A316430 Heinz numbers of integer partitions whose length is equal to the GCD of all the parts.

1, 2, 9, 21, 39, 57, 87, 91, 111, 125, 129, 159, 183, 203, 213, 237, 247, 267, 301, 303, 321, 325, 339, 377, 393, 417, 427, 453, 489, 519, 543, 551, 553, 559, 575, 579, 597, 669, 687, 689, 707, 717, 753, 789, 791, 813, 817, 843, 845, 879, 923, 925, 933, 951, 973

Offset: 1

Author

Gus Wiseman, Jul 02 2018

Keywords

• nonn

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

2 is the only even term in the sequence. 3k is in the sequence if and only if k is in A031215. 5k is in the sequence if and only if k = pq with p and q in A031336.

Examples

			Sequence of integer partitions whose length is equal to their GCD begins: (), (1), (2,2), (4,2), (6,2), (8,2), (10,2), (6,4), (12,2), (3,3,3), (14,2), (16,2), (18,2), (10,4), (20,2), (22,2), (8,6), (24,2), (14,4), (26,2), (28,2), (6,3,3).
		

Crossrefs

Subsequence of A004280.

Cf. A056239, A067538, A074761, A143773, A289508, A289509, A296150, A316413, A316431, A316432, A316433, A031215, A031336.

Programs

Mathematica

Select[Range[200],PrimeOmega[#]==GCD@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]]&]

PARI

is(n,f=factor(n))=gcd(apply(primepi,f[,1]))==vecsum(f[,2]) \\ Charles R Greathouse IV, Jul 25 2024

Formula

a(n) << n log^2 n, can this be improved? - Charles R Greathouse IV, Jul 25 2024