A372684 Least k such that prime(k) >= 2^n.

1, 3, 5, 7, 12, 19, 32, 55, 98, 173, 310, 565, 1029, 1901, 3513, 6543, 12252, 23001, 43391, 82026, 155612, 295948, 564164, 1077872, 2063690, 3957810, 7603554, 14630844, 28192751, 54400029, 105097566, 203280222, 393615807, 762939112, 1480206280, 2874398516, 5586502349

Offset: 1

Author

Gus Wiseman, May 30 2024

Keywords

• nonn

Examples

			The numbers prime(a(n)) together with their binary expansions and binary indices begin:
        2:                       10 ~ {2}
        5:                      101 ~ {1,3}
       11:                     1011 ~ {1,2,4}
       17:                    10001 ~ {1,5}
       37:                   100101 ~ {1,3,6}
       67:                  1000011 ~ {1,2,7}
      131:                 10000011 ~ {1,2,8}
      257:                100000001 ~ {1,9}
      521:               1000001001 ~ {1,4,10}
     1031:              10000000111 ~ {1,2,3,11}
     2053:             100000000101 ~ {1,3,12}
     4099:            1000000000011 ~ {1,2,13}
     8209:           10000000010001 ~ {1,5,14}
    16411:          100000000011011 ~ {1,2,4,5,15}
    32771:         1000000000000011 ~ {1,2,16}
    65537:        10000000000000001 ~ {1,17}
   131101:       100000000000011101 ~ {1,3,4,5,18}
   262147:      1000000000000000011 ~ {1,2,19}
   524309:     10000000000000010101 ~ {1,3,5,20}
  1048583:    100000000000000000111 ~ {1,2,3,21}
  2097169:   1000000000000000010001 ~ {1,5,22}
  4194319:  10000000000000000001111 ~ {1,2,3,4,23}
  8388617: 100000000000000000001001 ~ {1,4,24}
		

Crossrefs

The opposite (greatest k such that prime(k) <= 2^n) is A007053.

Positions of first appearances in A035100.

The distance from prime(a(n)) to 2^n is A092131.

Counting zeros instead of all bits gives A372474, firsts of A035103.

Counting ones instead of all bits gives A372517, firsts of A014499.

For primes between powers of 2:

- sum A293697

- length A036378

- min A104080 or A014210

- max A014234, delta A013603

For squarefree numbers between powers of 2:

- sum A373123

- length A077643, run-lengths of A372475

- min A372683, delta A373125, indices A372540

- max A372889, delta A373126, indices A143658

For squarefree numbers between primes:

- sum A373197

- length A373198 = A061398 - 1

- min A000040

- max A112925, opposite A112926

Cf. A000120, A029837, A029931, A030190, A049095, A069010, A070939, A077641, A080791, A211997.

Programs

Mathematica

Table[PrimePi[If[n==1,2,NextPrime[2^n]]],{n,30}]

PARI

a(n) = primepi(nextprime(2^n)); \\ Michel Marcus, May 31 2024

Formula

a(n>1) = A007053(n) + 1.

a(n) = A000720(A104080(n)).

prime(a(n)) = A104080(n).

prime(a(n)) - 2^n = A092131(n).

Extensions

More terms from Michel Marcus, May 31 2024