A331416 -id:A331416 - cp's OEIS frontend

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.

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.

A366839 Sum of even prime factors of 2n, counted with multiplicity.

Original entry on oeis.org

2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 10, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 12, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 10, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 14, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 10, 2, 4, 2, 6, 2

Offset: 1

Gus Wiseman, Oct 25 2023

nonn

			The prime factors of 2*60 are {2,2,2,3,5}, of which the even factors are {2,2,2}, so a(60) = 6.

A compound version is A001414, triangle A331416.
Dividing by 2 gives A001511.
Positions of 2's are A005408.
For count instead of sum we have A007814, odd version A087436.
The partition triangle for this statistic is A116598 aerated.
For indices we have A366531, halved A366533, triangle A113686/A174713.
The odd version is A366840.
A019507 lists numbers with (even factor sum) = (odd factor sum).
A066207 lists numbers with all even prime indices, counted by A035363.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A257992 counts even prime indices, odd A257991.
A366528 adds up odd prime indices, triangle A113685 (without zeros A365067).
Cf. A000720, A066208, A258117, A325698.

Table[2*Length[NestWhileList[#/2&,n,EvenQ]],{n,100}]

a(n) = 2 * valuation(n, 2) + 2; \\ Amiram Eldar, Sep 13 2024

a(n) = 2*A001511(n).
a(n) = A100006(n) - A366840(2n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4. - Amiram Eldar, Sep 13 2024

A366840 Sum of odd prime factors of n, counted with multiplicity.

Original entry on oeis.org

0, 0, 3, 0, 5, 3, 7, 0, 6, 5, 11, 3, 13, 7, 8, 0, 17, 6, 19, 5, 10, 11, 23, 3, 10, 13, 9, 7, 29, 8, 31, 0, 14, 17, 12, 6, 37, 19, 16, 5, 41, 10, 43, 11, 11, 23, 47, 3, 14, 10, 20, 13, 53, 9, 16, 7, 22, 29, 59, 8, 61, 31, 13, 0, 18, 14, 67, 17, 26, 12, 71, 6

Offset: 1

Gus Wiseman, Oct 27 2023

Contains all positive integers except 1, 2, 4.

			The prime factors of 60 are {2,2,2,3,5}, of which the odd factors are {3,5}, so a(60) = 8.

The compound version is A001414, triangle A331416.
For count instead of sum we have A087436, even version A007814.
Odd-indexed terms are A100005.
Positions of odd terms are A335657, even A036349.
For prime indices we have A366528, triangle A113685 (without zeros A365067)
The even version is A366839 = 2*A001511.
The partition triangle for this statistic is A366851, even version A116598.
A019507 lists numbers with (even factor sum) = (odd factor sum).
A066207 lists numbers with all even prime indices, counted by A035363.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A257992 counts even prime indices, odd A257991.
Cf. A000009, A066208, A113686, A174713, A258117, A325698, A366531, A366850.

Table[Total[Times@@@DeleteCases[If[n==1,{}, FactorInteger[n]],{2,_}]],{n,100}]

a(n) = my(f=factor(n), j=if(n%2, 1, 2)); sum(i=j, #f~, f[i,1]*f[i,2]); \\ Michel Marcus, Oct 30 2023

a(n) = A100006(n) - A366839(n).
a(2n) = a(n).
a(2n-1) = A001414(2n-1) = A100005(n).
Completely additive with a(2^e) = 0 and a(p^e) = e*p for an odd prime p. - Amiram Eldar, Nov 03 2023

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]]}]

A366841 Least positive integer whose odd prime factors sum to n, starting with n = 5.

Original entry on oeis.org

5, 9, 7, 15, 27, 21, 11, 35, 13, 33, 105, 39, 17, 65, 19, 51, 195, 57, 23, 95, 171, 69, 285, 115, 29, 161, 31, 87, 483, 93, 261, 155, 37, 217, 465, 111, 41, 185, 43, 123, 555, 129, 47, 215, 387, 141, 645, 235, 53, 329, 705, 159, 987, 265, 59, 371, 61, 177

Offset: 5

Gus Wiseman, Oct 27 2023

nonn

All terms are odd.

It seems that all composite terms not divisible by 3 form a supersequence of A292081. - Ivan N. Ianakiev, Oct 30 2023

			The terms together with their prime factors (which sum to n) begin:
    5 = 5
    9 = 3*3
    7 = 7
   15 = 3*5
   27 = 3*3*3
   21 = 3*7
   11 = 11
   35 = 5*7
   13 = 13
   33 = 3*11
  105 = 3*5*7

Michel Marcus, Table of n, a(n) for n = 5..1000

This is the odd case of A056240.
Positions of first appearances in A366840 (sum of odd prime factors).
The partition triangle for this statistic is A366851, even A116598.
A001414 adds up prime factors, triangle A331416.
A019507 lists numbers with (even factor sum) = (odd factor sum).
A027746 lists prime factors, length A001222.
A087436 counts odd prime factors, even A007814.
A366528 adds up odd prime indices, triangle A113685 (without zeros A365067).
Cf. A031368, A036349, A056239, A066208, A100005, A112798, A239261, A257991, A335657, A366839.

nn=1000;
w=Table[Total[Times@@@DeleteCases[If[n==1,{},FactorInteger[n]],{2,_}]],{n,nn}];
spnm[y_]:=Max@@Select[Union[y],Function[i,Union[Select[y,#<=i&]]==Range[i]]];
Table[Position[w,k][[1,1]],{k,5,spnm[Join[{1,2,3,4},Take[w,nn]/.(0->1)]]}]

f(n) = my(f=factor(n), j=if (n%2, 1, 2)); sum(i=j, #f~, f[i,1]*f[i,2]); \\ A366840
a(n) = my(k=1); while (f(k) != n, k++); k; \\ Michel Marcus, Nov 02 2023

cp's OEIS Frontend

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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A366839 Sum of even prime factors of 2n, counted with multiplicity.

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A366840 Sum of odd prime factors of n, counted with multiplicity.

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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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A366841 Least positive integer whose odd prime factors sum to n, starting with n = 5.

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