A059662 -id:A059662 - cp's OEIS frontend

A059661 Like A059459, but each term must be greater than the previous ones.

2, 3, 7, 23, 31, 4127, 4159, 20543, 134238271, 134238527, 167792959, 1241534783, 3389018431, 72108495167, 72108503359, 72108765503, 2722258935367507707706996859526254457151, 2722258935367507707708149781030861304127, 13611294676837538538536137218847444070719

Offset: 1

Antti Karttunen, Feb 03 2001

N. J. A. Sloane, Maple implementations of XORnos and DIFF

Cf. A033875, A059459, A059662.

flip_primes_asc_search := proc(a,upto_bit,upto_length) local i,n,t; if(nops(a) >= upto_length) then RETURN(a); fi; t := a[nops(a)]; for i from 0 to upto_bit do n := XORnos(t,(2^i)); if(isprime(n) and (n > t)) then print([op(a), n]); RETURN(flip_primes_asc_search([op(a), n],upto_bit,upto_length)); fi; od; RETURN([op(a),`and no more`]); end;
flip_primes_asc_search([2],512,21);

uptobit = 512; uptolength = 17; Clear[f]; f[a_] := f[a] = Module[{n, i, t}, If[Length[a] >= uptolength, Return[a]]; t = a[[-1]]; For[i = 0, i <= uptobit, i++, n = BitXor[t, 2^i]; If[PrimeQ[n] && n > t, Return[f[Append[ a, n]]]]]]; A059661 = f[{2}] (* Jean-François Alcover, Mar 07 2016, adapted from Maple *)

from sympy import isprime
from itertools import islice
def agen():
    an, bit = 2, 1
    while True:
        yield an
        while an&bit or not isprime(an+bit): bit <<= 1
        an += bit; bit = 1
print(list(islice(agen(), 17))) # Michael S. Branicky, Oct 01 2022

a(n) = 2 + Sum_{k=1..n-1} 2^A059662(k). - Pontus von Brömssen, Jan 07 2023

A139803 A033875(n) + 2^a(n) = A033875(n+1).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 3, 4, 5, 2, 7, 4, 1, 2, 3, 4, 9, 2, 955, 468

Offset: 1

Johan Särnbratt, May 22 2008

a(23) > 10^4. - Zak Seidov, Jan 24 2017

a(23) > 30000 (if it exists). - Pontus von Brömssen, Jan 08 2023

			a(10) = 4 because A033875(10) = 31, 31 + 2^4 = 47, which is prime.

Skipping from prime to prime by least powers of 2: A033875.

Cf. A059662, A067760.

p = 2; n = 0; While[true, x = 0; While[ ! PrimeQ[p + 2^x], x++ ]; p = p + 2^x; Print[x]; n++ ]

a(n) = A067760((A033875(n)-1)/2) for n >= 2. - Pontus von Brömssen, Jan 08 2023

A259630 a(n) is the smallest integer not occurring earlier such that 2^a(1) + 2^a(2) + ... + 2^a(n) is a prime.

Original entry on oeis.org

1, 0, 2, 4, 3, 12, 5, 14, 27, 8, 25, 30, 31, 36, 13, 18, 131, 60, 133, 458, 247, 1040, 21, 618, 283, 300, 209, 6282, 19107, 11792, 3401, 30214, 1211, 3044, 15989, 30194

Offset: 1

Thomas Ordowski, Sep 24 2015

Is this sequence infinite?
Associated primes: A059661.
Essentially the same as A059662: 1 followed by A059662. - R. J. Mathar, Oct 09 2015
a(37) > 145000. - Giovanni Resta, Jul 01 2019

			a(1) = 1 since 2^0 = 1 is not prime, but 2^1 = 2 is prime.
a(2) = 0 since 2^1 + 2^0 = 2 + 1 = 3 is prime.
a(3) = 2 since 2^1 + 2^0 + 2^2 = 2 + 1 + 4 = 7 is prime.

Cf. A059662, A113878.

findsm(va, n) = {m = 0; ok = 0; vs = vecsort(va); sa = sum(k=1, #va, 2^va[k]); while (!ok, if (! vecsearch(vs, m), ns = sa + 2^m; if (isprime(ns), ok = 1; break);); m++;); m;}
lista(nn) = {va = []; for (n=1, nn, m = findsm(va, n); va = concat(va, m); print1(m, ", "););} \\ Michel Marcus, Sep 26 2015

from sympy import isprime
A259630_list, A259630_set, k = [], set(), 0
while len(A259630_list) < 50:
    n, m = 0,1
    k += m
    while n in A259630_set or not isprime(k):
        n += 1
        k += m
        m *= 2
    A259630_list.append(n)
    A259630_set.add(n) # Chai Wah Wu, Jun 27 2019

a(18)-a(25) from Michel Marcus, Sep 26 2015
a(26)-a(28) from Joerg Arndt (with ispseudoprime() in Pari), Sep 28 2015
a(29)-a(34) from Chai Wah Wu, Jun 27 2019
a(35)-a(36) from Giovanni Resta, Jun 30 2019

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A059661 Like A059459, but each term must be greater than the previous ones.

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A139803 A033875(n) + 2^a(n) = A033875(n+1).

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A259630 a(n) is the smallest integer not occurring earlier such that 2^a(1) + 2^a(2) + ... + 2^a(n) is a prime.

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