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Total number of noncomposite parts in all partitions of n.
2

%I #22 Nov 15 2020 08:58:32

%S 0,1,3,6,11,19,32,50,77,115,170,244,348,486,675,923,1253,1682,2246,

%T 2968,3904,5094,6616,8533,10962,13997,17808,22538,28426,35689,44670,

%U 55678,69199,85692,105826,130261,159935,195778,239092,291191,353854,428925,518848

%N Total number of noncomposite parts in all partitions of n.

%F a(n) = A037032(n) + A000070(n-1), n >= 1.

%F a(n) = A006128(n) - A326981(n).

%e For n = 6 we have:

%e --------------------------------------

%e . Number of

%e Partitions noncomposite

%e of 6 parts

%e --------------------------------------

%e 6 .......................... 0

%e 3 + 3 ...................... 2

%e 4 + 2 ...................... 1

%e 2 + 2 + 2 .................. 3

%e 5 + 1 ...................... 2

%e 3 + 2 + 1 .................. 3

%e 4 + 1 + 1 .................. 2

%e 2 + 2 + 1 + 1 .............. 4

%e 3 + 1 + 1 + 1 .............. 4

%e 2 + 1 + 1 + 1 + 1 .......... 5

%e 1 + 1 + 1 + 1 + 1 + 1 ...... 6

%e ------------------------------------

%e Total ..................... 32

%e So a(6) = 32.

%p b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n], b(n, i-1)+

%p (p-> p+[0, `if`(isprime(i), p[1], 0)])(b(n-i, min(n-i, i))))

%p end:

%p a:= n-> b(n$2)[2]:

%p seq(a(n), n=0..50); # _Alois P. Heinz_, Aug 13 2019

%t b[n_] := Sum[PrimeNu[k] PartitionsP[n-k], {k, 1, n}];

%t c[n_] := SeriesCoefficient[Product[1/(1-x^k), {k, 1, n}]/(1-x), {x, 0, n}];

%t a[n_] := b[n] + c[n-1];

%t a /@ Range[0, 50] (* _Jean-François Alcover_, Nov 15 2020 *)

%Y First differs from A183088 at a(13).

%Y Cf. A000041, A000070, A006128, A008578 (noncomposites), A037032, A144115, A144116, A144119, A326958, A326981.

%K nonn

%O 0,3

%A _Omar E. Pol_, Aug 08 2019