cp's OEIS Frontend

A371451 Number of connected components of the binary indices of the prime indices of n.

%I A371451 #11 Jan 29 2025 17:30:53
%S A371451 0,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,2,1,1,2,1,1,2,1,2,1,2,1,1,1,1,2,1,
%T A371451 2,2,1,2,1,1,1,3,1,1,1,1,1,2,1,1,1,2,1,2,1,2,2,2,1,1,1,1,2,1,1,2,1,1,
%U A371451 2,2,1,2,1,2,1,2,1,2,1,1,1,1,1,3,1,2,1,1,1,1,1,1,1,1,2,2,1,2,2,1,1,1,1,2,2
%N A371451 Number of connected components of the binary indices of the prime indices of n.
%C A371451 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.
%C A371451 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%H A371451 Antti Karttunen, <a href="/A371451/b371451.txt">Table of n, a(n) for n = 1..65537</a>
%H A371451 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.
%H A371451 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%e A371451 The binary indices of prime indices of 805 are {{1,2},{3},{1,4}}, with 2 connected components {{1,2},{1,4}} and {{3}}, so a(805) = 2.
%t A371451 csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]},If[c=={},s, csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A371451 bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A371451 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A371451 Table[Length[csm[bix/@prix[n]]],{n,100}]
%o A371451 (PARI)
%o A371451 zero_first_elem_and_bitmask_connected_elems(ys) = { my(cs = List([ys[1]]), i=1); ys[1] = 0; while(i<=#cs, for(j=2, #ys, if(ys[j]&&(0!=bitand(cs[i], ys[j])), listput(cs, ys[j]); ys[j] = 0)); i++); (ys); };
%o A371451 A371451(n) = if(1==n, 0, my(cs = apply(p -> primepi(p), factor(n)[, 1]~), s=0); while(#cs, cs = select(c -> c, zero_first_elem_and_bitmask_connected_elems(cs)); s++); (s)); \\ _Antti Karttunen_, Jan 29 2025
%Y A371451 For prime indices of prime indices we have A305079, ones A305078.
%Y A371451 Positions of ones are A325118.
%Y A371451 Positions of first appearances are A325782.
%Y A371451 For prime indices of binary indices we have A371452, ones A371291.
%Y A371451 For binary indices of binary indices we have A326753, ones A326749.
%Y A371451 A001187 counts connected graphs.
%Y A371451 A007718 counts non-isomorphic connected multiset partitions.
%Y A371451 A048143 counts connected antichains of sets.
%Y A371451 A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
%Y A371451 A070939 gives length of binary expansion.
%Y A371451 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A371451 A326964 counts connected set-systems, covering A323818.
%Y A371451 Cf. A000720, A019565, A087086, A096111, A325097, A326782, A368109, A371292, A371294, A371445, A371447.
%K A371451 nonn
%O A371451 1,6
%A A371451 _Gus Wiseman_, Apr 01 2024
%E A371451 Data section extended to a(105) by _Antti Karttunen_, Jan 29 2025