A330953 -id:A330953 - cp's OEIS frontend

A331415 Sum of prime factors minus sum of prime indices of n.

0, 1, 1, 2, 2, 2, 3, 3, 2, 3, 6, 3, 7, 4, 3, 4, 10, 3, 11, 4, 4, 7, 14, 4, 4, 8, 3, 5, 19, 4, 20, 5, 7, 11, 5, 4, 25, 12, 8, 5, 28, 5, 29, 8, 4, 15, 32, 5, 6, 5, 11, 9, 37, 4, 8, 6, 12, 20, 42, 5, 43, 21, 5, 6, 9, 8, 48, 12, 15, 6, 51, 5, 52, 26, 5, 13, 9, 9

Offset: 1

Gus Wiseman, Jan 17 2020

nonn

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.

			The prime factors of 12 are {2,2,3}, while the prime indices are {1,1,2}, so a(12) = 7 - 4 = 3.

The number of k's is A331387(k) = sum of k-th column of A331385.
The sum of prime factors of n is A001414(n).
The sum of prime indices of n is A056239(n).
Numbers divisible by the sum of their prime factors are A036844.
Sum of prime factors is divisible by sum of prime indices: A331380
Product of prime indices equals sum of prime factors: A331384.
Cf. A000040, A000720, A001222, A014689, A056239, A112798, A301987, A318995, A325036, A330953, A331378, A331379, A331416, A331418.

Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>k*(p-PrimePi[p])]],{n,30}]

Totally additive with a(prime(k)) = prime(k) - k = A014689(k).

a(n) = A001414(n) - A056239(n).

A331380 Numbers whose sum of prime factors is divisible by their sum of prime indices.

Original entry on oeis.org

2, 4, 8, 16, 32, 33, 39, 55, 64, 65, 66, 74, 77, 78, 86, 91, 110, 128, 130, 132, 154, 156, 164, 182, 188, 220, 256, 260, 264, 308, 312, 364, 371, 411, 440, 459, 512, 513, 520, 528, 530, 616, 624, 636, 689, 728, 746, 755, 765, 766, 855, 880, 906, 915, 918, 1007

Offset: 1

Gus Wiseman, Jan 16 2020

nonn

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.

			The sequence of terms together with their prime indices begins:
     2: {1}
     4: {1,1}
     8: {1,1,1}
    16: {1,1,1,1}
    32: {1,1,1,1,1}
    33: {2,5}
    39: {2,6}
    55: {3,5}
    64: {1,1,1,1,1,1}
    65: {3,6}
    66: {1,2,5}
    74: {1,12}
    77: {4,5}
    78: {1,2,6}
    86: {1,14}
    91: {4,6}
   110: {1,3,5}
   128: {1,1,1,1,1,1,1}
   130: {1,3,6}
   132: {1,1,2,5}
For example, 132 has prime factors {2,2,3,11} and prime indices {1,1,2,5}, and 18 is divisible by 9, so 132 is in the sequence.

These are the Heinz numbers of the partitions counted by A331380.
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.
Heinz number is divisible by sum of primes: A330953.
Partitions whose product divides their sum of primes are A331381.
Partitions whose product is equal to their sum of primes are A331383.
Product of prime indices equals sum of prime factors: A331384.
Cf. A000040, A001414, A056239, A330954, A331378, A331379, A331382, A331415, A331416.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],Divisible[Plus@@Prime/@primeMS[#],Plus@@primeMS[#]]&]

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

Gus Wiseman, Jan 16 2020

nonn

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.

			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.

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.

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

Gus Wiseman, Jan 17 2020

			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)

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.

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

Gus Wiseman, Jan 17 2020

nonn

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

			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.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

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.

Table[Sum[Length[Select[IntegerPartitions[k],Total[Prime/@#]==k+n&]],{k,0,2*n}],{n,0,10}]

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

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

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

Gus Wiseman, Jan 17 2020

nonn

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

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.

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

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

A331417 Maximum sum of primes of the parts of an integer partition of n.

Original entry on oeis.org

0, 2, 4, 6, 8, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 0

Gus Wiseman, Jan 17 2020

nonn

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

Converges to A000040.
Row lengths of A331416.
The corresponding minimum is A007494.
Partitions whose sum of primes is divisible by their sum are A331379.
Cf. A000607, A001414, A014689, A056239, A179437, A330950, A330953, A330954, A331381, A331383, A331385, A331387, A331415, A331418.

Table[Max@@Table[Total[Prime/@y],{y,IntegerPartitions[n]}],{n,0,10}]

a(n) = A331418(n) + n - 1.

A330952 Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by all parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 8, 11, 14, 20, 25, 32, 42, 54, 69, 87, 109, 137, 172, 215, 269, 331, 409, 499, 612, 751, 917, 1111, 1344, 1626, 1963, 2359, 2834, 3396, 4065, 4849, 5779, 6865, 8146, 9658, 11424, 13483, 15898, 18710, 21999, 25823, 30272, 35417, 41397

Offset: 0

Gus Wiseman, Jan 15 2020

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The a(1) = 1 through a(9) = 11 partitions:
  1  11  21   211   221    321     2221     3221      621
         111  1111  2111   411     3211     4211      3321
                    11111  2211    4111     22211     22221
                           21111   22111    32111     32211
                           111111  211111   41111     42111
                                   1111111  221111    222111
                                            2111111   321111
                                            11111111  411111
                                                      2211111
                                                      21111111
                                                      111111111

The Heinz numbers of these partitions are given by A120383.
Partitions whose product is divisible by their sum are A057568.
Partitions whose Heinz number is divisible by their product are A324925.
Partitions whose Heinz number is divisible by their sum are A330950.
Cf. A056239, A112798, A324756, A326149, A326155, A330953, A330954, A331383.

Table[Length[Select[IntegerPartitions[n],And@@Table[Divisible[Times@@Prime/@#,i],{i,#}]&]],{n,0,30}]

A366851 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n such that the sum of primes indexed by all parts greater than one is k.

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 2, 0, 2, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 2, 2, 2, 2, 1, 1, 0, 0, 1

Offset: 0

Gus Wiseman, Nov 01 2023

To illustrate the definition, the sum of primes indexed by all parts greater than one of the partition (5,2,2,1) is prime(5) + prime(2) + prime(2) = 17.

			Triangle begins:
  1
  1
  1 0 0 1
  1 0 0 1 0 1
  1 0 0 1 0 1 1 1
  1 0 0 1 0 1 1 1 1 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 1 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 1 1 1 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 2 0 2 1 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 3 2 2 2 2 1 1 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 3 3 4 4 2 3 2 0 3 1 0 0 0 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 3 3 4 5 4 4 3 3 3 2 3 0 1 0 0 1 0 1
The T(8,13) = 3 partitions are: (6,1,1), (4,2,2), (3,3,2).
The T(10,17) = 4 partitions are: (7,1,1,1), (5,2,2,1), (4,4,2), (4,3,3).

Row lengths are A055670.
Columns appear to converge to A099773.
A bisected even version is A116598 (counts partitions by number of 1's).
Counting all parts (not just > 1) gives A331416, shifted A331385.
A000041 counts integer partitions, strict A000009 (also into odds).
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A330953 counts partitions with Heinz number divisible by sum of primes.
A331381 counts partitions with (product)|(sum of primes), equality A331383.
Cf. A000837, A014689, A113686, A239261, A331417, A331418, A365067, A366842.

Table[Length[Select[IntegerPartitions[n], Total[Select[Prime/@#,OddQ]]==k&]], {n,0,10}, {k,0,If[n<=1,0,Prime[n]]}]

cp's OEIS Frontend

A331415 Sum of prime factors minus sum of prime indices of n.

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A331380 Numbers whose sum of prime factors is divisible by their sum of prime indices.

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

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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.

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

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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.

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A331417 Maximum sum of primes of the parts of an integer partition of n.

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Formula

A330952 Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by all parts.

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A366851 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n such that the sum of primes indexed by all parts greater than one is k.

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