A007516 -id:A007516 - cp's OEIS frontend

A004249 a(n) = (2^2^...^2) (with n 2's) + 1.

2, 3, 5, 17, 65537

Offset: 0

N. J. A. Sloane, Robert G. Wilson v, David W. Wilson

nonn

a(0) could equally well be taken to be 1 rather than 2, which gives A007516. - N. J. A. Sloane, Sep 14 2009
A subsequence of the Fermat numbers 2^2^n + 1 = A000215.
a(0) through a(4) are primes; a(5) = 2^65536 + 1 is divisible by 825753601.
a(5) = 20035299...19156737 has 19729 decimal digits. - Alois P. Heinz, Jun 15 2022
It is unknown if a(6) = A000215(65536) is composite. - Jeppe Stig Nielsen, Jun 15 2022

P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 73.

Y. Bugeaud and M. Queffélec, On Rational Approximation of the Binary Thue-Morse-Mahler Number, Journal of Integer Sequences, 16 (2013), #13.2.3.
Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m

Cf. Fermat numbers 2^2^n + 1 = A000215. A007516 is another version.

a(0) = 2, a(n) = 2^a(n-1)/2 + 1 for n >= 1.

a(n) = A014221(n) + 1. - Leroy Quet, Jun 10 2009, updated by Jeppe Stig Nielsen, Jun 15 2022

A066814 Smallest prime p such that (p-1) has n divisors, or 0 if no such prime exists.

Original entry on oeis.org

2, 3, 5, 7, 17, 13, 0, 31, 37, 113, 0, 61, 0, 193, 401, 211, 65537, 181, 0, 241, 577, 13313, 0, 421, 1297, 12289, 4357, 2113, 0, 1009, 0, 1321, 25601, 2424833, 752734097, 1801, 0, 786433, 495617, 2161, 0, 4801, 0, 15361, 7057, 155189249, 0

Offset: 1

Wouter Meeussen, Jan 20 2002

nonn

The only primes p for which p-1 has a prime number of divisors are Fermat primes A019434.

			a(17)=65537 because DivisorSigma[0,65536]=17.

Cf. A004249, A007516, A066529.

it=Table[ p=Prime[ n ]; DivisorSigma[ 0, p-1 ], {n, 400000} ]; Flatten[ Position[ it, #, 1, 1 ]&/@Range[ 100 ]/.{}- > 0 ]

Comment clarified by T. D. Noe, Nov 06 2009

Edited by Max Alekseyev, Nov 10 2009

A258429 Primes p such that p - 1 = (tau(p - 1) - 1)^k for some k >= 0, where tau(n) is the number of divisors of n (A000005).

Original entry on oeis.org

2, 5, 17, 65537

Offset: 1

Jaroslav Krizek, May 29 2015

nonn

Conjecture: the sequence is finite.
Corresponding values of numbers k: 0, 2, 2, 4, ...
A Fermat prime from A019434 of the form F(n) = 2^(2^n) + 1 is a term if k = 2^n * log(2) / log(2^n) is an integer.

			65537 (prime) is in the sequence because 65537 - 1 = (tau(65536) - 1)^4 = 16^4.

Cf. A000005, A019434, A219338, A007516, A004249, A249759.

[2] cat [n+1: n in [A219338(n)] | IsPrime(n+1)];

Set(Sort([n: n in[1..1000000], k in [0..100] | IsPrime(n) and (n-1) eq (NumberOfDivisors(n-1) - 1)^k]));

listp(nn) = {print1(p=2, ", "); forprime(p=5, nn, expo = valuation(x=(p-1), y=(numdiv(p-1)-1)); if (x == y^expo, print1(p, ", ")););} \\ Michel Marcus, Jun 04 2015

A066808 a(n) = F(n)-1 mod 2^n+1 with F(n) = n-th Fermat number = 1+2^2^n.

Original entry on oeis.org

0, 1, 1, 4, 1, 4, 16, 4, 1, 256, 16, 4, 4081, 4, 16, 256, 1, 4, 261121, 4, 65536, 256, 16, 4, 65536, 33554305, 16, 67108864, 65536, 4, 16, 4, 1, 256, 16, 262144, 68451041281, 4, 16, 256, 65536, 4, 4398042316801, 4, 65536, 35184371957761, 16, 4, 281474976645121

Offset: 0

Wouter Meeussen, Jan 19 2002

All terms except n=12,18,25,36,42,45,48,55 result in a(n) that are powers of 2, whereas these exceptions (4081, 261121, 33554305, 68451041281, 4398042316801, 35184371957761, 281474976645121, 36020000925941761) are all odd.

Alois P. Heinz, Table of n, a(n) for n = 0..3323
Chris Caldwell, Fermat Number, The Prime Glossary.
Eric Weisstein's World of Mathematics, Fermat Number.

Cf. A004249, A007516, A000215, A019434.

a:= n-> 2&^(2^n) mod (2^n+1):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 04 2022

Table[ PowerMod[ 2, 2^n, 2^n+1 ], {n, 64} ]

F(n)-1=1 mod (2^n+1) for all n=2^k because F(n)=2+ F(1)F(2)..F(n-1)

a(0)=0 prepended by Alois P. Heinz, Jul 04 2022

A333132 a(n) = n for n <= 3; thereafter a(n) = 2^(a(n-1)-1) + a(n-1).

Original entry on oeis.org

1, 2, 3, 7, 71, 1180591620717411303495

Offset: 1

Ilya Gutkovskiy, Mar 08 2020

nonn

The next term is too large to include.

a(n) = number of compositions of a(1) + number of compositions of a(2) + ... + number of compositions of a(n-1) for n > 2.

			a(5) = 71, 71 in base 2 (reverse order of digits) = 1110001.
                                                    |||   |
                                                    123   7

Cf. A004249, A007516, A011782, A014221, A034797, A103527.

a[n_] := a[n] = If[n <= 3, n, 2^(a[n - 1] - 1) + a[n - 1]]; Table[a[n], {n, 1, 6}]

a(n) = n for n <= 2; thereafter a(n) = Sum_{k=1..n-1} 2^(a(k)-1).

cp's OEIS Frontend

A004249 a(n) = (2^2^...^2) (with n 2's) + 1.

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A066814 Smallest prime p such that (p-1) has n divisors, or 0 if no such prime exists.

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A258429 Primes p such that p - 1 = (tau(p - 1) - 1)^k for some k >= 0, where tau(n) is the number of divisors of n (A000005).

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A066808 a(n) = F(n)-1 mod 2^n+1 with F(n) = n-th Fermat number = 1+2^2^n.

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A333132 a(n) = n for n <= 3; thereafter a(n) = 2^(a(n-1)-1) + a(n-1).

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