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A373123 Sum of all squarefree numbers from 2^(n-1) to 2^n - 1.

%I A373123 #10 May 29 2024 07:05:17
%S A373123 1,5,18,63,218,891,3676,15137,60580,238672,953501,3826167,15308186,
%T A373123 61204878,244709252,979285522,3917052950,15664274802,62663847447,
%U A373123 250662444349,1002632090376,4010544455838,16042042419476,64168305037147,256675237863576
%N A373123 Sum of all squarefree numbers from 2^(n-1) to 2^n - 1.
%e A373123 This is the sequence of row sums of A005117 treated as a triangle with row-lengths A077643:
%e A373123    1
%e A373123    2   3
%e A373123    5   6   7
%e A373123   10  11  13  14  15
%e A373123   17  19  21  22  23  26  29  30  31
%e A373123   33  34  35  37  38  39  41  42  43  46  47  51  53  55  57  58  59  61  62
%t A373123 Table[Total[Select[Range[2^(n-1),2^n-1],SquareFreeQ]],{n,10}]
%o A373123 (PARI) a(n) = my(s=0); forsquarefree(i=2^(n-1), 2^n-1, s+=i[1]); s; \\ _Michel Marcus_, May 29 2024
%Y A373123 Counting all numbers (not just squarefree) gives A010036.
%Y A373123 For the sectioning of A005117:
%Y A373123 Row-lengths are A077643, partial sums A143658.
%Y A373123 First column is A372683, delta A373125, indices A372540, firsts of A372475.
%Y A373123 Last column is A372889, delta A373126, indices A143658, diffs A077643.
%Y A373123 For primes instead of powers of two:
%Y A373123 - sum A373197
%Y A373123 - length A373198 = A061398 - 1
%Y A373123 - maxima A112925, opposite A112926
%Y A373123 For prime instead of squarefree:
%Y A373123 - sum A293697 (except initial terms)
%Y A373123 - length A036378
%Y A373123 - min A104080 or A014210, indices A372684 (firsts of A035100)
%Y A373123 - max A014234, delta A013603
%Y A373123 A000120 counts ones in binary expansion (binary weight), zeros A080791.
%Y A373123 A005117 lists squarefree numbers, first differences A076259.
%Y A373123 A030190 gives binary expansion, reversed A030308.
%Y A373123 A070939 or (preferably) A029837 gives length of binary expansion.
%Y A373123 Cf. A372473 (firsts of A372472), A372541 (firsts of A372433).
%Y A373123 Cf. A029931, A048793, A049093, A049094, A059015, A069010, A077641.
%K A373123 nonn
%O A373123 1,2
%A A373123 _Gus Wiseman_, May 27 2024