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Irregular triangle read by rows where row n lists the binary indices of the n-th prime number.
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%I #7 May 10 2024 09:27:40

%S 2,1,2,1,3,1,2,3,1,2,4,1,3,4,1,5,1,2,5,1,2,3,5,1,3,4,5,1,2,3,4,5,1,3,

%T 6,1,4,6,1,2,4,6,1,2,3,4,6,1,3,5,6,1,2,4,5,6,1,3,4,5,6,1,2,7,1,2,3,7,

%U 1,4,7,1,2,3,4,7,1,2,5,7,1,4,5,7,1,6,7

%N Irregular triangle read by rows where row n lists the binary indices of the n-th prime number.

%C A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

%e We have prime(12) = (2^1 + 2^3 + 2^6)/2, so row 12 is (1,3,6).

%e Each prime followed by its binary indices:

%e 2: 2

%e 3: 1 2

%e 5: 1 3

%e 7: 1 2 3

%e 11: 1 2 4

%e 13: 1 3 4

%e 17: 1 5

%e 19: 1 2 5

%e 23: 1 2 3 5

%e 29: 1 3 4 5

%e 31: 1 2 3 4 5

%e 37: 1 3 6

%e 41: 1 4 6

%e 43: 1 2 4 6

%e 47: 1 2 3 4 6

%t Table[Join@@Position[Reverse[IntegerDigits[Prime[n],2]],1],{n,15}]

%Y Row lengths are A014499.

%Y Second column is A023506(n) + 1.

%Y Final column is A035100.

%Y Prime-indexed rows of A048793.

%Y Row-sums are A372429, restriction of A029931 (sum of binary indices).

%Y A019565 gives Heinz number of binary indices, adjoint A048675.

%Y A029837 gives greatest binary index, least A001511.

%Y A048793 lists binary indices, length A000120, reverse A272020.

%Y A070939 gives length of binary expansion.

%Y Cf. A000040, A005940, A056239, A071814, A096111, A191232, A230877, A231204, A372427-A372442.

%K nonn,tabf,base

%O 1,1

%A _Gus Wiseman_, May 07 2024