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 1-4 of 4 results.

A033178 Number of multisets of n positive integers with equal sum and product.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 2, 2, 2, 3, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 1, 5, 4, 3, 3, 5, 2, 4, 3, 5, 2, 3, 2, 6, 3, 3, 4, 7, 2, 5, 2, 4, 4, 5, 2, 5, 4, 4, 3, 7, 2, 5, 4, 5, 4, 4, 2, 9, 3, 4, 4, 7, 2, 5, 5, 4, 3, 6, 3, 9, 4, 3, 3, 6, 3, 5, 2, 7, 4, 5, 2, 10, 5, 4, 5, 8, 2, 6, 3, 6, 3, 6, 5, 6, 5, 4, 5, 8, 3, 6, 3, 5
Offset: 2

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Author

David W. Wilson

Keywords

Comments

The multiset {n^1, 2^1, 1^(n-2)} has n elements and sum = product = 2n. Hence a(n) >= 1.

Examples

			a(5) = 3: {2,2,2,1,1}, {3,3,1,1,1}, {5,2,1,1,1}.
a(7) = 2: {4,3,1,1,1,1,1}, {7,2,1,1,1,1,1}.

References

  • R. K. Guy, 'Unsolved Problems in Number Theory' (Section D24).

Links

Crossrefs

Cf. A033179, A104173.

A341865 The cardinality of the largest multiset of positive integers whose product and sum equals n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 5, 5, 5, 1, 8, 1, 7, 9, 12, 1, 13, 1, 14, 13, 11, 1, 19, 17, 13, 21, 20, 1, 23, 1, 27, 21, 17, 25, 30, 1, 19, 25, 33, 1, 33, 1, 32, 37, 23, 1, 42, 37, 41, 33, 38, 1, 47, 41, 47, 37, 29, 1, 52, 1, 31, 53, 58
Offset: 1

Views

Author

Nathaniel Gregg, Feb 22 2021

Keywords

Comments

The largest multisets are given by the prime factorization of n and 1s added until the sum equals the product.

Examples

			For n = 12, the set of size a(12) = 8 is {1,1,1,1,1,2,2,3}.

Crossrefs

Cf. A059975, A104173, A033178, A033179, A341866.

Programs

  • PARI
    a(n) = my(f=factor(n)); n - sum(k=1, #f~, f[k,2]*(f[k,1]-1)); \\ Michel Marcus, Feb 26 2021

Formula

a(n) = n - Sum_(d_i*(p_i-1)), where n = Product_(p_i^d_i).
a(n) = n - A059975(n). - Joerg Arndt, Feb 22 2021

A341866 The cardinality of the smallest (nontrivial, except for prime n) multiset of positive integers whose product and sum equal n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 5, 5, 1, 6, 1, 7, 9, 8, 1, 9, 1, 10, 13, 11, 1, 12, 17, 13, 17, 14, 1, 15, 1, 16, 21, 17, 25, 18, 1, 19, 25, 20, 1, 21, 1, 22, 29, 23, 1, 24, 37, 25, 33, 26, 1, 27, 41, 28, 37, 29, 1, 30, 1, 31, 41, 32
Offset: 1

Views

Author

Nathaniel Gregg, Feb 22 2021

Keywords

Comments

The smallest set is obtained by taking the largest such multiset (A341865(n)) and replacing the largest proper subset that is also a product-sum multiset with its product. A singleton would always be the smallest product-sum multiset, so those are excluded except for prime n where no nontrivial multisets exist.

Examples

			For n = 12, the set of size a(n) = 6 is {1,1,1,1,2,6}.

Crossrefs

Cf. A104173, A033178, A033179, A341865.
Equals A330492 + 1. - Hugo Pfoertner, Feb 23 2021

Programs

  • PARI
    a(n) = if (n==1, 1, my(p=vecmin(factor(n)[,1])); (n/p-1)*(p-1) + 1); \\ Michel Marcus, Feb 26 2021

Formula

a(n) = (n/p - 1)*(p-1) + 1, where p is the smallest factor of n.
a(n) = A341865(n) - A341865(n/p) + 1, where p is the smallest prime factor of n.

A309230 Positions of records in A033178.

Original entry on oeis.org

2, 5, 13, 25, 37, 41, 61, 85, 113, 181, 361, 421, 433, 613, 793, 1009, 1121, 1261, 2053, 2161, 3421, 4001, 5441, 6481, 7141, 7561, 8033, 9361
Offset: 1

Views

Author

Onno M. Cain, Jul 16 2019

Keywords

Comments

Terms appear to have relatively few prime factors compared to their neighbors. a(28)=9361=11*23*37 is the first term with 3 factors.

Links

Crossrefs

Cf. A033178, A033179, A104173.
Showing 1-4 of 4 results.

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