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A282467
Number of partitions of n which are not the partitions into (one or more) consecutive parts.
1
0, 1, 1, 4, 5, 9, 13, 21, 27, 40, 54, 75, 99, 133, 172, 230, 295, 382, 488, 625, 788, 1000, 1253, 1573, 1955, 2434, 3006, 3716, 4563, 5600, 6840, 8348, 10139, 12308, 14879, 17974, 21635, 26013, 31181, 37336, 44581, 53170, 63259, 75173, 89128, 105556, 124752, 147271, 173522, 204223, 239939, 281587
OFFSET
1,4
COMMENTS
Also number of partitions of n minus the number of odd divisors of n.
LINKS
M. D. Hirschhorn and P. M. Hirschhorn, Partitions into Consecutive Parts, Mathematics Magazine: 2003, Volume 76, Number 4, Pages: 306-308.
FORMULA
a(n) = A000041(n) - A001227(n).
EXAMPLE
For n = 6, the number of partitions of 6 is A000041(6) = 11. There are two partitions of 6 into (one or more) consecutive parts: [6] and [3, 2, 1], so a(6) = 11 - 2 = 9. On the other hand, 6 has two odd divisors: 1 and 3, so a(6) = 11 - 2 = 9.
For n = 15, the number of partitions of 15 is A000041(15) = 176. There are four partitions of 15 into (one or more) consecutive parts: [15], [8, 7], [6, 5, 4] and [5, 4, 3, 2, 1], so a(15) = 176 - 4 = 172. On the other hand, 15 has four odd divisors: 1, 3, 5 and 15, so a(15) = 176 - 4 = 172.
MAPLE
a:= n-> combinat[numbpart](n)-mul(`if`(i[1]=2, 1, i[2]+1), i=ifactors(n)[2]):
seq(a(n), n=1..100); # Alois P. Heinz, Feb 27 2020
MATHEMATICA
Table[PartitionsP@ n - DivisorSum[n, Boole[# > 0] &, OddQ@ # &], {n, 52}] (* Michael De Vlieger, Feb 27 2017 *)
PROG
(Sage)
A282467 = lambda n: number_of_partitions(n) - len(list(filter(is_odd, divisors(n))))
[A282467(n) for n in (1..52)] # Peter Luschny, Feb 28 2017
(PARI) a(n)=numbpart(n) - numdiv(n>>valuation(n, 2)) \\ Charles R Greathouse IV, Mar 01 2017
CROSSREFS
Sequence in context: A363282 A116045 A096818 * A318980 A226622 A332302
KEYWORD
nonn
AUTHOR
Omar E. Pol, Feb 25 2017
STATUS
approved