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A316430
Heinz numbers of integer partitions whose length is equal to the GCD of all the parts.
10
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.
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 02 2018
STATUS
approved