A203074 - OEIS

A203074

a(0)=1; for n > 0, a(n) = next prime after 2^(n-1).

1, 2, 3, 5, 11, 17, 37, 67, 131, 257, 521, 1031, 2053, 4099, 8209, 16411, 32771, 65537, 131101, 262147, 524309, 1048583, 2097169, 4194319, 8388617, 16777259, 33554467, 67108879, 134217757, 268435459, 536870923, 1073741827, 2147483659

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OFFSET

0,2

COMMENTS

Equals {1} union A014210. Unlike A014210, every positive integer can be written in one or more ways as a sum of terms of this sequence. See A203075, A203076.

LINKS

M. F. Hasler & Bill McEachen, Table of n, a(n) for n = 0..1300 (missing lines n = 1159..1165 from Bill McEachen)

Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - N. J. A. Sloane, May 20 2023]

FORMULA

A203074(n) = 2^(n-1) + A013597(n-1), for n > 0. - M. F. Hasler, Mar 15 2012

a(n) = A104080(n-1) for n > 2. - Georg Fischer, Oct 23 2018

EXAMPLE

a(5) = 17, since this is the next prime after 2^(5-1) = 2^4 = 16.

MATHEMATICA

nextprime[n_Integer] := (k=n+1; While[!PrimeQ[k], k++]; k); aprime[m_Integer] := (If[m==0, 1, nextprime[2^(m-1)]]); Table[aprime[l], {l, 0, 100}]

nxt[{n_, a_}]:={n+1, NextPrime[2^n]}; NestList[nxt, {0, 1}, 40][[All, 2]] (* Harvey P. Dale, Oct 10 2017 *)

PROG

(PARI) a(n)=if(n, nextprime(2^n/2+1), 1) \\ Charles R Greathouse IV

(PARI) A203074(n)=nextprime(2^(n-1)+1)-!n \\ M. F. Hasler, Mar 15 2012

(Magma) [1] cat [NextPrime(2^(n-1)): n in [1..40]]; // Vincenzo Librandi, Feb 23 2018