A000215 -id:A000215 - cp's OEIS frontend

A066466 Numbers having just one anti-divisor.

3, 4, 6, 96, 393216

Offset: 1

Robert G. Wilson v, Jan 02 2002

See A066272 for definition of anti-divisor.
Jon Perry calls these anti-primes.
A066272(a(n)) = 1.
From Max Alekseyev, Jul 23 2007; updated Jun 25 2025: (Start)
Except for a(2) = 4, the terms of A066466 have form 2^k*p where p is odd prime and both 2^(k+1)*p-1, 2^(k+1)*p+1 are prime (i.e., twin primes). In other words, this sequence, omitting 4, is a subsequence of A040040 containing elements of the form 2^k*p with prime p.
Furthermore, since 2^(k+1)*p-1, 2^(k+1)*p+1 must equal -1 and +1 modulo 3, the number 2^(k+1)*p must be 0 modulo 3, implying that p=3. Therefore every term, except 4, must be of the form 3*2^k such that 3*2^(k+1)-1, 3*2^(k+1)+1 are twin primes. In other words, k+1 belongs to the intersection of A002253 and A002235.
According to Ballinger and Keller's lists, there are no other such k up to 22*10^6. Therefore a(6) (if it exists) is greater than 3*2^(22*10^6) ~= 10^6622660. (End)
From Daniel Forgues, Nov 23 2009: (Start)
The 2 last known anti-primes seem to relate to the Fermat primes (coincidence?):
96 = 3 * 2^5 = 3 * 2^F_1 = 3 * 2^[2^(2^1) + 1] and
393216 = 3 * 2^17 = 3 * 2^F_2 = 3 * 2^[2^(2^2) + 1],
where F_k is the k-th Fermat prime. (End)

Ray Ballinger and Wilfrid Keller, List of primes k·2^n + 1 for k < 300
Wilfrid Keller, List of primes k·2^n - 1 for k < 300
Jon Perry, The Anti-divisor [Cached copy]
Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]

Cf. A000215, A002253, A002235, A066272, A019434.

antid[n_] := Select[ Union[ Join[ Select[ Divisors[2n - 1], OddQ[ # ] && # != 1 & ], Select[ Divisors[2n + 1], OddQ[ # ] && # != 1 & ], 2n/Select[ Divisors[ 2n], OddQ[ # ] && # != 1 &]]], # < n & ]; Select[ Range[10^5], Length[ antid[ # ]] == 1 & ]

Edited by Max Alekseyev, Oct 13 2009

A070592 Largest prime factor of the n-th Fermat number F(n) = 2^(2^n) + 1.

Original entry on oeis.org

3, 5, 17, 257, 65537, 6700417, 67280421310721, 5704689200685129054721, 93461639715357977769163558199606896584051237541638188580280321

Offset: 0

Benoit Cloitre, May 12 2002

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 72.

Eric M. Schmidt, Table of n, a(n) for n = 0..11
C. L. Stewart, On Divisors of Fermat, Fibonacci, Lucas, and Lehmer Numbers, Proceedings of the London Mathematical Society, Vol. s3-35, No. 3 (1977), pp. 425-447. See p. 430.
Eric Weisstein's World of Mathematics, Fermat Number.

Cf. A000215, A006530, A050922, A093179.

a(n) = vecmax(factor(2^(2^n) + 1)[,1]); \\ Michel Marcus, Jul 05 2017

From Amiram Eldar, Oct 25 2024: (Start)
a(n) = A006530(A000215(n)).
a(n) > c * n * 2^n for n >= 1, where c is a positive absolute constant (Stewart, 1977). (End)

Offset changed by Arkadiusz Wesolowski, Jul 09 2011

A080307 Multiples of the Fermat numbers 2^(2^n)+1.

Original entry on oeis.org

3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 21, 24, 25, 27, 30, 33, 34, 35, 36, 39, 40, 42, 45, 48, 50, 51, 54, 55, 57, 60, 63, 65, 66, 68, 69, 70, 72, 75, 78, 80, 81, 84, 85, 87, 90, 93, 95, 96, 99, 100, 102, 105, 108, 110, 111, 114, 115, 117, 119, 120, 123, 125, 126, 129, 130, 132

Offset: 1

Matthew Vandermast, Feb 16 2003

Since all the Fermat numbers are relatively prime to each other (see link), the probability that a given integer is not a multiple of the first k Fermat numbers is 2^((2^k)-1) / 2^(2^k)-1, the limit of which is 0.5 as k increases infinitely; therefore the probability that an integer is a Fermat multiple, as well as the probability that it is not, is 0.5.

R. Munafo, Notes on Fermat numbers.

Cf. A000215, A080308, A080309.

A095077 Primes with four 1-bits in their binary expansion.

Original entry on oeis.org

23, 29, 43, 53, 71, 83, 89, 101, 113, 139, 149, 163, 197, 263, 269, 277, 281, 293, 337, 353, 389, 401, 449, 523, 547, 593, 643, 673, 773, 1031, 1049, 1061, 1091, 1093, 1097, 1217, 1283, 1289, 1297, 1409, 1553, 1601, 2069, 2083, 2089, 2129

Offset: 1

Antti Karttunen, Jun 01 2004

T. D. Noe, Table of n, a(n) for n = 1..1000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Subsequence of A027699. First differs from A085448 at n = 19, where a(n)=337, while A085448 continues from there with 311, whose binary expansion has six 1-bits, not four. Cf. A095057.
Cf. A000215 (primes having two bits set), A081091 (three bits set).
Cf. A264908.

Select[Prime[Range[320]], Plus@@IntegerDigits[#, 2] == 4 &] (* Alonso del Arte, Jan 11 2011 *)
Select[ Flatten[ Table[2^i + 2^j + 2^k + 1, {i, 3, 11}, {j, 2, i - 1}, {k, j - 1}]], PrimeQ] (* Robert G. Wilson v, Jul 30 2016 *)

bits1_4(x) = { nB = floor(log(x)/log(2)); z = 0;
for(i=0,nB,if(bittest(x,i),z++;if(z>4,return(0););););
if(z == 4, return(1);, return(0););};
forprime(x=17,2129,if(bits1_4(x),print1(x, ", ");););
\\ Washington Bomfim, Jan 11 2011

is(n)=isprime(n) && hammingweight(n)==4 \\ Charles R Greathouse IV, Jul 30 2016

list(lim)=my(v=List(),t); for(a=3,logint(lim\=1,2), for(b=2,a-1, for(c=1,b-1, t=1<lim, return(Vec(v))); if(isprime(t), listput(v,t))))); Vec(v) \\ Charles R Greathouse IV, Jul 30 2016

from itertools import count, islice
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A095077_gen(): # generator of terms
    return filter(isprime,map(lambda s:int('1'+''.join(s)+'1',2),(s for l in count(2) for s in multiset_permutations('0'*(l-2)+'11'))))
A095077_list = list(islice(A095077_gen(),30)) # Chai Wah Wu, Jul 19 2022

A123599 Smallest generalized Fermat prime of the form a^(2^n) + 1, where base a>1 is an integer; or -1 if no such prime exists.

Original entry on oeis.org

3, 5, 17, 257, 65537, 185302018885184100000000000000000000000000000001

Offset: 0

Alexander Adamchuk, Nov 14 2006

nonn

First 5 terms {3, 5, 17, 257, 65537} = A019434 are the Fermat primes of the form 2^(2^n) + 1. Note that for all currently known a(n) up to n = 17 last digit is 7 or 1 (except a(0) = 3 and a(1) = 5). Corresponding least bases a>1 such that a^(2^n) + 1 is prime are listed in A056993.

The last-digit behavior clearly continues since, for any a, we have that a^(2^2) will be either 0 or 1 modulo 5. So for n >= 2, a(n) is 1 or 2 modulo 5, and odd. - Jeppe Stig Nielsen, Nov 16 2020

Charles R Greathouse IV, Table of n, a(n) for n = 0..9
Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Cf. A000215, A019434, A056993.

Cf. A006093, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002.

Do[f=Min[Select[ Table[ i^(2^n) + 1, {i, 2, 500} ],PrimeQ]];Print[{n,f}],{n,0,9}]

A138888 Non-Fermat numbers.

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Omar E. Pol, Apr 03 2008

Numbers that are not members of A000215.

Cf. A000215, A001690, A018252.

Complement[Range[10000],{3,5,17,257}] (* Harvey P. Dale, Aug 04 2023 *)

def A138888(n): return n+(m:=((n-1).bit_length()-1).bit_length() if n>2 else 0)+(n+m>=(1<<(1<Chai Wah Wu, Oct 19 2024

For n > 2, a(n) = n+m+2 if n+m>=4^(2^m) and a(n) = n+m+1 otherwise where m = floor(log_2(log_2(n-1))). - Chai Wah Wu, Oct 19 2024

A001543 a(0) = 1, a(n) = 5 + Product_{i=0..n-1} a(i) for n > 0.

Original entry on oeis.org

1, 6, 11, 71, 4691, 21982031, 483209576974811, 233491495280173380882643611671, 54518278368171228201482876236565907627201914279213829353891

Offset: 0

N. J. A. Sloane

nonn

This is the special case k=5 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058 and k=2 Fermat sequence A000215. - Seppo Mustonen, Sep 04 2005

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..12
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
S. W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly 70 (1963), 403-405.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - _N. J. A. Sloane_, Jun 13 2012
S. Mustonen, On integer sequences with mutual k-residues
Seppo Mustonen, On integer sequences with mutual k-residues [Local copy]
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Column k=5 of A177888.

Flatten[{1,RecurrenceTable[{a[1]==6, a[n]==a[n-1]*(a[n-1]-5)+5}, a, {n, 1, 10}]}] (* Vaclav Kotesovec, Dec 17 2014 *)
Join[{1},NestList[#(#-5)+5&,6,10]] (* Harvey P. Dale, Oct 10 2016 *)

{
  print1("1, 6");
  n=6;
  m=Mod(5,6);
  for(i=2,9,
    n=m.mod+lift(m);
    m=chinese(m,Mod(5,n));
    print1(", "n)
  )
} \\ Charles R Greathouse IV, Dec 09 2011

a(n) = a(n-1) * (a(n-1) - 5) + 5. - Charles R Greathouse IV, Dec 09 2011

a(n) ~ c^(2^n), where c = 1.696053774403103324180661918166106455311376345474042496749974632237971081462... . - Vaclav Kotesovec, Dec 17 2014

New name from Alonso del Arte, Dec 09 2011

A001544 A nonlinear recurrence: a(n) = a(n-1)^2 - 6*a(n-1) + 6, with a(0) = 1, a(1) = 7.

Original entry on oeis.org

1, 7, 13, 97, 8833, 77968897, 6079148431583233, 36956045653220845240164417232897, 1365749310322943329964576677590044473746108255675592519835615233

Offset: 0

N. J. A. Sloane

nonn

This is the special case k=6 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058 and k=2 Fermat sequence A000215. - Seppo Mustonen, Sep 04 2005

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Indranil Ghosh, Table of n, a(n) for n = 0..11
S. W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly 70 (1963), 403-405.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - _N. J. A. Sloane_, Jun 13 2012
Seppo Mustonen, On integer sequences with mutual k-residues
Seppo Mustonen, On integer sequences with mutual k-residues [Local copy]
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Column k=6 of A177888. - Alois P. Heinz, Nov 07 2012

Flatten[{1,RecurrenceTable[{a[1]==7, a[n]==a[n-1]*(a[n-1]-6)+6}, a, {n, 1, 10}]}] (* Vaclav Kotesovec, Dec 17 2014 *)
Join[{1},NestList[#^2-6#+6&,7,10]] (* Harvey P. Dale, Nov 19 2024 *)

a(n)=if(n<1, n==0, if(n==1, 7, n=a(n-1); n^2-6*n+6))

a(n) ~ c^(2^n), where c = 1.76450357631319101484804524709844019487003729926754942591419313922841785792... . - Vaclav Kotesovec, Dec 17 2014

A070816 Solutions to phi(gpf(x)) - gpf(phi(x)) = 65534 = c are special multiples of 65537, x=65537*k, where the largest prime factors of factor k were observed in {2, 3, 5, 17, 257}.

Original entry on oeis.org

65537, 131074, 196611, 262148, 327685, 393222, 524296, 655370, 786444, 983055, 1048592, 1114129, 1310740, 1572888, 1966110, 2097184, 2228258, 2621480, 3145776, 3342387, 3932220, 4194368, 4456516, 5242960, 5570645, 6291552

Offset: 1

Labos Elemer, May 09 2002

nonn

See solutions to other even cases of c [=A070813]: A007283 for 0, A070004 for 2, A070814 for 14, A070815 for 254.

			For n = 572662306 = 2*17*257*65537, gpf(n) = 65537, phi(n) = 268435456, commutator[572662306] = phi(65537) - gpf(268435456) = 65536 - 2 = 65534.

Cf. A000010, A000215, A006530, A007283, A070002, A070003, A070004, A070777, A070812, A070813.

pf[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] Do[s=EulerPhi[pf[n]]-pf[EulerPhi[n]]; If[Equal[s, 65534], Print[{n, n/65537, pf[n/65537]}]], {n, 3, 1000000}] (* Terms of sequence are n *)

A070969 a(n) = 2^(2^n + 1) + 1.

Original entry on oeis.org

5, 9, 33, 513, 131073, 8589934593, 36893488147419103233, 680564733841876926926749214863536422913, 231584178474632390847141970017375815706539969331281128078915168015826259279873

Offset: 0

Anonymous, May 17 2002

nonn

			a(0)=5 because 2^(2^0 + 1) + 1 = 2^(1 + 1) + 1 = 2^2 + 1 = 4 + 1 = 5.

Vincenzo Librandi, Table of n, a(n) for n = 0..13

Cf. A000215, A006485, A077585.

[(2^((2^n)+1))+1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011

2^(2^Range[0, 9] + 1) + 1 (* Vladimir Joseph Stephan Orlovsky, Jun 22 2011 *)

A070969(n):=(2^(2^n+1))+1$ makelist(A070969(n),n,0,10); /* Martin Ettl, Dec 10 2012 */

a(n)=1<<(2^n+1)+1 \\ Charles R Greathouse IV, Jul 30 2011

Edited by Robert G. Wilson v, May 20 2002

cp's OEIS Frontend

A066466 Numbers having just one anti-divisor.

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A070592 Largest prime factor of the n-th Fermat number F(n) = 2^(2^n) + 1.

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A080307 Multiples of the Fermat numbers 2^(2^n)+1.

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A095077 Primes with four 1-bits in their binary expansion.

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A123599 Smallest generalized Fermat prime of the form a^(2^n) + 1, where base a>1 is an integer; or -1 if no such prime exists.

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A138888 Non-Fermat numbers.

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A001543 a(0) = 1, a(n) = 5 + Product_{i=0..n-1} a(i) for n > 0.

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A001544 A nonlinear recurrence: a(n) = a(n-1)^2 - 6*a(n-1) + 6, with a(0) = 1, a(1) = 7.

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