A309426 - OEIS

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A309426

Number of prime parts in the partitions of n into 4 parts.

0, 0, 0, 0, 0, 1, 3, 5, 11, 12, 19, 23, 32, 36, 47, 53, 68, 77, 92, 103, 123, 134, 157, 173, 197, 216, 245, 265, 299, 323, 357, 385, 425, 454, 499, 534, 580, 619, 671, 711, 770, 816, 875, 926, 993, 1044, 1116, 1175, 1249, 1314, 1396, 1462, 1552, 1625, 1714

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,7

LINKS

Table of n, a(n) for n=0..54.

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)} (A010051(i) + A010051(j) + A010051(k) + A010051(n-i-j-k)).

EXAMPLE

Figure 1: The partitions of n into 4 parts for n = 8, 9, ..

1+1+1+9

1+1+2+8

1+1+3+7

1+1+4+6

1+1+1+8 1+1+5+5

1+1+2+7 1+2+2+7

1+1+1+7 1+1+3+6 1+2+3+6

1+1+2+6 1+1+4+5 1+2+4+5

1+1+3+5 1+2+2+6 1+3+3+5

1+1+1+6 1+1+4+4 1+2+3+5 1+3+4+4

1+1+1+5 1+1+2+5 1+2+2+5 1+2+4+4 2+2+2+6

1+1+2+4 1+1+3+4 1+2+3+4 1+3+3+4 2+2+3+5

1+1+3+3 1+2+2+4 1+3+3+3 2+2+2+5 2+2+4+4

1+2+2+3 1+2+3+3 2+2+2+4 2+2+3+4 2+3+3+4

2+2+2+2 2+2+2+3 2+2+3+3 2+3+3+3 3+3+3+3

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n | 8 9 10 11 12 ...

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a(n) | 11 12 19 23 32 ...

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- Wesley Ivan Hurt, Sep 08 2019

MATHEMATICA

Table[Sum[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) + (PrimePi[j] - PrimePi[j - 1]) + (PrimePi[k] - PrimePi[k - 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, 50}]

Table[Count[Flatten[IntegerPartitions[n, {4}]], _?PrimeQ], {n, 0, 60}] (* Harvey P. Dale, Oct 04 2019 *)

CROSSREFS

Cf. A010051, A026810.

Sequence in context: A066281 A072063 A242269 * A115398 A014597 A357369

Adjacent sequences: A309423 A309424 A309425 * A309427 A309428 A309429

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Aug 01 2019

STATUS

approved