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A353429 Number of integer compositions of n with all prime parts and all prime run-lengths.

%I A353429 #12 May 18 2022 18:06:37
%S A353429 1,0,0,0,1,0,2,0,0,1,4,0,2,2,5,4,9,1,5,12,20,11,19,18,31,43,54,37,63,
%T A353429 95,121,124,154,178,261,353,393,417,565,770,952,1138,1326,1647,2186,
%U A353429 2824,3261,3917,4941,6423,7935,9719,11554,14557,18536,23380,27985
%N A353429 Number of integer compositions of n with all prime parts and all prime run-lengths.
%H A353429 Alois P. Heinz, <a href="/A353429/b353429.txt">Table of n, a(n) for n = 0..4000</a>
%e A353429 The a(13) = 2 through a(16) = 9 compositions:
%e A353429   (22333)  (77)       (555)     (3355)
%e A353429   (33322)  (2255)     (33333)   (5533)
%e A353429            (5522)     (222333)  (22255)
%e A353429            (223322)   (333222)  (55222)
%e A353429            (2222222)            (332233)
%e A353429                                 (2222233)
%e A353429                                 (2223322)
%e A353429                                 (2233222)
%e A353429                                 (3322222)
%p A353429 b:= proc(n, h) option remember; `if`(n=0, 1, add(`if`(i<>h and isprime(i),
%p A353429       add(`if`(isprime(j), b(n-i*j, i), 0), j=2..n/i), 0), i=2..n/2))
%p A353429     end:
%p A353429 a:= n-> b(n, 0):
%p A353429 seq(a(n), n=0..70);  # _Alois P. Heinz_, May 18 2022
%t A353429 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@PrimeQ/@#&&And@@PrimeQ/@Length/@Split[#]&]],{n,0,15}]
%Y A353429 The first condition only is A023360, partitions A000607.
%Y A353429 For partitions we have A351982, only run-lens A100405, only parts A008483.
%Y A353429 The second condition only is A353401, partitions A055923.
%Y A353429 A003242 counts anti-run compositions, ranked by A333489.
%Y A353429 A011782 counts compositions.
%Y A353429 A052284 counts compositions into nonprimes, partitions A002095.
%Y A353429 A106356 counts compositions by number of adjacent equal parts.
%Y A353429 A114901 counts compositions with no runs of length 1, ranked by A353427.
%Y A353429 A329738 counts uniform compositions, partitions A047966.
%Y A353429 Cf. A005811, A128695, A137200, A167606, A333755, A335464, A353428.
%K A353429 nonn
%O A353429 0,7
%A A353429 _Gus Wiseman_, May 16 2022
%E A353429 a(26)-a(56) from _Alois P. Heinz_, May 18 2022