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A326981
Total number of composite parts in all partitions of n.
2
0, 0, 0, 0, 1, 1, 3, 4, 9, 13, 22, 31, 51, 70, 105, 145, 210, 283, 398, 530, 726, 958, 1283, 1673, 2212, 2854, 3714, 4756, 6119, 7764, 9893, 12457, 15728, 19674, 24636, 30615, 38079, 47034, 58109, 71396, 87692, 107179, 130943, 159278, 193619, 234486, 283720
OFFSET
0,7
FORMULA
a(n) = A144119(n) - A000070(n-1), n >= 1.
a(n) = A006128(n) - A326957(n).
EXAMPLE
For n = 6 we have:
--------------------------------------
. Number of
Partitions composite
of 6 parts
--------------------------------------
6 .......................... 1
3 + 3 ...................... 0
4 + 2 ...................... 1
2 + 2 + 2 .................. 0
5 + 1 ...................... 0
3 + 2 + 1 .................. 0
4 + 1 + 1 .................. 1
2 + 2 + 1 + 1 .............. 0
3 + 1 + 1 + 1 .............. 0
2 + 1 + 1 + 1 + 1 .......... 0
1 + 1 + 1 + 1 + 1 + 1 ...... 0
------------------------------------
Total ...................... 3
So a(6) = 3.
MAPLE
b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, 0], b(n, i-1)+
(p-> p+[0, `if`(isprime(i), 0, p[1])])(b(n-i, min(n-i, i))))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50); # Alois P. Heinz, Aug 13 2019
MATHEMATICA
b[n_, i_] := b[n, i] = If[n==0 || i==1, {1, 0}, b[n, i-1] + # + {0, If[PrimeQ[i], 0, #[[1]]]}&[b[n-i, Min[n-i, i]]]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 50] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)
KEYWORD
nonn
AUTHOR
Omar E. Pol, Aug 09 2019
STATUS
approved