A059919 -id:A059919 - cp's OEIS frontend

A273945 Odd prime factors of generalized Fermat numbers of the form 3^(2^m) + 1 with m >= 0.

5, 17, 41, 193, 257, 12289, 59393, 65537, 275201, 786433, 790529, 8972801, 13631489, 21523361, 134382593, 155189249, 448524289, 524455937, 847036417, 3221225473, 12348030977, 22320686081, 77309411329, 206158430209, 4638564679681, 6597069766657, 12079910333441

Offset: 1

Arkadiusz Wesolowski, Jun 05 2016

nonn

Odd primes p such that the multiplicative order of 3 (mod p) is a power of 2.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..35
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.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=3)
Harvey Dubner and Wilfrid Keller, Factors of Generalized Fermat Numbers, Math. Comp. 64 (1995), no. 209, pp. 397-405.
OEIS Wiki, Generalized Fermat numbers
Hans Riesel, Common prime factors of the numbers A_n=a^(2^n)+1, BIT 9 (1969), pp. 264-269.

Cf. A023394, A059919, A072982, A268657, A268661, A273946 (base 5), A273947 (base 6), A273948 (base 7), A273949 (base 11), A273950 (base 12).

Select[Prime@Range[2, 10^5], IntegerQ@Log[2, MultiplicativeOrder[3, #]] &]

A059918 a(n) = (3^(2^n)-1)/2.

Original entry on oeis.org

1, 4, 40, 3280, 21523360, 926510094425920, 1716841910146256242328924544640, 5895092288869291585760436430706259332839105796137920554548480

Offset: 0

Henry Bottomley, Feb 08 2001

Denominator of b(n) where b(n) = 1/2*(b(n-1) + 1/b(n-1)), b(0)=2. - Vladeta Jovovic, Aug 15 2002

Harry J. Smith, Table of n, a(n) for n = 0..11

Cf. A059917 (numerators).
Cf. A059723, A059917, A059919, A011764.
Cf. A007089, A136308.

Array[(3^(2^#) - 1)/2 &, 8, 0] (* Michael De Vlieger, Feb 05 2022 *)

{ for (n=0, 11, write("b059918.txt", n, " ", (3^(2^n) - 1)/2); ) } \\ Harry J. Smith, Jun 30 2009

a(n) = a(n-1)*(3^(2^(n-1))+1) with a(0) = 1.
a(n) = (3^(2^n)-1)/2 = (A059723(n+1)-A059723(n))/A059723(n) = A059917(n)-1 = a(n-1)*A059919(n-1) = a(n-1)*(A011764(n-1)+1)
1 = Sum_{n>=0} 3^(2^n)/a(n+1). 1 = 3/4 + 9/40 + 81/3280 + 6561/21523360 + ...; with partial sums: 3/4, 39/40, 3279/3280, 21523359/21523360, ..., (a(n)-1)/a(n), ... . - Gary W. Adamson, Jun 22 2003
A136308(n) = A007089(a(n)). - Jason Kimberley, Dec 19 2012

A155877 Sums of three Fermat numbers.

Original entry on oeis.org

9, 11, 13, 15, 23, 25, 27, 37, 39, 51, 263, 265, 267, 277, 279, 291, 517, 519, 531, 771, 65543, 65545, 65547, 65557, 65559, 65571, 65797, 65799, 65811, 66051, 131077, 131079, 131091, 131331, 196611, 4294967303, 4294967305, 4294967307

Offset: 1

Jonathan Vos Post, Jan 29 2009

nonn

			a(1) = 3 + 3 + 3 = 9.
a(2) = 3 + 3 + 5 = 11.
a(3) = 3 + 5 + 5 = 13.
a(4) = 5 + 5 + 5 = 15.
a(5) = 3 + 3 + 17 = 23.
a(6) = 3 + 5 + 17 = 25.
a(7) = 5 + 5 + 17 = 27.
a(8) = 3 + 17 + 17 = 37.
a(9) = 5 + 17 + 17 = 39.
a(10) = 17 + 17 + 17 = 51.
a(11) = 3 + 3 + 257 = 263.

Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Cf. A000215, A019434, A050922, A051179, A059919, A063486, A073617, A078303, A078304, A085866.

{(2^(2^a) + 1) + (2^(2^b) + 1) + (2^(2^c) + 1)} = {A000215(a) + A000215(b) + A000215(c)}.

More terms from R. J. Mathar, Feb 06 2009

A275377 Number of odd prime factors (with multiplicity) of generalized Fermat number 3^(2^n) + 1.

Original entry on oeis.org

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

Offset: 0

Arkadiusz Wesolowski, Jul 25 2016

			b(n) = (3^(2^n) + 1)/2.
Complete Factorizations
b(0) = 2
b(1) = 5
b(2) = 41
b(3) = 17*193
b(4) = 21523361
b(5) = 926510094425921
b(6) = 1716841910146256242328924544641
b(7) = 257*275201*138424618868737*3913786281514524929*P21
b(8) = 12289*8972801*891206124520373602817*P90
b(9) = 134382593*22320686081*12079910333441*100512627347897906177*P93*P101

Arkadiusz Wesolowski, A 93-digit prime factor of b(9)

Cf. A059919, A273945.

a001222(n) = bigomega(n)
a059919(n) = 3^(2^n)+1
a(n) = if(n==0, 0, a001222(a059919(n))-1) \\ Felix Fröhlich, Jul 25 2016

a(n) = A001222(A059919(n)) - 1 for n > 0. - Felix Fröhlich, Jul 25 2016

a(9) was found in 2008 by Tom Womack

A152582 Numbers of the form 9^(2^n) + 2.

Original entry on oeis.org

11, 83, 6563, 43046723, 1853020188851843, 3433683820292512484657849089283, 11790184577738583171520872861412518665678211592275841109096963

Offset: 1

Cino Hilliard, Dec 08 2008

nonn

Except for the first term, this sequence is the same as A057727. There appears to be no divisibility rule for this sequence.

Cf. A059919 (-1), A011764 (-2).

g(a,n) = if(a%2,b=2,b=1);for(x=0,n,y=a^(2^x)+b;print1(y","))

cp's OEIS Frontend

A273945 Odd prime factors of generalized Fermat numbers of the form 3^(2^m) + 1 with m >= 0.

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A059918 a(n) = (3^(2^n)-1)/2.

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A155877 Sums of three Fermat numbers.

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A275377 Number of odd prime factors (with multiplicity) of generalized Fermat number 3^(2^n) + 1.

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PARI

Formula

Extensions

A152582 Numbers of the form 9^(2^n) + 2.

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