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 1-8 of 8 results.

A341983 Number of ways to write n as an ordered sum of 4 primes (counting 1 as a prime).

Original entry on oeis.org

1, 4, 10, 16, 23, 28, 38, 44, 55, 52, 62, 60, 84, 80, 106, 88, 123, 108, 160, 128, 184, 136, 214, 168, 261, 172, 270, 168, 304, 204, 352, 200, 382, 232, 442, 264, 470, 232, 502, 268, 557, 300, 608, 292, 672, 340, 722, 372, 789, 356, 856, 396, 900, 432, 968, 380, 1024, 432
Offset: 4

Views

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

Links

Crossrefs

Cf. A008578, A280917, A283762, A340960, A341947, A341975, A341982, 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, 5)
    end:
a:= n-> coeff(b(n), x, 4):
seq(a(n), n=4..61);  # Alois P. Heinz, Feb 24 2021
  • Mathematica
    nmax = 61; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^4, {x, 0, nmax}], x] // Drop[#, 4] &

Formula

G.f.: ( x + Sum_{k>=1} x^prime(k) )^4.

A341984 Number of ways to write n as an ordered sum of 5 primes (counting 1 as a prime).

Original entry on oeis.org

1, 5, 15, 30, 50, 71, 100, 130, 170, 195, 231, 250, 310, 340, 420, 430, 525, 535, 685, 680, 851, 800, 1025, 970, 1280, 1145, 1470, 1250, 1685, 1440, 1991, 1600, 2230, 1790, 2615, 2070, 2985, 2190, 3250, 2410, 3700, 2665, 4125, 2840, 4560, 3200, 5135, 3470, 5670, 3705, 6226, 4120
Offset: 5

Views

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

Crossrefs

Cf. A008578, A280917, A283762, A340961, A341948, A341976, A341982, A341983.

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, 6)
    end:
a:= n-> coeff(b(n), x, 5):
seq(a(n), n=5..56);  # Alois P. Heinz, Feb 24 2021
  • Mathematica
    nmax = 56; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^5, {x, 0, nmax}], x] // Drop[#, 5] &

Formula

G.f.: ( x + Sum_{k>=1} x^prime(k) )^5.

A341985 Number of ways to write n as an ordered sum of 6 primes (counting 1 as a prime).

Original entry on oeis.org

1, 6, 21, 50, 96, 156, 237, 336, 465, 596, 747, 882, 1077, 1260, 1536, 1736, 2067, 2286, 2761, 3030, 3627, 3842, 4578, 4806, 5826, 6000, 7167, 7116, 8562, 8430, 10318, 9906, 12093, 11396, 14286, 13386, 16868, 15270, 19242, 17180, 22218, 19536, 25393, 21750, 28680, 24456
Offset: 6

Views

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

Crossrefs

Cf. A008578, A280917, A283762, A340962, A341949, A341977, A341982, A341983, A341984, A341986, A341987, A341988, A341989.

Programs

  • Mathematica
    nmax = 51; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^6, {x, 0, nmax}], x] // Drop[#, 6] &

Formula

G.f.: ( x + Sum_{k>=1} x^prime(k) )^6.

A341986 Number of ways to write n as an ordered sum of 7 primes (counting 1 as a prime).

Original entry on oeis.org

1, 7, 28, 77, 168, 308, 511, 785, 1155, 1603, 2142, 2723, 3430, 4207, 5202, 6216, 7497, 8729, 10451, 12061, 14350, 16205, 19033, 21182, 24934, 27482, 32109, 34587, 40139, 42714, 49791, 52290, 60718, 62699, 73297, 75278, 88571, 89488, 104993, 104482, 123760, 122066
Offset: 7

Views

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

Links

Crossrefs

Cf. A008578, A280917, A283762, A340963, A341950, A341978, A341982, A341983, A341984, A341985, A341987, A341988, A341989.

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, 8)
    end:
a:= n-> coeff(b(n), x, 7):
seq(a(n), n=7..48);  # Alois P. Heinz, Feb 25 2021
  • Mathematica
    nmax = 48; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^7, {x, 0, nmax}], x] // Drop[#, 7] &

Formula

G.f.: ( x + Sum_{k>=1} x^prime(k) )^7.

A347744 Number of compositions (ordered partitions) of n into at most 2 prime parts (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 3, 3, 3, 3, 3, 4, 2, 3, 1, 4, 3, 5, 2, 4, 1, 6, 3, 6, 2, 5, 1, 8, 2, 5, 0, 4, 1, 8, 3, 6, 2, 7, 0, 8, 1, 5, 2, 6, 1, 10, 3, 8, 2, 7, 1, 12, 2, 8, 0, 6, 1, 12, 2, 6, 0, 7, 1, 14, 3, 7, 2, 10, 0, 12, 1, 6, 2, 10, 1, 14, 3, 11, 2, 10, 0, 14, 1, 10
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 11 2021

Keywords

Crossrefs

Cf. A001031, A008578, A096139, A280917, A341982, A347739, A347745.

Programs

  • Mathematica
    Table[Length@Flatten[Permutations/@IntegerPartitions[n,2,Join[{1},Prime@Range@PrimePi@n]],1],{n,0,80}] (* Giorgos Kalogeropoulos, Sep 12 2021 *)
  • PARI
    { a(n) = local(s(n) = if(n<2,1,isprime(n))); if(n==0,1,sum(i=1,n,s(i)*s(n-i))); } \\ Christian Krause, Dec 06 2022

A341987 Number of ways to write n as an ordered sum of 8 primes (counting 1 as a prime).

Original entry on oeis.org

1, 8, 36, 112, 274, 560, 1016, 1688, 2647, 3928, 5580, 7568, 9990, 12832, 16332, 20336, 25167, 30472, 37004, 44136, 53054, 62272, 73788, 85240, 100276, 114752, 134072, 151144, 174834, 194616, 224304, 247240, 283467, 308448, 352668, 381032, 436368, 467272, 533520
Offset: 8

Views

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

Crossrefs

Cf. A008578, A280917, A283762, A340964, A341951, A341979, A341982, A341983, A341984, A341985, A341986, A341988, A341989.

Programs

  • Mathematica
    nmax = 46; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^8, {x, 0, nmax}], x] // Drop[#, 8] &

Formula

G.f.: ( x + Sum_{k>=1} x^prime(k) )^8.

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

Views

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

Crossrefs

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

Programs

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

Formula

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

A341989 Number of ways to write n as an ordered sum of 10 primes (counting 1 as a prime).

Original entry on oeis.org

1, 10, 55, 210, 625, 1542, 3310, 6390, 11400, 19090, 30353, 46060, 67210, 94780, 130230, 174862, 230650, 298800, 382115, 482090, 603373, 746860, 918770, 1118100, 1355110, 1626742, 1949190, 2312380, 2740220, 3212640, 3769784, 4375900, 5092485, 5854680, 6758935, 7703112
Offset: 10

Views

Author

Ilya Gutkovskiy, Feb 24 2021

Keywords

Crossrefs

Cf. A008578, A280917, A283762, A340966, A341972, A341981, A341982, A341983, A341984, A341985, A341986, A341987, A341988.

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, 11)
    end:
a:= n-> coeff(b(n), x, 10):
seq(a(n), n=10..45);  # Alois P. Heinz, Feb 25 2021
  • Mathematica
    nmax = 45; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^10, {x, 0, nmax}], x] // Drop[#, 10] &

Formula

G.f.: ( x + Sum_{k>=1} x^prime(k) )^10.
Showing 1-8 of 8 results.

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

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