A282467 Number of partitions of n which are not the partitions into (one or more) consecutive parts.

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

G. C. Greubel, Table of n, a(n) for n = 1..5000

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

Cf. A000041, A001227.

Sequence in context: A363282 A116045 A096818 * A318980 A226622 A332302

Adjacent sequences: A282464 A282465 A282466 * A282468 A282469 A282470

Keyword

nonn

Author

Omar E. Pol, Feb 25 2017

Status

approved