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A379318 Odd numbers whose product of prime indices is a multiple of their sum of prime indices.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 165, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239
Offset: 1

Views

Author

Gus Wiseman, Jan 16 2025

Keywords

Comments

Contains all odd primes.
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 sum and product of prime indices are A056239 and A003963 respectively.

Examples

			The terms together with their prime indices begin:
     2: {1}         53: {16}           109: {29}
     3: {2}         59: {17}           113: {30}
     5: {3}         61: {18}           125: {3,3,3}
     7: {4}         63: {2,2,4}        127: {31}
     9: {2,2}       65: {3,6}          131: {32}
    11: {5}         67: {19}           137: {33}
    13: {6}         71: {20}           139: {34}
    17: {7}         73: {21}           149: {35}
    19: {8}         79: {22}           150: {1,2,3,3}
    23: {9}         81: {2,2,2,2}      151: {36}
    29: {10}        83: {23}           154: {1,4,5}
    30: {1,2,3}     84: {1,1,2,4}      157: {37}
    31: {11}        89: {24}           163: {38}
    37: {12}        97: {25}           165: {2,3,5}
    41: {13}       101: {26}           167: {39}
    43: {14}       103: {27}           169: {6,6}
    47: {15}       107: {28}           173: {40}
    49: {4,4}      108: {1,1,2,2,2}    179: {41}

Crossrefs

Including evens gives A326149, counted by A057568.
For nonprime instead of odd we get A326150.
For even instead of odd we get A379319, counted by A379320.
Partitions of this type are counted by A379734, strict A379735, see A379733.
For squarefree instead of odd we get A379844, even case A379845.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A069016, A301988, A318950, A319000, A319916, A324851, A325041, A326152, A379671, A379678.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],OddQ[#]&&Divisible[Times@@prix[#],Total[prix[#]]]&]

A380411 Number of integer partitions of n such that the product of parts is greater than the sum of primes indexed by the parts.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 4, 8, 14, 23, 39, 58, 85, 121, 168, 228, 308, 404, 533, 691, 892, 1136, 1449, 1820, 2291, 2857, 3553, 4387, 5418, 6646, 8144, 9931, 12086, 14649, 17733, 21379, 25747, 30905, 37049, 44282, 52863, 62936, 74841, 88792, 105202, 124387
Offset: 0

Views

Author

Gus Wiseman, Jan 26 2025

Keywords

Examples

			The partition y = (4,3,2) has product of parts 4*3*2 = 24 and sum of corresponding primes 7+5+3 = 15, so y is counted under a(9).
The a(0) = 1 through a(10) = 14 partitions:
  ()  .  .  .  .  .  .  (322)  (44)    (54)     (55)
                               (332)   (333)    (64)
                               (422)   (432)    (433)
                               (2222)  (522)    (442)
                                       (3222)   (532)
                                       (3321)   (622)
                                       (4221)   (3322)
                                       (22221)  (3331)
                                                (4222)
                                                (4321)
                                                (5221)
                                                (22222)
                                                (32221)
                                                (33211)

Crossrefs

For parts instead of primes on the RHS we have A114324.
The version for divisibility instead of inequality is A330954.
The version for equality is A331383, ranks A331384.
These partitions are ranked by A380410.
A000040 lists the primes, differences A001223.
A000041 counts integer partitions, strict A000009.
A001414 gives sum of prime factors.
A003963 gives product of prime indices
A379666 counts partitions by sum and product.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568 (strict A379733), ranks A326149, see A379319, A380217.
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A175508, A318950, A319000, A319916, A326152, A379320, A379734, A380218, A380219, A380344.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Times@@#>Plus@@Prime/@#&]],{n,0,30}]

A380343 Number of strict integer partitions of n whose product of parts is a multiple of n + 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 3, 0, 3, 5, 5, 0, 8, 0, 15, 11, 8, 0, 42, 8, 12, 26, 49, 0, 100, 0, 90, 56, 27, 105, 246, 0, 41, 108, 414, 0, 450, 0, 332, 651, 81, 0, 1341, 210, 693, 366, 754, 0, 1869, 1044, 2579, 634, 206, 0, 5695, 0, 278, 4850, 5927, 2802
Offset: 0

Views

Author

Gus Wiseman, Jan 22 2025

Keywords

Examples

			The a(5) = 1 through a(17) = 8 partitions (A=10, C=12):
  32  .  421  .  54  .  83   .  76    95    843   .  98
                        632     742   653   852      863
                        641     7321  A31   861      962
                                      5432  6432     C32
                                      6521  8421     7631
                                                     9431
                                                     9521
                                                     65321

Crossrefs

The non-strict version is A379320, ranked by A380217 = A379319/2.
For n instead of n+1 we have A379733, non-strict A057568.
The case of equality for non-strict partitions is A380218.
A000041 counts integer partitions, strict A000009.
A379666 counts partitions by sum and product.
A380219 counts partitions of n whose product is a proper multiple of n, ranks A380216.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568, ranks A326149
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A003963, A069016, A319000, A319916, A325042, A326152, A326156, A379671, A379734.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Times@@#,n+1]&]],{n,0,30}]
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