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 1-6 of 6 results.

A223893 Number of partitions of n into at most three distinct primes.

Original entry on oeis.org

0, 1, 1, 0, 2, 0, 2, 1, 1, 2, 1, 2, 2, 2, 2, 3, 1, 4, 3, 4, 3, 4, 3, 5, 3, 5, 3, 4, 4, 5, 6, 5, 5, 5, 5, 7, 6, 5, 7, 4, 7, 7, 8, 7, 7, 6, 10, 8, 9, 9, 8, 7, 12, 8, 12, 8, 10, 6, 14, 9, 15, 8, 13, 7, 14, 11, 16, 8, 14, 7, 19, 11, 19, 10, 15, 9, 21, 12, 20, 11, 18
Offset: 1

Views

Author

Frank M Jackson, Mar 28 2013

Keywords

Comments

The sequence shows a stronger version of the Goldbach conjecture that for n > 6, n has partitions with at most three distinct primes.

Examples

			a(21)=3 as 21 = 2+19 = 3+5+13 = 3+7+11.

Links

Crossrefs

Cf. A000586, A000607, A071335, A185101, A218007, A218469.

Programs

  • Mathematica
    a[n_] := Length@Select[IntegerPartitions[n, 3, Prime@Range@PrimePi@n],
Sort@#==Union@# &]; Array[a, 100] (* Giovanni Resta, Mar 29 2013 *)

A347622 Number of partitions of n into at most 2 prime parts (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 3, 2, 3, 1, 3, 1, 4, 1, 3, 0, 2, 1, 4, 2, 3, 1, 4, 0, 4, 1, 3, 1, 3, 1, 5, 2, 4, 1, 4, 1, 6, 1, 4, 0, 3, 1, 6, 1, 3, 0, 4, 1, 7, 2, 4, 1, 5, 0, 6, 1, 3, 1, 5, 1, 7, 2, 6, 1, 5
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 09 2021

Keywords

Crossrefs

Cf. A008578, A034891, A218007, A341945, A347552, A347623, A347624, A347625.

A347623 Number of partitions of n into at most 4 prime parts (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 9, 10, 10, 12, 11, 14, 13, 15, 14, 18, 15, 20, 17, 22, 18, 25, 19, 26, 20, 26, 21, 30, 23, 33, 24, 35, 26, 38, 27, 40, 28, 41, 31, 48, 32, 50, 34, 53, 36, 58, 37, 60, 39, 61, 40, 67, 40, 68, 41, 72, 45, 79, 47, 82, 49
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 09 2021

Keywords

Crossrefs

Cf. A008578, A034891, A218007, A341947, A347578, A347622, A347624, A347625.

A347624 Number of partitions of n into at most 5 prime parts (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 9, 10, 12, 13, 15, 15, 18, 18, 21, 21, 25, 24, 29, 28, 33, 32, 38, 36, 43, 40, 48, 44, 53, 47, 58, 50, 64, 56, 70, 61, 77, 65, 85, 71, 91, 75, 99, 81, 109, 89, 117, 95, 126, 101, 138, 108, 146, 114, 156, 119, 168, 126, 175, 133, 187, 140, 204
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 09 2021

Keywords

Crossrefs

Cf. A008578, A034891, A218007, A341948, A347609, A347622, A347623, A347625.

A347625 Number of partitions of n into at most 6 prime parts (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 10, 12, 14, 17, 19, 22, 24, 27, 29, 33, 35, 40, 41, 47, 48, 56, 56, 65, 64, 75, 73, 85, 82, 96, 91, 106, 101, 119, 111, 133, 123, 147, 136, 162, 149, 180, 162, 196, 177, 217, 194, 238, 209, 259, 228, 282, 247, 307, 263, 330, 284, 357, 303, 383
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 09 2021

Keywords

Crossrefs

Cf. A008578, A034891, A218007, A341949, A347610, A347622, A347623, A347624.

A218469 Number of partitions of n into at most three distinct primes (including 1).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 3, 2, 3, 2, 3, 3, 4, 3, 4, 3, 5, 5, 6, 5, 5, 5, 6, 6, 6, 5, 4, 6, 6, 9, 7, 7, 6, 8, 7, 10, 6, 8, 5, 10, 8, 12, 9, 10, 7, 13, 9, 14, 10, 12, 7, 15, 9, 17, 9, 13, 6, 17, 10, 21, 10, 15, 8, 19, 11, 22, 9, 16, 8, 24, 12, 25, 12, 19, 10, 26, 12
Offset: 1

Views

Author

Frank M Jackson, Mar 26 2013

Keywords

Comments

Using {1 union primes} as the base, the above sequence relies on the strong Goldbach's conjecture that any positive integer is the sum of at most three distinct terms.

Examples

			a(21)=5 as 21 = 2+19 = 1+3+17 = 1+7+13 = 3+5+13 = 3+7+11.

Links

Crossrefs

Cf. A002375, A025583, A068307, A071335, A185101, A218007.

Programs

  • Mathematica
    primeQ[p0_] := If[p0==1, True, PrimeQ[p0]]; SetAttributes[primeQ, Listable]; goldbachcount[p1_] := (parts=IntegerPartitions[p1, 3]; count=0; n=1; While[n<=Length[parts], If[Intersection[Flatten[primeQ
  [parts[[n]]]]][[1]]&&Total[Intersection[parts[[n]]]]==Total[parts
  [[1]]], count++]; n++]; count); Table[goldbachcount[i], {i, 1, 100}]
Showing 1-6 of 6 results.

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