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A340572 Number of partitions of n into 4 parts with at least one prime part. 1
0, 0, 0, 0, 0, 1, 2, 2, 5, 5, 8, 10, 13, 16, 21, 24, 31, 35, 41, 49, 57, 64, 75, 84, 95, 107, 119, 133, 147, 164, 179, 198, 215, 236, 256, 281, 300, 329, 349, 382, 407, 441, 465, 506, 531, 575, 603, 652, 681, 733, 765, 822, 853, 919, 952, 1019, 1057, 1128, 1166, 1247, 1284 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

LINKS

Table of n, a(n) for n=0..60.

Index entries for sequences related to partitions

FORMULA

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} sign( c(k) + c(j) + c(i) + c(n-i-j-k) ), where c is the prime characteristic (A010051).

MAPLE

b:= proc(n, i, t) option remember; series(

     `if`(n=0, t, `if`(i<1, 0, expand(x*b(n-i, min(n-i, i),

     `if`(isprime(i), 1, t)))+b(n, i-1, t))), x, 5)

    end:

a:= n-> coeff(b(n$2, 0), x, 4):

seq(a(n), n=0..60);  # Alois P. Heinz, Oct 24 2021

MATHEMATICA

Table[Sum[Sum[Sum[Sign[(PrimePi[k] - PrimePi[k - 1]) + (PrimePi[j] - PrimePi[j - 1]) + (PrimePi[i] - PrimePi[i - 1]) + (PrimePi[n - i - j - k] - PrimePi[n - i - j - k - 1])], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]

CROSSREFS

Cf. A010051, A026810, A340571.

Sequence in context: A340223 A073707 A238945 * A091609 A183563 A222706

Adjacent sequences:  A340569 A340570 A340571 * A340573 A340574 A340575

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jan 11 2021

STATUS

approved

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Last modified December 8 12:28 EST 2021. Contains 349596 sequences. (Running on oeis4.)