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%I #15 Jun 05 2024 08:53:05
%S 1,3,5,7,12,19,32,55,98,173,310,565,1029,1901,3513,6543,12252,23001,
%T 43391,82026,155612,295948,564164,1077872,2063690,3957810,7603554,
%U 14630844,28192751,54400029,105097566,203280222,393615807,762939112,1480206280,2874398516,5586502349
%N Least k such that prime(k) >= 2^n.
%F a(n>1) = A007053(n) + 1.
%F a(n) = A000720(A104080(n)).
%F prime(a(n)) = A104080(n).
%F prime(a(n)) - 2^n = A092131(n).
%e The numbers prime(a(n)) together with their binary expansions and binary indices begin:
%e 2: 10 ~ {2}
%e 5: 101 ~ {1,3}
%e 11: 1011 ~ {1,2,4}
%e 17: 10001 ~ {1,5}
%e 37: 100101 ~ {1,3,6}
%e 67: 1000011 ~ {1,2,7}
%e 131: 10000011 ~ {1,2,8}
%e 257: 100000001 ~ {1,9}
%e 521: 1000001001 ~ {1,4,10}
%e 1031: 10000000111 ~ {1,2,3,11}
%e 2053: 100000000101 ~ {1,3,12}
%e 4099: 1000000000011 ~ {1,2,13}
%e 8209: 10000000010001 ~ {1,5,14}
%e 16411: 100000000011011 ~ {1,2,4,5,15}
%e 32771: 1000000000000011 ~ {1,2,16}
%e 65537: 10000000000000001 ~ {1,17}
%e 131101: 100000000000011101 ~ {1,3,4,5,18}
%e 262147: 1000000000000000011 ~ {1,2,19}
%e 524309: 10000000000000010101 ~ {1,3,5,20}
%e 1048583: 100000000000000000111 ~ {1,2,3,21}
%e 2097169: 1000000000000000010001 ~ {1,5,22}
%e 4194319: 10000000000000000001111 ~ {1,2,3,4,23}
%e 8388617: 100000000000000000001001 ~ {1,4,24}
%t Table[PrimePi[If[n==1,2,NextPrime[2^n]]],{n,30}]
%o (PARI) a(n) = primepi(nextprime(2^n)); \\ _Michel Marcus_, May 31 2024
%Y The opposite (greatest k such that prime(k) <= 2^n) is A007053.
%Y Positions of first appearances in A035100.
%Y The distance from prime(a(n)) to 2^n is A092131.
%Y Counting zeros instead of all bits gives A372474, firsts of A035103.
%Y Counting ones instead of all bits gives A372517, firsts of A014499.
%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 For squarefree numbers between primes:
%Y - sum A373197
%Y - length A373198 = A061398 - 1
%Y - min A000040
%Y - max A112925, opposite A112926
%Y Cf. A000120, A029837, A029931, A030190, A049095, A069010, A070939, A077641, A080791, A211997.
%K nonn
%O 1,2
%A _Gus Wiseman_, May 30 2024
%E More terms from _Michel Marcus_, May 31 2024
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