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%I #16 Dec 06 2016 10:58:04
%S 0,1,0,1,1,0,0,1,1,1,1,0,0,1,1,1,1,1,1,0,0,2,1,1,1,1,2,1,0,2,2,1,1,1,
%T 2,2,2,0,0,2,3,2,1,1,1,2,3,3,1,0,0,3,4,3,1,1,1,2,4,4,2,1,0,3,5,5,2,1,
%U 1,1,3,5,5,3,2,0,0,3,6,7,3,2,1,1,1,3,7,7,4,3,1,0
%N Irregular triangle read by rows: T(n,k) = number of partitions of n into prime parts in which the largest part is the k-th prime.
%H Alois P. Heinz, <a href="/A261013/b261013.txt">Rows n = 1..500, flattened</a>
%H O. P. Gupta and S. Luthra, <a href="http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_2/20005ad0_181.pdf">Partitions into primes</a>, Proc. Nat. Inst. Sci. India. Part A. 21 (1955), 181-184.
%e Triangle begins:
%e 0,
%e 1,
%e 0,1,
%e 1,0,
%e 0,1,1,
%e 1,1,0,
%e 0,1,1,1,
%e 1,1,1,0,
%e 0,2,1,1,
%e 1,1,2,1,
%e ...
%p with(numtheory):
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1)+(p-> `if`(p>n, 0, b(n-p, i)))(ithprime(i))))
%p end:
%p T:= n-> `if`(n=1, 0, seq(b(n-ithprime(k), k), k=1..pi(n))):
%p seq(T(n), n=1..25); # _Alois P. Heinz_, Aug 16 2015
%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i-1] + Function[p, If[p>n, 0, b[n-p, i]]][Prime[i]]]]; T[n_] := If[n == 1, 0, Table[b[n - Prime[k], k], {k, 1, PrimePi[n]}]]; Table[T[n], {n, 1, 25}] // Flatten (* _Jean-François Alcover_, Dec 06 2016 after _Alois P. Heinz_ *)
%Y Row sums are A000607.
%K nonn,tabf,look
%O 1,22
%A _N. J. A. Sloane_, Aug 16 2015
%E More terms from _Alois P. Heinz_, Aug 16 2015
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