A220294 -id:A220294 - cp's OEIS frontend

A255772 Start with 7; thereafter, in order of appearance, the prime factors of A220294.

7, 3, 13, 241, 97, 673, 193, 22253377, 18446744069414584321, 769, 442499826945303593556473164314770689, 349621839326921795694385454593, 331192380488114152600457428497953408512758882817, 212780015855109121

Offset: 1

Hans Havermann, Mar 06 2015

nonn

A220161(n-1) = the product of the first A255770(n) terms.

Arthur Engel, Problem-Solving Strategies, Springer, 1998, pages 121-122 (E3, said to be a "recent competition problem from the former USSR").

Hans Havermann, Table of n, a(n) for n = 1..22
factordb, First 20 factorizations of A220294

Cf. A220161, A220294, A255770, A255771.

A255771 Number of distinct prime factors of A220294(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 2, 4, 2, 2

Offset: 0

Hans Havermann, Mar 06 2015

These are the first differences of A255770.

			A220294(0) = 3 so a(0) = 1.
A220294(1) = 13 so a(1) = 1.
A220294(2) = 241 so a(2) = 1.
A220294(3) = 97*673 so a(3) = 2.
A220294(4) = 193*22253377 so a(4) = 2.

Arthur Engel, Problem-Solving Strategies, Springer, 1998, pages 121-122 (E3, said to be a "recent competition problem from the former USSR").

factordb, First 20 factorizations of A220294

Cf. A220161, A220294, A255770, A255772, A275528.

Offset changed by Arkadiusz Wesolowski, Aug 01 2016

a(9) was found in 2008 by Geoffrey Reynolds. a(10) was found by Anders Björn and Hans Riesel. - Arkadiusz Wesolowski, Aug 02 2016

A220161 a(n) = 1 + 2^(2^n) + 2^(2^(n+1)).

Original entry on oeis.org

7, 21, 273, 65793, 4295032833, 18446744078004518913, 340282366920938463481821351505477763073, 115792089237316195423570985008687907853610267032561502502920958615344897851393

Offset: 0

Michel Marcus, Dec 06 2012

nonn

For n >= 1, W. Sierpiński proves that a(n) is divisible by 21.

For n >= 1, A. Engel shows that a(n) = a(n-1) * A220294(n-1). - Hans Havermann, Mar 07 2015

Arthur Engel, Problem-Solving Strategies, Springer, 1998, pages 121-122 (E3, said to be a "recent competition problem from the former USSR").
W. Sierpiński, 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Problem #123.

G. C. Greubel, Table of n, a(n) for n = 0..10
K. Zelator, On a theorem on sums of the form 1+2^(2^n)+2^(2^n+1)+...+2^(2^n+m) and a result linking Fermat with Mersenne numbers, arXiv:0806.1514 [math.GM], 2008.

Cf. A000215, A002061, A070969, A087046, A220294, A255770, A255771, A255772.

[1 + 2^(2^n) + 2^(2^(n+1)): n in [0..10]]; // G. C. Greubel, Aug 10 2018

Table[1+2^(2^n)+2^(2^(n+1)),{n,0,7}] (* Harvey P. Dale, Dec 16 2015 *)

A220161(n):=1 + 2^(2^n) + 2^(2^(n+1))$ makelist(A220161(n),n,0,10); /* Martin Ettl, Dec 10 2012 */

vector(10, n, n--; 1 + 2^(2^n) + 2^(2^(n+1))) \\ G. C. Greubel, Aug 10 2018

def a(n): return 1 + 2**(2**n) + 2**(2**(n+1))
print([a(n) for n in range(8)]) # Michael S. Branicky, Jul 21 2021

a(n) = A000215(n+1) + A000215(n) - 1.
A070969(n) = sqrt(4*a(n) - 3). a(n+1) = a(n) * (1 + a(n) - A070969(n)) = a(n) * (1 + A087046(n+2)) hence a(n) divides a(n+1). - Michael Somos, Dec 10 2012
a(n) = A002061(A000215(n)). - Pontus von Brömssen, Aug 31 2021

A255770 Number of distinct prime factors of A220161(n).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 11, 13, 17, 19, 21

Offset: 0

Hans Havermann, Mar 06 2015

This strictly increasing sequence proves (yet again) the infinitude of primes.

			A220161(0) = 7 so a(0) = 1.
A220161(1) = 3*7 so a(1) = 2.
A220161(2) = 3*7*13 so a(2) = 3.
A220161(3) = 3*7*13*241 so a(3) = 4.
A220161(4) = 3*7*13*97*241*673 so a(4) = 6.

Arthur Engel, Problem-Solving Strategies, Springer, 1998, pages 121-122 (E3, said to be a "recent competition problem from the former USSR").

Alexander Bogomolny, Infinitude of Primes via Powers of 2
factordb, First 20 factorizations of A220161

Cf. A220161, A220294, A255771, A255772.

Offset changed by Arkadiusz Wesolowski, Aug 01 2016

A275528 Prime factors of numbers of the form 4^(2^m) - 2^(2^m) + 1 with m >= 0.

Original entry on oeis.org

3, 13, 97, 193, 241, 673, 769, 12289, 786433, 22253377, 39714817, 152371201, 597688321, 1107296257, 3221225473, 7348420609, 11560943617, 29796335617, 74490839041, 77309411329, 206158430209, 246423748609, 448203325441, 2422022479873, 5469640851457, 28114855919617

Offset: 1

Arkadiusz Wesolowski, Jul 31 2016

nonn

Primes p other than 3 such that one third of the multiplicative order of 2 (mod p) is a power of 2.

Primes in A255772 (except 7), sorted.

			3 divides 2^2 - 2^1 + 1 = 3.
13 divides 2^4 - 2^2 + 1 = 13.
97 divides 2^16 - 2^8 + 1 = 65281.
193 divides 2^32 - 2^16 + 1 = 4294901761.
241 divides 2^8 - 2^4 + 1 = 241.
673 divides 2^16 - 2^8 + 1 = 65281.
769 divides 2^128 - 2^64 + 1 = 340282366920938463444927863358058659841.
12289 divides 2^2048 - 2^1024 + 1.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..27
Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.

Cf. A220294, A255772.

forprime(p=3, 10^15, o=znorder(Mod(2, p))/3; x=ispower(2*o); if(p==3||2^(x-1)==o, print1(p, ", ")));

A002716 An infinite coprime sequence defined by recursion.

Original entry on oeis.org

3, 5, 13, 17, 241, 257, 65281, 65537, 4294901761, 4294967297, 18446744069414584321, 18446744073709551617, 340282366920938463444927863358058659841

Offset: 0

N. J. A. Sloane

nonn

Every term is relatively prime to all others. - Michael Somos, Feb 01 2004

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).

A. W. F. Edwards, Infinite coprime sequences, Math. Gaz., 48 (1964), 416-422.
A. W. F. Edwards, Infinite coprime sequences, Math. Gaz., 48 (1964), 416-422. [Annotated scanned copy]
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], 2012-2018.

Cf. A000215, A001685, A002715, A003686, A064526, A220294.

a[0] = 3; a[1] = 5;
a[n_] := a[n] = If[OddQ[n], a[n-1] + a[n-2] - 1, a[n-1]^2 - 3*a[n-1] + 3];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Aug 16 2018, after Michel Somos *)

{a(n) = if( n<2, 3 * (n>=0) + 2 * (n>0), if( n%2, a(n-1) + a(n-2) - 1, a(n-1)^2 - 3 * a(n-1) + 3))} /* Michael Somos, Feb 01 2004 */

a(2*n + 1) = a(2*n) + a(2*n - 1) - 1, a(2*n) = a(2*n - 1)^2 - 3 * a(2*n - 1) + 3, a(0) = 3, a(1) = 5. - Michael Somos, Feb 01 2004
Conjecture: a(2n+1)=A001146(n+1)+1. - R. J. Mathar, May 15 2007
a(2*n) = A220294(n). a(2*n + 1) = A000215(n+1). - Michael Somos, Dec 10 2012

More terms from Jeffrey Shallit

Edited by Michael Somos, Feb 01 2004

cp's OEIS Frontend

A255772 Start with 7; thereafter, in order of appearance, the prime factors of A220294.

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A255771 Number of distinct prime factors of A220294(n).

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A220161 a(n) = 1 + 2^(2^n) + 2^(2^(n+1)).

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A255770 Number of distinct prime factors of A220161(n).

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A275528 Prime factors of numbers of the form 4^(2^m) - 2^(2^m) + 1 with m >= 0.

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PARI

A002716 An infinite coprime sequence defined by recursion.

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Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions