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A373124 Sum of indices of primes between powers of 2.

%I A373124 #9 May 31 2024 06:12:58
%S A373124 1,2,7,11,45,105,325,989,3268,10125,33017,111435,369576,1277044,
%T A373124 4362878,15233325,53647473,189461874,676856245,2422723580,8743378141,
%U A373124 31684991912,115347765988,421763257890,1548503690949,5702720842940,21074884894536,78123777847065
%N A373124 Sum of indices of primes between powers of 2.
%C A373124 Sum of k such that 2^n+1 <= prime(k) <= 2^(n+1).
%e A373124 Row-sums of the sequence of all positive integers as a triangle with row-lengths A036378:
%e A373124    1
%e A373124    2
%e A373124    3  4
%e A373124    5  6
%e A373124    7  8  9 10 11
%e A373124   12 13 14 15 16 17 18
%e A373124   19 20 21 22 23 24 25 26 27 28 29 30 31
%e A373124   32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
%t A373124 Table[Total[PrimePi/@Select[Range[2^(n-1)+1,2^n],PrimeQ]],{n,10}]
%o A373124 (PARI) ip(n) = primepi(1<<n); \\ A007053
%o A373124 t(n) = n*(n+1)/2; \\ A000217
%o A373124 a(n) = t(ip(n+1)) - t(ip(n)); \\ _Michel Marcus_, May 31 2024
%Y A373124 For indices of primes between powers of 2:
%Y A373124 - sum A373124 (this sequence)
%Y A373124 - length A036378
%Y A373124 - min A372684 (except initial terms), delta A092131
%Y A373124 - max A007053
%Y A373124 For primes between powers of 2:
%Y A373124 - sum A293697
%Y A373124 - length A036378
%Y A373124 - min A104080 or A014210
%Y A373124 - max A014234, delta A013603
%Y A373124 For squarefree numbers between powers of 2:
%Y A373124 - sum A373123
%Y A373124 - length A077643, run-lengths of A372475
%Y A373124 - min A372683, delta A373125, indices A372540
%Y A373124 - max A372889, delta A373126, indices A143658
%Y A373124 Cf. A000040, A000120, A014499, A029837, A029931, A035100, A069010, A070939, A112925, A112926, A211997.
%K A373124 nonn
%O A373124 0,2
%A A373124 _Gus Wiseman_, May 31 2024