A024941 - OEIS

A024941

Number of partitions of n into distinct primes of the form 4k + 1.

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 2, 0, 0, 0, 2, 1, 0, 0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 3, 0, 1, 0, 1, 1, 0, 3, 1, 0, 0, 2, 3, 1, 1, 1, 2, 1, 0, 3, 2, 0, 0, 1, 5, 1, 0, 1, 2, 3, 1, 3, 3, 1, 0, 2, 5, 3, 1, 1, 2, 3, 2, 4, 4, 1, 1, 2, 7, 4, 1, 2, 3

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

OFFSET

0,43

COMMENTS

a(0) = 1 corresponds to the empty partition {}.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

EXAMPLE

a(41) = 1 since it can be expressed as a sum of primes of the form 4k + 1 in only one way: a trivial partition containing just itself.

a(42) = 2 since 42 = 5 + 37 = 13 + 29.

Although 43 = 2 * 13 + 17 = 6 * 5 + 13, none of those consist of distinct primes only. Hence a(43) = 0.

MATHEMATICA

searchMax = 120; primes4kp1 = Select[4Range[Floor[searchMax/4]] + 1, PrimeQ]; Table[Length[Select[IntegerPartitions[n, All, primes4kp1], DuplicateFreeQ]], {n, 0, searchMax}] (* Alonso del Arte, Apr 17 2019 *)

PROG

(PARI) { my(V=select(x->x%4==1, primes(40))); my(x='x+O('x^V[#V])); Vec(prod(k=1, #V, 1+x^V[k])) } \\ Joerg Arndt, Apr 19 2019

CROSSREFS

Cf. A024942 (4k - 1).

Sequence in context: A305566 A326813 A137347 * A219492 A285796 A317241

Adjacent sequences: A024938 A024939 A024940 * A024942 A024943 A024944

KEYWORD

nonn

AUTHOR

Clark Kimberling

EXTENSIONS

Definition clarified by Felix Fröhlich, Apr 17 2019

a(0) = 1 prepended by Joerg Arndt, Apr 19 2019

STATUS

approved