login
Sum of indices of primes between powers of 2.
2

%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