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

1,2

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.

Links

Table of n, a(n) for n=1..55.

Formula

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

Example

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).

Mathematica

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

Prog

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

Crossrefs

Subsequence of A004280.

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

Sequence in context: A298912 A368187 A005476 * A131476 A023549 A192971

Adjacent sequences: A316427 A316428 A316429 * A316431 A316432 A316433

Keyword

nonn

Author

Gus Wiseman, Jul 02 2018

Status

approved