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%I #26 Apr 22 2019 17:35:04
%S 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,
%T 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,
%U 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
%N Number of partitions of n into distinct primes of the form 4k + 1.
%C a(0) = 1 corresponds to the empty partition {}.
%H Alois P. Heinz, <a href="/A024941/b024941.txt">Table of n, a(n) for n = 0..10000</a>
%e 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.
%e a(42) = 2 since 42 = 5 + 37 = 13 + 29.
%e Although 43 = 2 * 13 + 17 = 6 * 5 + 13, none of those consist of distinct primes only. Hence a(43) = 0.
%t 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 *)
%o (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
%Y Cf. A024942 (4k - 1).
%K nonn
%O 0,43
%A _Clark Kimberling_
%E Definition clarified by _Felix Fröhlich_, Apr 17 2019
%E a(0) = 1 prepended by _Joerg Arndt_, Apr 19 2019
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