A006285 -id:A006285 - cp's OEIS frontend

A232460 a(n) = 2^(2^n) - 5.

-3, -1, 11, 251, 65531, 4294967291, 18446744073709551611, 340282366920938463463374607431768211451, 115792089237316195423570985008687907853269984665640564039457584007913129639931

Offset: 0

Arkadiusz Wesolowski, Nov 24 2013

For n >= 3, a(n) is not of the form 2^k + p, where p is a prime. Therefore every term greater than 11 is in A006285 (de Polignac numbers).

Wacław Sierpiński, Elementary Theory of Numbers, Monografie Matematyczne 42 (1964), p. 415.

Cf. A006285.

Magma
```
[2^(2^n)-5 : n in [0..8]]
```
Mathematica
```
Table[2^(2^n) - 5, {n, 0, 8}]
```
PARI
```
for(n=0, 8, print1(2^(2^n)-5, ", "));
```

def A232460(n): return (1<<(1<Chai Wah Wu, Jul 19 2022

a(n) = A000215(n) - 6.

a(0) = - 3; a(n) = (a(n-1) + 5)^2 - 5, n >= 1.

A232565 a(n) is the smallest k such that 2^(2^n) - 2^k - 1 is prime, or -1 if no such k exists.

Original entry on oeis.org

0, 1, 2, 4, 2, 8, 18, 76, 32, 151, 692, 592, 154, 580, 27365, 11267

Offset: 1

Charles R Greathouse IV, Nov 26 2013

Crocker showed that 2^(2^n) - 1 - 2^a - 2^b is not prime (with n > 2) if a and b are distinct. This sequence demonstrates that the theorem is sharp in the sense that distinctness is required.

If n > 2, then the (largest) prime P(n) = 2^(2^n)-2^a(n)-1 is a de Polignac number (A065381); i.e., P(n)-2^m is not prime. It seems that if n > 6, then |P(n)-2^m| is composite for every natural m and P(n)*2^m-1 is composite (by the dual Riesel conjecture). So if n > 6, then the prime P(n) may be a Riesel number (A182296). For example, the prime P(7) = 2^(2^7)-(2^18+1) is the first candidate (note that 2^18+1 is the smallest de Polignac number of form 2^k+1). Also, by Crocker's theorem, the smallest number of form 2^(2^n)-2^m-1, namely 2^(2^n-1)-1 is a de Polignac number (A006285) and for n > 6 may be a dual Riesel number (A101036). For example, the double Mersenne prime 2^(2^7-1)-1 probably is a dual Riesel number. It is not known whether these are Riesel numbers with a covering set. - Thomas Ordowski, Jan 24 2024

Roger Crocker, "On the sum of a prime and of two powers of two", Pacific Journal of Mathematics 36:1 (1971), pp. 103-107.

Cf. A156695.

a(n)=my(N=2^2^n-1);for(a=1,2^n-1,if(ispseudoprime(N-2^a), return(a)));0

a(15) from Charles R Greathouse IV, Dec 02 2013
a(16) from Daniel Suteu, Oct 11 2020
Name edited by Thomas Ordowski, Jan 24 2024

A337487 De Polignac numbers k > 1 such that k - 2^m is a de Polignac number for every 1 < 2^m < k.

Original entry on oeis.org

1117175145, 2544265305, 3147056235, 3366991695, 3472109835, 3621922845, 3861518805, 4447794915, 4848148485, 5415281745, 5693877405, 7525056375, 7602256605, 9055691835, 9217432215, 13431856995, 16819230075, 19373391165, 21468020835, 24358769685, 27002844795, 30252463305, 33359739795

Offset: 1

Thomas Ordowski, Aug 29 2020

nonn

Odd integers k > 3 that are not of the form p + 2^m + 2^n with m,n >= 0, where p is a prime.
These are de Polignac numbers k > 1 in A156695. Numbers k in A156695 such that k - 2 is composite.
Problem: are there infinitely many such numbers?

Amiram Eldar, Table of n, a(n) for n = 1..204 (terms below 10^12, calculated using the b-file at A156695)

An intersection of A006285 > 1 and A156695 (with m,n >= 1).

A156695 = Cases[Import["https://oeis.org/A156695/b156695.txt", "Table"], {, }][[;; , 2]]; dePolQ[n_] := n > 3 && AllTrue[n - 2^Range[Floor[Log[2, n]]], !PrimeQ[#] &]; Select[A156695, dePolQ] (* Amiram Eldar, Aug 29 2020 *)

A212292 Odd numbers not of the form p^2 + q^2 + r with p, q, and r prime.

Original entry on oeis.org

1, 3, 5, 7, 9, 17, 33, 43, 83, 179, 623, 713, 1019

Offset: 1

Charles R Greathouse IV, Jun 21 2012

The corresponding sequence with the restriction to primes removed is empty.
Wang shows that all but x^{9/20+e} members of this sequence up to x are congruent to 2 mod 3, for any e > 0.
There are no more terms < 10^7. - Donovan Johnson, Jun 27 2012
There are no more terms < 4*10^9. - Jud McCranie, Jun 09 2013
There are no more terms < 10^11. - Giovanni Resta, Jun 09 2013

Wang Mingqiang, On sums of a prime, and a square of prime, and a k-power of prime, Northeastern Mathematical Journal 18:4 (2002), pp. 283-286.

Cf. A045636, A006285, A118955, A156695, A226484.

list(lim)=my(p1=vector(primepi(sqrt(lim-5.5)),i,prime(i)^2), p2=List(), v=List(), u=List([1,3,5,7,9]), t); for(i=1,#p1, for(j=i,#p1,t=p1[i]+p1[j]; if(t>lim, break, listput(p2,t)))); p2=vecsort(Vec(p2),,8); for(i=1,#p2,forprime(p=2,lim-p2[i],listput(v,p2[i]+p))); v=select(n->n%2, vecsort(Vec(v),,8)); for(i=2,#v,forstep(j=v[i-1]+2,v[i]-2,2,listput(u,j))); Vec(u)

A252168 Smallest k > 0 such that |(2n-1) - 2^k| is prime, or -1 if no such k exists.

Original entry on oeis.org

2, 3, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 1, 2, 4, 1, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 4, 1, 2, 4, 3, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 3, 4, 4, 47, 1, 2, 1, 2, 6, 1, 1, 2, 3, 3, 8, 1, 1, 2, 3, 1, 2, 5, 1, 2, 1, 2, 4

Offset: 1

Eric Chen, Dec 14 2014

It is known that a(254602) = -1, because |509203-2^k| is always divisible by 3, 5, 7, 13, 17, or 241. a(1147) is the first unknown term.
a((A101036(n)+1)/2) = -1, so there are infinitely many n such that a(n) = -1.
a((A133122(n)+1)/2) = A096502((A133122(n)-1)/2).

			a(12) = 2 because 2*12-1 = 23 and that 23-2^1 = 21 is not prime but 23-2^2 = 19 is.
a(69) = 6 because 2*69-1 = 137, |137-2^k| is composite for k = 1, 2, 3, 4, 5 and prime for k = 6.
Even the smallest k can be also very large. For example, a(169) = 791.
a(1147) > 65536.

Jinyuan Wang, Table of n, a(n) for n = 1..1146

Cf. A006285, A096502, A133122, A188903.

Cf. A046067, A046069, A067760.

Table[k = 1; While[!PrimeQ[Abs[(2*n-1) - 2^k]], k++]; k, {n, 1, 1000}]

A252168(n)={ my(k=1); n=2*n-1; while(!ispseudoprime(abs(n-2^k)), k++); k }

a(19) corrected by Jinyuan Wang, Mar 25 2023

A256237 Primes p such that for all 2^k < p the numbers p + 2^k, p - 2^k, p2^k + 1, and p2^k - 1 are composite.

Original entry on oeis.org

8923, 24943, 35437, 42533, 52783, 83437, 105953, 116437, 126631, 133241, 145589, 164729, 172331, 192173, 204013, 215279, 254329, 304709, 308899, 398833, 430499, 436687, 454351, 476869, 479909, 483443, 497597, 522479, 527729, 529103, 545257, 561439, 562651

Offset: 1

Arkadiusz Wesolowski, Mar 20 2015

nonn

Subsequence of A256163.

Cf. A006285, A065381, A153352, A180247, A255967.

lst:=[]; for p in [3..562651 by 2] do if IsPrime(p) then t:=0; k:=0; while 2^k lt p do if IsPrime(p-2^k) or IsPrime(p+2^k) or IsPrime(p*2^k-1) or IsPrime(p*2^k+1) then t:=1; break; end if; k+:=1; end while; if IsZero(t) then Append(~lst, p); end if; end if; end for; lst;

A276417 a(n) = least positive k such that (2*n + 1) - 2^k is prime, or 0 if no such k exists.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 1, 2, 4, 1, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 4, 1, 2, 4, 3, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 3, 4, 4, 0, 1, 2, 1, 2, 6, 1, 1, 2, 3, 3, 0, 1, 1, 2, 3, 1, 2, 5, 1, 2, 1, 2, 4

Offset: 0

Arkadiusz Wesolowski, Sep 02 2016

nonn

a(n) = 1 iff n is in A006254. - Robert Israel, Sep 02 2016

For n > 1, a(n) = 0 iff 2n+1 is de Polignac number, A006285. - Thomas Ordowski, Apr 13 2017

			a(14) = 4 because (2*14 + 1) - 2^k is composite for k = 1, 2, 3 and prime for k = 4.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A006254, A006285, A101036, A133122, A188903, A232460, A252168.

lst:=[]; for n in [1..173 by 2] do k:=0; c:=k; repeat k+:=1; c+:=1; a:=n-2^k; until a lt 1 or IsPrime(a); if a lt 1 then Append(~lst, 0); else Append(~lst, c); end if; end for; lst;

f:= proc(n) local k;
     for k from 1 do
       if 2*n < 2^k then return 0
       elif isprime(2*n+1-2^k) then return k
       fi
     od
end proc:
map(f, [$0..100]); # Robert Israel, Sep 02 2016

Table[If[n <= 2, 0, k = 1; While[! PrimeQ[2 n + 1 - 2^k], k++]; k], {n, 0, 120}] (* Michael De Vlieger, Sep 03 2016 *)

a(n) = my(k=1); while(2^k < 2*n+1, if(ispseudoprime((2*n+1)-2^k), return(k)); k++); return(0) \\ Felix Fröhlich, Sep 02 2016

If A188903(n) >= 2, then a(n) = log_2(A188903(n)), otherwise a(n) = 0.

A350629 Positive even numbers that cannot be written as 3^i (i >= 1) plus a prime.

Original entry on oeis.org

2, 4, 18, 24, 36, 42, 48, 54, 60, 66, 72, 78, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 252, 258, 264, 270, 276, 282, 288, 294, 300, 306, 312, 318, 324, 328, 330, 336, 342, 348, 354, 360, 366

Offset: 1

N. J. A. Sloane, Feb 05 2022

nonn

Indices of zeros in A350628 multiplied by 2.

Cf. A006285, A058517, A350628, A350630, A350957, A350958.

A350959 Number of ways to write 2*n+1 as 2^i (i >= 0) plus a prime.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 3, 1, 3, 4, 2, 2, 3, 1, 2, 3, 3, 2, 4, 1, 2, 5, 2, 3, 3, 1, 3, 2, 1, 3, 4, 1, 2, 5, 2, 2, 6, 3, 2, 3, 3, 2, 4, 1, 3, 3, 2, 1, 3, 2, 2, 6, 2, 2, 5, 2, 3, 4, 1, 2, 2, 2, 0, 3, 3, 3, 5, 1, 4, 4, 3, 3, 4, 0, 2, 4, 2, 2, 4, 2, 1, 5, 3, 3, 5, 2, 3, 2, 1, 4, 4, 1, 2, 4, 1, 1, 6, 2, 4, 5

Offset: 0

N. J. A. Sloane, Feb 06 2022, answering a question from David W. Lewis

nonn

Cf. A006285, A350957, A109925.

a(n) = A109925(2*n+1). - R. J. Mathar, Mar 07 2022

A006286 Numbers not of form p + 2^x + 2^y.

Original entry on oeis.org

1, 2, 3, 128, 150, 252, 332, 338, 374, 510, 600, 702, 758, 810, 878, 906, 908, 960, 978, 998, 1020, 1088, 1200, 1208, 1212, 1244, 1260, 1272, 1478, 1530, 1542, 1550, 1590, 1598, 1620, 1650, 1658, 1720, 1760, 1778, 1784, 1808, 1830, 1860, 1868

Offset: 1

N. J. A. Sloane

Jeffrey Shallit, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Sean A. Irvine, Table of n, a(n) for n = 1..1000
Thomas Bloom, Problem 9, Erdős Problems.
Roger Crocker, On the sum of a prime and of two powers of two, Pacific J. Math. 36(1): 103-107 (1971).
Hao Pan, On the integers not of the form {$p+2^a+2^b$},, arXiv:0905.3809 [math.NT], 2009.
Terence Tao, Erdős problem database, see entry no. 9.

Cf. A006285, A156695.

More terms from Sean A. Irvine, Feb 22 2017

cp's OEIS Frontend

A232460 a(n) = 2^(2^n) - 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

A232565 a(n) is the smallest k such that 2^(2^n) - 2^k - 1 is prime, or -1 if no such k exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A337487 De Polignac numbers k > 1 such that k - 2^m is a de Polignac number for every 1 < 2^m < k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A212292 Odd numbers not of the form p^2 + q^2 + r with p, q, and r prime.

Views

Author

Keywords

Comments

References

Crossrefs

Programs

PARI

A252168 Smallest k > 0 such that |(2n-1) - 2^k| is prime, or -1 if no such k exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A256237 Primes p such that for all 2^k < p the numbers p + 2^k, p - 2^k, p*2^k + 1, and p*2^k - 1 are composite.

Views

Author

Keywords

Crossrefs

Programs

Magma

A276417 a(n) = least positive k such that (2*n + 1) - 2^k is prime, or 0 if no such k exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A350629 Positive even numbers that cannot be written as 3^i (i >= 1) plus a prime.

Views

Author

Keywords

Comments

A256237 Primes p such that for all 2^k < p the numbers p + 2^k, p - 2^k, p2^k + 1, and p2^k - 1 are composite.