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A078869 Number of n-tuples with elements in {2,4,6} which can occur as the differences between n+1 consecutive primes > n+1. (Values of a(11), ..., a(18) are conjectured to be correct, but are only known to be upper bounds.)

%I A078869 #18 Feb 16 2025 08:32:48
%S A078869 3,7,15,26,38,48,67,92,105,108,109,118,130,128,112,80,36,6,0,0,0,0,0,
%T A078869 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A078869 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A078869 Number of n-tuples with elements in {2,4,6} which can occur as the differences between n+1 consecutive primes > n+1. (Values of a(11), ..., a(18) are conjectured to be correct, but are only known to be upper bounds.)
%C A078869 The ">n+1" rules out n-tuples like (2,2), which only occurs for the primes 3, 5, 7. All terms from a(19) on equal 0.
%C A078869 An n-tuple (a_1,a_2,...,a_n) is counted iff the partial sums 0, a_1, a_1+a_2, ..., a_1+...+a_n do not contain a complete residue system (mod p) for any prime p.
%H A078869 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/k-TupleConjecture.html">k-Tuple Conjecture</a>
%t A078869 test[tuple_] := Module[{r, sums, i, j}, r=Length[tuple]; sums=Prepend[tuple.Table[If[j>=i, 1, 0], {i, 1, r}, {j, 1, r}], 0]; For[i=1, Prime[i]<=r+1, i++, If[Length[Union[Mod[sums, Prime[i]]]]==Prime[i], Return[False]]]; True]; tuples[0]={{}}; tuples[n_] := tuples[n]=Select[Flatten[Outer[Append, tuples[n-1], {2, 4, 6}, 1], 1], test]; a[n_] := Length[tuples[n]]
%Y A078869 The 26 4-tuples and 38 5-tuples are in A078868 and A078870. Cf. A001359, A008407, A029710, A031924, A022004-A022007, A078852, A078858, A078946-A078969, A020497.
%K A078869 nonn
%O A078869 1,1
%A A078869 _Labos Elemer_, Dec 19 2002
%E A078869 Edited by _Dean Hickerson_, Dec 20 2002