A033178 -id:A033178 - cp's OEIS frontend

A038007 Even numbers k such that b(k) is greater than b(k-1) and b(k+1); b(k) = A033178(k).

166, 248, 298, 518, 536, 586, 766, 946, 956, 1076, 1126, 1148, 1208, 1216, 1366, 1376, 1418, 1544, 1558, 1618, 1678, 1688, 1726, 1756, 1786, 1796, 1898, 1918, 1948, 1978, 2138, 2168, 2188, 2206, 2228, 2296, 2338, 2426, 2458, 2476

Offset: 1

Christian G. Bower

nonn

Cf. A033178, A038008, A038009, A038011.

A038005 Odd k such that b(k) is less than b(k-1) and b(k+1). b(k): A033178.

Original entry on oeis.org

87, 99, 135, 159, 195, 207, 227, 231, 279, 315, 335, 339, 399, 405, 427, 507, 539, 549, 567, 619, 627, 639, 675, 699, 747, 759, 819, 843, 867, 879, 927, 1071, 1087, 1095, 1127, 1147, 1155, 1179, 1209, 1239, 1245, 1267, 1287, 1347, 1371

Offset: 0

Christian G. Bower

nonn

Cf. A038006, A038013.

A343297 Numbers k such that there are exactly two multisets of cardinality k where the sum equals the product (A033178(k)=2).

Original entry on oeis.org

7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 30, 34, 36, 42, 44, 48, 54, 60, 66, 80, 84, 90, 112, 126, 142, 192, 210, 234, 252, 258, 330, 350, 354, 440, 594, 654, 714, 720, 780, 966, 1102, 2400, 2820, 4350, 4354, 5274, 6174, 6324

Offset: 1

Nathaniel Gregg, Apr 11 2021

nonn

At most one of a(n) - 1 and 2*a(n)-1 are composite. More precisely, a(n) are those positive integers such that exactly one of product(s)*(a(n)+sum(s)-k-2)+1 can be factored as (product(s)*p-1)*(product(s)*q-1), where s varies over all multisets of k positive integers and 1 < p <= q < a(n). The first statement is given by considering s = {} and s = {2}. a(50) is greater than 10^4.

			a(5) = 12 because {2,2,2,2,1,1,1,1,1,1,1,1} and {12,2,1,1,1,1,1,1,1,1,1,1} are the only multisets of size 12 where the sum equals the product.

Michael W. Ecker, When Does a Sum of Positive Integers Equal Their Product?, Mathematics Magazine 75(1), 2002, pp. 41-47.

Cf. A033178, A033179, A343298.

A038004 Least k such that A033178(k)=n.

Original entry on oeis.org

2, 7, 5, 13, 25, 37, 41, 89, 61, 85, 113, 221, 253, 281, 181, 361, 421, 433, 761, 841, 613, 793, 1009, 1541, 1201, 1901, 1121, 1261, 2801, 2881, 2053, 3173, 2773, 2521, 2161, 3421, 4001, 6865, 5581, 6553, 5441, 6481, 7141, 10193

Offset: 1

Christian G. Bower

nonn

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

Onno M. Cain, Jul 16 2019

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.

Onno M. Cain, Bioperational Multisets in Various Semi-rings, arXiv:1908.03235 [math.RA], 2019.

Cf. A033178, A033179, A104173.

A033179 Numbers k such that exactly one multiset of k positive integers has equal sum and product.

Original entry on oeis.org

2, 3, 4, 6, 24, 114, 174, 444

Offset: 1

David W. Wilson

No other terms below 10^11 (Weingartner, 2012). Probably finite and complete.
For any m, there is the multiset {m, 2, 1^(m-2)} with sum and product 2m.
(A) If m-1 is composite (m-1=ab), then {a+1, b+1, 1^(m-2)} is another multiset with sum = product. (Hugo van der Sanden)
(B) If 2m-1 is composite (2m-1=ab), then {2, (a+1)/2, (b+1)/2, 1^(m-3)} is another such multiset. (Don Reble)
(C) If m = 30j+12, then {2, 2, 2, 2, 2j+1, 1^(30j+7)} is another such multiset. (Don Reble)
Conditions (A), (B), (C) eliminate all k's except for 2, 3, 4, 6, 30j+0, and 30j+24.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 174, p. 54, Ellipses, Paris 2008.
R. K. Guy, 'Unsolved Problems in Number Theory' (Section D24).

Michael W. Ecker, When Does a Sum of Positive Integers Equal Their Product? Mathematics Magazine 75(1), 2002, pp. 41-47.
Hlib Husarov and Eberhard Mayerhofer, An algorithm for the Product-Sum Equality, arXiv:2508.09647 [math.NT], 2025. See p. 2.
Piotr Miska and Maciej Ulas, On the Diophantine equation sigma_2(Xn)=sigma_n(Xn), arXiv:2203.03942 [math.NT], 2022.
Michael A. Nyblom, Sophie Germain Primes and the Exceptional Values of the Equal-Sum-And-Product Problem, Fib. Q. 50(1), 2012, 58-61.
Burkard Polster, What's the next freak identity? A new deep connection with Sophie Germain primes, YouTube Mathologer video, 2024.
Andreas Weingartner, On the Diophantine equation Prod x_i = Sum x_i, INTEGERS 12 (2012), #A57.

Cf. A033178.

Revised by Don Reble, Jun 11 2005

Edited by Max Alekseyev, Nov 13 2013

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

Nathaniel Gregg, Feb 22 2021

nonn

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

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

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

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

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

Nathaniel Gregg, Feb 22 2021

nonn

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.

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

Cf. A104173, A033178, A033179, A341865.

Equals A330492 + 1. - Hugo Pfoertner, Feb 23 2021

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

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.

A343298 a(n) is the smallest m such that the only non-basic multiset of positive integers of cardinality m where the sum equals the product has n nonunit elements, or zero if no such m exists.

Original entry on oeis.org

0, 7, 8, 12, 42

Offset: 1

Nathaniel Gregg, Apr 11 2021

A multiset where the sum equals the product is called a bioperational multiset. A bioperational multiset is called basic if it is of the form {2,n,1,...,1}, because these exist of size n for all n > 1. Nonzero entries are a subset of A343297. a(6) > 10^4 or zero.

			a(5) = 42 because {2,2,2,2,3; 37} and {42,2; 40} are the only bioperational multisets of size 42, where the number after the semicolon is the number of repeated 1's.

Onno M. Cain, Bioperational Multisets in Various Semi-rings, arXiv:1908.03235 [math.RA], 2019.
Michael W. Ecker, When Does a Sum of Positive Integers Equal Their Product? Mathematics Magazine 75(1), 2002, pp. 41-47.
Michael A. Nyblom, Sophie Germain Primes and the Exceptional Values of the Equal-Sum-And-Product Problem, Fib. Q. 50(1), 2012, 58-61.

Cf. A343297, A033178, A033179.

cp's OEIS Frontend

A038007 Even numbers k such that b(k) is greater than b(k-1) and b(k+1); b(k) = A033178(k).

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A038005 Odd k such that b(k) is less than b(k-1) and b(k+1). b(k): A033178.

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A343297 Numbers k such that there are exactly two multisets of cardinality k where the sum equals the product (A033178(k)=2).

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A038004 Least k such that A033178(k)=n.

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A309230 Positions of records in A033178.

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A033179 Numbers k such that exactly one multiset of k positive integers has equal sum and product.

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

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

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A343298 a(n) is the smallest m such that the only non-basic multiset of positive integers of cardinality m where the sum equals the product has n nonunit elements, or zero if no such m exists.

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