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Sum of all squarefree numbers from 2^(n-1) to 2^n - 1.
11

%I #10 May 29 2024 07:05:17

%S 1,5,18,63,218,891,3676,15137,60580,238672,953501,3826167,15308186,

%T 61204878,244709252,979285522,3917052950,15664274802,62663847447,

%U 250662444349,1002632090376,4010544455838,16042042419476,64168305037147,256675237863576

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

%e This is the sequence of row sums of A005117 treated as a triangle with row-lengths A077643:

%e 1

%e 2 3

%e 5 6 7

%e 10 11 13 14 15

%e 17 19 21 22 23 26 29 30 31

%e 33 34 35 37 38 39 41 42 43 46 47 51 53 55 57 58 59 61 62

%t Table[Total[Select[Range[2^(n-1),2^n-1],SquareFreeQ]],{n,10}]

%o (PARI) a(n) = my(s=0); forsquarefree(i=2^(n-1), 2^n-1, s+=i[1]); s; \\ _Michel Marcus_, May 29 2024

%Y Counting all numbers (not just squarefree) gives A010036.

%Y For the sectioning of A005117:

%Y Row-lengths are A077643, partial sums A143658.

%Y First column is A372683, delta A373125, indices A372540, firsts of A372475.

%Y Last column is A372889, delta A373126, indices A143658, diffs A077643.

%Y For primes instead of powers of two:

%Y - sum A373197

%Y - length A373198 = A061398 - 1

%Y - maxima A112925, opposite A112926

%Y For prime instead of squarefree:

%Y - sum A293697 (except initial terms)

%Y - length A036378

%Y - min A104080 or A014210, indices A372684 (firsts of A035100)

%Y - max A014234, delta A013603

%Y A000120 counts ones in binary expansion (binary weight), zeros A080791.

%Y A005117 lists squarefree numbers, first differences A076259.

%Y A030190 gives binary expansion, reversed A030308.

%Y A070939 or (preferably) A029837 gives length of binary expansion.

%Y Cf. A372473 (firsts of A372472), A372541 (firsts of A372433).

%Y Cf. A029931, A048793, A049093, A049094, A059015, A069010, A077641.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 27 2024