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A353401 Number of integer compositions of n with all prime run-lengths.

%I A353401 #9 May 18 2022 18:20:46
%S A353401 1,0,1,1,1,1,4,3,6,9,10,18,27,35,54,83,107,176,242,354,515,774,1070,
%T A353401 1648,2332,3429,4984,7326,10521,15591,22517,32908,48048,70044,101903,
%U A353401 149081,216973,316289,461959,672664,981356,1431256,2086901,3041577,4439226,6467735
%N A353401 Number of integer compositions of n with all prime run-lengths.
%H A353401 Alois P. Heinz, <a href="/A353401/b353401.txt">Table of n, a(n) for n = 0..1000</a>
%H A353401 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%e A353401 The a(0) = 1 through a(9) = 9 compositions (empty column indicated by dot, 0 is the empty composition):
%e A353401   0   .  11   111   22   11111   33     11122     44       333
%e A353401                                  222    22111     1133     11133
%e A353401                                  1122   1111111   3311     33111
%e A353401                                  2211             11222    111222
%e A353401                                                   22211    222111
%e A353401                                                   112211   1111122
%e A353401                                                            1112211
%e A353401                                                            1122111
%e A353401                                                            2211111
%p A353401 b:= proc(n, h) option remember; `if`(n=0, 1, add(`if`(i<>h, add(
%p A353401      `if`(isprime(j), b(n-i*j, i), 0), j=2..n/i), 0), i=1..n/2))
%p A353401     end:
%p A353401 a:= n-> b(n, 0):
%p A353401 seq(a(n), n=0..60);  # _Alois P. Heinz_, May 18 2022
%t A353401 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[Length/@Split[#],_?(!PrimeQ[#]&)]&]],{n,0,15}]
%Y A353401 The case of runs equal to 2 is A003242 aerated.
%Y A353401 The <= 1 version is A003242 ranked by A333489.
%Y A353401 The version for parts instead of run-lengths is A023360, both A353429.
%Y A353401 The version for partitions is A055923.
%Y A353401 The > 1 version is A114901, ranked by A353427.
%Y A353401 The <= 2 version is A128695, matching A335464.
%Y A353401 The > 2 version is A353400, partitions A100405.
%Y A353401 Words with all distinct run-lengths: A032020, A044813, A098859, A130091, A329739, A351013, A351017.
%Y A353401 A005811 counts runs in binary expansion.
%Y A353401 A008466 counts compositions with some part > 2.
%Y A353401 A011782 counts compositions.
%Y A353401 A167606 counts compositions with adjacent parts coprime.
%Y A353401 A329738 counts uniform compositions, partitions A047966.
%Y A353401 Cf. A078012, A165413, A175413, A274174, A333381, A333755, A353390, A353391, A353392, A353402, A353403.
%K A353401 nonn
%O A353401 0,7
%A A353401 _Gus Wiseman_, May 15 2022
%E A353401 a(21)-a(45) from _Alois P. Heinz_, May 18 2022