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 1-3 of 3 results.

A380344 Product of prime indices minus sum of prime factors of n.

Original entry on oeis.org

1, -1, -1, -3, -2, -3, -3, -5, -2, -4, -6, -5, -7, -5, -2, -7, -10, -4, -11, -6, -2, -8, -14, -7, -1, -9, -1, -7, -19, -4, -20, -9, -4, -12, 0, -6, -25, -13, -4, -8, -28, -4, -29, -10, 1, -16, -32, -9, 2, -3, -6, -11, -37, -3, -1, -9, -6, -21, -42, -6, -43
Offset: 1

Views

Author

Gus Wiseman, Jan 24 2025

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, with product A003963.

Examples

			72 has prime factors {2,2,2,3,3} and prime indices {1,1,1,2,2}, so a(72) = 4 - 12 = -8.

Crossrefs

Positions of 0 are A331384.
For plus instead of minus we have A380409.
Positions of positives are A380410.
Triangles:
- A027746 = prime factors
- A112798 = prime indices
Statistics:
- A000027 = product of prime factors = row products of A027746
- A001414 = sum of prime factors = row sums of A027746
- A003963 = product of prime indices = row products of A112798
- A056239 = sum of prime indices = row sums of A112798
Combinations:
- A075254 = product of factors + sum of factors = A000027 + A001414
- A075255 = product of factors - sum of factors = A000027 - A001414
- A178503 = product of factors - sum of indices = A000027 - A056239
- A325036 = product of indices - sum of indices = A003963 - A056239
- A379681 = product of indices + sum of indices = A003963 + A056239
- A380344 = product of indices - sum of factors = A003963 - A001414
- A380345 = product of factors + sum of indices = A000027 + A056239
- A380409 = product of indices + sum of factors = A003963 + A001414
A000040 lists the primes, differences A001223.
A001222 counts prime factors with multiplicity.
A055396 gives least prime index, greatest A061395.
Cf. A000720, A175508, A319000, A325032, A325033, A325034, A325035, A325040, A379682, A380220.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@prix[n]-Plus@@Prime/@prix[n],{n,100}]

Formula

a(n) = A003963(n) - A001414(n).

A380409 Product of prime indices plus sum of prime factors.

Original entry on oeis.org

1, 3, 5, 5, 8, 7, 11, 7, 10, 10, 16, 9, 19, 13, 14, 9, 24, 12, 27, 12, 18, 18, 32, 11, 19, 21, 17, 15, 39, 16, 42, 11, 24, 26, 24, 14, 49, 29, 28, 14, 54, 20, 57, 20, 23, 34, 62, 13, 30, 21, 34, 23, 69, 19, 31, 17, 38, 41, 76, 18, 79, 44, 29, 13, 36, 26, 86
Offset: 1

Views

Author

Gus Wiseman, Jan 25 2025

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, with product A003963.

Examples

			72 has prime factors {2,2,2,3,3} and prime indices {1,1,1,2,2}, so a(72) = 12 + 4 = 16.

Crossrefs

For factors instead of indices we have A075254.
For indices instead of factors we have A379681.
For minus instead of plus we have A380344, zeros A331384.
Triangles:
- A027746 = prime factors
- A112798 = prime indices
Statistics:
- A000027 = product of prime factors = row products of A027746
- A001414 = sum of prime factors = row sums of A027746
- A003963 = product of prime indices = row products of A112798
- A056239 = sum of prime indices = row sums of A112798
Combinations:
- A075254 = product of factors + sum of factors = A000027 + A001414
- A075255 = product of factors - sum of factors = A000027 - A001414
- A178503 = product of factors - sum of indices = A000027 - A056239
- A325036 = product of indices - sum of indices = A003963 - A056239
- A379681 = product of indices + sum of indices = A003963 + A056239
- A380344 = product of indices - sum of factors = A003963 - A001414
- A380345 = product of factors + sum of indices = A000027 + A056239
- A380409 = product of indices + sum of factors = A003963 + A001414
A000040 lists the primes, differences A001223.
A001222 counts prime factors with multiplicity.
A055396 gives least prime index, greatest A061395.
Cf. A000720, A175508, A319000, A325032, A325033, A325034, A325035, A325040, A379682, A380220, A380410.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Prime/@prix[n]]+Times@@prix[n],{n,100}]

Formula

a(n) = A003963(n) + A001414(n).

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}]
Showing 1-3 of 3 results.

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