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