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.

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A331382 Numbers whose sum of prime factors is divisible by their product of prime indices.

Original entry on oeis.org

1, 2, 4, 8, 16, 18, 20, 32, 35, 44, 60, 62, 64, 65, 68, 72, 92, 95, 98, 128, 154, 160, 168, 256, 264, 288, 291, 303, 324, 364, 400, 476, 480, 512, 618, 623, 624, 642, 706, 763, 791, 812, 816, 826, 938, 994, 1024, 1036, 1064, 1068, 1106, 1144, 1148, 1152, 1162
Offset: 1

Views

Author

Gus Wiseman, Jan 16 2020

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     8: {1,1,1}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    32: {1,1,1,1,1}
    35: {3,4}
    44: {1,1,5}
    60: {1,1,2,3}
    62: {1,11}
    64: {1,1,1,1,1,1}
    65: {3,6}
    68: {1,1,7}
    72: {1,1,1,2,2}
    92: {1,1,9}
    95: {3,8}
    98: {1,4,4}
   128: {1,1,1,1,1,1,1}
For example, 60 has prime factors {2,2,3,5} and prime indices {1,1,2,3}, and 12 is divisible by 6, so 60 is in the sequence.

Crossrefs

These are the Heinz numbers of the partitions counted by A331381.
Numbers divisible by the sum of their prime factors are A036844.
Partitions whose product is divisible by their sum are A057568.
Numbers divisible by the sum of their prime indices are A324851.
Product of prime indices is divisible by sum of prime indices: A326149.
Partitions whose Heinz number is divisible by their sum are A330950.
Sum of prime factors is divisible by sum of prime indices: A331380
Partitions whose product is equal to the sum of primes are A331383.
Product of prime indices equals sum of prime factors: A331384.
Cf. A000040, A001414, A324850, A330953, A330954, A331378, A331379.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Divisible[Plus@@Prime/@primeMS[#],Times@@primeMS[#]]&]

A331385 Irregular triangle read by rows where T(n,k) is the number of integer partitions y of n such that Sum_i prime(y_i) = n + k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 3, 1, 0, 0, 0, 2, 3, 1, 1, 0, 0, 0, 1, 4, 3, 1, 2, 0, 0, 0, 0, 2, 5, 3, 2, 2, 0, 1, 0, 0, 0, 0, 1, 4, 6, 3, 4, 2, 0, 2, 0, 0, 0, 0, 0, 2, 6, 6, 4, 6, 2, 1, 2, 0, 1, 0, 0, 0, 0, 0, 1, 4, 8, 6, 6, 7, 2, 4, 2, 0, 1, 0, 0, 0, 1
Offset: 0

Views

Author

Gus Wiseman, Jan 17 2020

Keywords

Examples

			Triangle begins:
  1
  0 1
  0 1 1
  0 0 2 1
  0 0 1 3 1
  0 0 0 2 3 1 1
  0 0 0 1 4 3 1 2
  0 0 0 0 2 5 3 2 2 0 1
  0 0 0 0 1 4 6 3 4 2 0 2
  0 0 0 0 0 2 6 6 4 6 2 1 2 0 1
  0 0 0 0 0 1 4 8 6 6 7 2 4 2 0 1 0 0 0 1
  0 0 0 0 0 0 2 6 9 7 9 7 3 7 2 1 1 0 0 0 2
Row n = 8 counts the following partitions (empty column not shown):
  (2222)  (332)    (44)      (41111)    (53)        (611)   (8)
          (422)    (431)     (311111)   (62)        (5111)  (71)
          (3221)   (3311)    (2111111)  (521)
          (22211)  (4211)               (11111111)
                   (32111)
                   (221111)
Column k = 5 counts the following partitions:
  (11111)  (411)    (43)     (332)    (3222)   (22222)
           (3111)   (331)    (422)    (22221)
           (21111)  (421)    (3221)
                    (3211)   (22211)
                    (22111)

Crossrefs

Row lengths are A331418.
Row sums are A000041.
Column sums are A331387.
Shifting row n to the right n times gives A331416.
Partitions whose sum of primes is divisible by their sum are A331379.
Partitions whose product divides their sum of primes are A331381.
Partitions whose product equals their sum of primes are A331383.
Cf. A000040, A001414, A014689, A056239, A330950, A330953, A330954, A331378, A331415.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Total[Prime/@#]==m&]],{n,0,10},{m,n,Max@@Table[Total[Prime/@y],{y,IntegerPartitions[n]}]}]

A331387 Number of integer partitions whose sum of primes of parts equals their sum of parts plus n.

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 24, 34, 47, 64, 86, 113, 148, 191, 245, 310, 390, 486, 602, 740, 907, 1104, 1338, 1613, 1937, 2315, 2758, 3272, 3871, 4562, 5362, 6283, 7344, 8558, 9952, 11542, 13356, 15419, 17766, 20425, 23440, 26846, 30696, 35032, 39917, 45406
Offset: 0

Views

Author

Gus Wiseman, Jan 17 2020

Keywords

Comments

Primes of parts means the prime counting function applied to the part sizes. Equivalently, a(n) is the number of integer partitions with part sizes in A014689(n) interpreted as a multiset. - Andrew Howroyd, Apr 17 2021

Examples

			The a(0) = 1 through a(5) = 16 partitions:
  ()  (1)  (3)   (4)    (33)    (43)
      (2)  (11)  (31)   (41)    (331)
           (21)  (32)   (42)    (332)
           (22)  (111)  (311)   (411)
                 (211)  (321)   (421)
                 (221)  (322)   (422)
                 (222)  (1111)  (3111)
                        (2111)  (3211)
                        (2211)  (3221)
                        (2221)  (3222)
                        (2222)  (11111)
                                (21111)
                                (22111)
                                (22211)
                                (22221)
                                (22222)
For example, the partition (3,2,2,1) is counted under n = 5 because it has sum of primes 5+3+3+2 = 13 and its sum of parts plus n is also 3+2+2+1+5 = 13.

Links

Crossrefs

Column sums of A331385.
Partitions into primes are A000607.
Partitions whose sum of primes is divisible by their sum are A331379.
Partitions whose product divides their sum of primes are A331381.
Partitions whose product equals their sum of primes are A331383.
Cf. A000040, A001414, A014689, A056239, A330950, A330953, A330954, A331378, A331415, A331416, A331418.

Programs

  • Mathematica
    Table[Sum[Length[Select[IntegerPartitions[k],Total[Prime/@#]==k+n&]],{k,0,2*n}],{n,0,10}]
  • PARI
    seq(n)={my(m=1); while(prime(m)-m<=n, m++); Vec(1/prod(k=1, m, 1 - x^(prime(k)-k) + O(x*x^n)))} \\ Andrew Howroyd, Apr 16 2021

Formula

G.f.: 1/Product_{k>=1} 1 - x^(prime(k)-k). - Andrew Howroyd, Apr 16 2021

Extensions

Terms a(31) and beyond from Andrew Howroyd, Apr 16 2021

A331418 If A331417(n) is the maximum sum of primes of the parts of an integer partition of n, then a(n) = A331417(n) - n + 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 11, 12, 15, 20, 21, 26, 29, 30, 33, 38, 43, 44, 49, 52, 53, 58, 61, 66, 73, 76, 77, 80, 81, 84, 97, 100, 105, 106, 115, 116, 121, 126, 129, 134, 139, 140, 149, 150, 153, 154, 165, 176, 179, 180, 183, 188, 189, 198, 203, 208, 213, 214, 219
Offset: 0

Views

Author

Gus Wiseman, Jan 17 2020

Keywords

Comments

For n > 4, a(n) = A014692(n).

Crossrefs

Converges to A014692.
Row lengths of A331385.
Sum of prime factors is A001414.
Partitions into primes are A000607.
Partitions whose sum of primes is divisible by their sum are A331379.
Cf. A000040, A014689, A056239, A330950, A330953, A330954, A331378, A331381, A331383, A331387, A331415, A331416.

Programs

  • Mathematica
    Table[Max@@Table[Total[Prime/@y],{y,IntegerPartitions[n]}]-n+1,{n,0,30}]

Formula

a(n) = A331417(n) - n + 1.
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