A273949 -id:A273949 - cp's OEIS frontend

A274513 a(n) is the only number m such that 11^(2^m) + 1 is divisible by A273949(n).

0, 3, 1, 5, 5, 2, 7, 4, 15, 14, 3, 8, 19, 11, 10, 24, 27, 8, 19, 23, 7, 16, 31, 35, 4, 29, 28, 11, 11, 28, 35

Offset: 1

Arkadiusz Wesolowski, Jun 25 2016

forstep(p=3, 10^15, 2, if(!Mod(p, 11)==0, if(isprime(p), o=znorder(Mod(11, p)); x=ispower(2*o); if(2^(x-1)==o, print1(x-2, ", ")))));

A273950 Prime factors of generalized Fermat numbers of the form 12^(2^m) + 1 with m >= 0.

Original entry on oeis.org

5, 13, 17, 29, 89, 97, 233, 257, 769, 36097, 40961, 65537, 81281, 153953, 163841, 260753, 1724417, 4550657, 5767169, 8253953, 11304961, 13631489, 21495809, 69619841, 77651969, 147849217, 158334977, 159522817, 1711276033, 6528575489, 27286044673, 52613349377

Offset: 1

Arkadiusz Wesolowski, Jun 05 2016

nonn

Primes p such that the multiplicative order of 12 (mod p) is a power of 2.

Hans Riesel, Common prime factors of the numbers A_n=a^(2^n)+1, BIT 9 (1969), pp. 264-269.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..44
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=12)
Harvey Dubner and Wilfrid Keller, Factors of Generalized Fermat Numbers, Math. Comp. 64 (1995), no. 209, pp. 397-405.
OEIS Wiki, Generalized Fermat numbers

Cf. A023394, A072982, A152585, A268660, A268664, A273945 (base 3), A273946 (base 5), A273947 (base 6), A273948 (base 7), A273949 (base 11).

Select[Prime@Range[2, 10^5], IntegerQ@Log[2, MultiplicativeOrder[12, #]] &]

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

Original entry on oeis.org

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, #]] &]

A273946 Odd prime factors of generalized Fermat numbers of the form 5^(2^m) + 1 with m >= 0.

Original entry on oeis.org

3, 13, 17, 257, 313, 641, 769, 2593, 11489, 19457, 65537, 163841, 786433, 1503233, 1655809, 7340033, 14155777, 18395137, 23606273, 29423041, 39714817, 75068993, 167772161, 2483027969, 4643094529, 6616514561, 47148957697, 241931001601, 2748779069441

Offset: 1

Arkadiusz Wesolowski, Jun 05 2016

nonn

Odd primes p such that the multiplicative order of 5 (mod p) is a power of 2.

Hans Riesel, Common prime factors of the numbers A_n=a^(2^n)+1, BIT 9 (1969), pp. 264-269.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..37
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=5)
Harvey Dubner and Wilfrid Keller, Factors of Generalized Fermat Numbers, Math. Comp. 64 (1995), no. 209, pp. 397-405.
OEIS Wiki, Generalized Fermat numbers

Cf. A023394, A072982, A199591, A268658, A268662, A273945 (base 3), A273947 (base 6), A273948 (base 7), A273949 (base 11), A273950 (base 12).

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

A273947 Prime factors of generalized Fermat numbers of the form 6^(2^m) + 1 with m >= 0.

Original entry on oeis.org

7, 17, 37, 257, 353, 1297, 1697, 2753, 18433, 65537, 80897, 98801, 145601, 763649, 3360769, 4709377, 13631489, 50307329, 376037377, 2483027969, 3191106049, 4926056449, 51808043009, 152605556737, 916326983681, 1268357529601, 6597069766657, 40711978221569

Offset: 1

Arkadiusz Wesolowski, Jun 05 2016

nonn

Primes p other than 5 such that the multiplicative order of 6 (mod p) is a power of 2.

Hans Riesel, Common prime factors of the numbers A_n=a^(2^n)+1, BIT 9 (1969), pp. 264-269.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..34
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=6)
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, Some factors of the numbers G_n=6^(2^n)+1 and H_n=10^(2^n)+1, Math. Comp. 23 (1969), no. 106, pp. 413-415.

Cf. A023394, A072982, A078303, A268663, A273945 (base 3), A273946 (base 5), A273948 (base 7), A273949 (base 11), A273950 (base 12).

Select[Prime@Range[4, 10^5], IntegerQ@Log[2, MultiplicativeOrder[6, #]] &]

A273948 Odd prime factors of generalized Fermat numbers of the form 7^(2^m) + 1 with m >= 0.

Original entry on oeis.org

5, 17, 257, 353, 769, 1201, 12289, 13313, 35969, 65537, 114689, 163841, 169553, 7699649, 9379841, 11886593, 28667393, 64749569, 70254593, 134818753, 197231873, 4643094529, 19847446529, 47072139617, 206158430209, 452850614273, 531968664833, 943558259713

Offset: 1

Arkadiusz Wesolowski, Jun 05 2016

nonn

Odd primes p other than 3 such that the multiplicative order of 7 (mod p) is a power of 2.
From Robert Israel, Jun 16 2016: (Start)
If p is in the sequence, then for each m either p | 7^(2^k)+1 for some k < m or 2^m | p-1.  Thus all members except 5, 17, 353, 1201, 169553, 7699649, 134818753, 47072139617 are congruent to 1 mod 2^7.
The intersection of this sequence and A019337 is A019434 minus {3}. (End)

Hans Riesel, Common prime factors of the numbers A_n=a^(2^n)+1, BIT 9 (1969), pp. 264-269.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..34
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.
Harvey Dubner and Wilfrid Keller, Factors of Generalized Fermat Numbers, Math. Comp. 64 (1995), no. 209, pp. 397-405.
OEIS Wiki, Generalized Fermat numbers

Cf. A023394, A072982, A078304, A273945 (base 3), A273946 (base 5), A273947 (base 6), A273949 (base 11), A273950 (base 12).

filter:= proc(t)
  if not isprime(t) then return false fi;
  7 &^ (2^padic:-ordp(t-1,2)) mod t = 1
end proc:
select(filter, [seq(i,i=5..10^6,2)]); # Robert Israel, Jun 16 2016

Select[Prime@Range[3, 10^5], IntegerQ@Log[2, MultiplicativeOrder[7, #]] &]

A275382 Number of odd prime factors (with multiplicity) of generalized Fermat number 11^(2^n) + 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 5, 6

Offset: 0

Arkadiusz Wesolowski, Jul 25 2016

			b(n) = (11^(2^n) + 1)/2.
Complete Factorizations
b(0) = 2*3
b(1) = 61
b(2) = 7321
b(3) = 17*6304673
b(4) = 51329*447600088289
b(5) = 193*257*21283620033217629539178799361
b(6) = 316955440822738177*P49
b(7) = 15361*111489577217*574341646346402207998363393*
       4018529583345312964042058778793458689*P55
b(8) = 15190529*4696846849*19618834249745000485889*
       4393553986026616439660661873903822389581313*
       290103547098489711747952055517085778590240759297*P138

Cf. A199592, A273949.

a001222(n) = bigomega(n)
a199592(n) = 11^(2^n)+1
a(n) = if(n==0, 1, a001222(a199592(n))-1) \\ Felix Fröhlich, Jul 25 2016

a(n) = A001222(A199592(n)) - 1 for n > 0. - Felix Fröhlich, Jul 25 2016

a(8) was found in 2006 by Bruce Dodson

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A274513 a(n) is the only number m such that 11^(2^m) + 1 is divisible by A273949(n).

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A273950 Prime factors of generalized Fermat numbers of the form 12^(2^m) + 1 with m >= 0.

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

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A273946 Odd prime factors of generalized Fermat numbers of the form 5^(2^m) + 1 with m >= 0.

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A273947 Prime factors of generalized Fermat numbers of the form 6^(2^m) + 1 with m >= 0.

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A273948 Odd prime factors of generalized Fermat numbers of the form 7^(2^m) + 1 with m >= 0.

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A275382 Number of odd prime factors (with multiplicity) of generalized Fermat number 11^(2^n) + 1.

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