

2, 3, 2, 5, 3, 7, 5, 7, 7, 11, 7, 13, 11, 13, 13, 17, 13, 19, 17, 19, 19, 23, 19, 23, 23, 23, 23, 29, 23, 31, 29, 31, 31, 31, 31, 37, 31, 37, 37, 41, 37, 43, 41, 43, 43, 47, 43, 47, 47, 47, 47, 53, 47, 53, 53, 53, 53, 59, 53, 61, 59, 61, 61, 61, 61, 67, 61, 67
Offset: 2

Views

Author

David James Sycamore, Michael De Vlieger May 30 2020

Keywords

Comments

For n odd, a(n) is the greatest prime <= n; for n even > 2, a(n) is the greatest prime < n-1.
For composite n = p + 1, p does not appear since 2 is the smallest prime. - Michael De Vlieger, May 31 2020

Examples

			a(9) = 7, the greatest prime < 9 (prime partitions of 9 are: [2,7], [2,2,5], [2,2,2,3] and [3,3,3], in which 7 is greatest of greatest parts).
a(12) = 7 (greatest prime < 11).

Links

Crossrefs

Cf. A331634, A335106.

Programs

  • Mathematica
    Array[Max@ Select[IntegerPartitions[#], AllTrue[#, PrimeQ] &][[All, 1]] &, 68, 2] (* Michael De Vlieger, May 30 2020 *)
Array[Prime[PrimePi@ # - Boole[And[PrimeQ[# - 1], # != 3]]] &, 68, 2] (* Michael De Vlieger, May 31 2020 *)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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