A273947 -id:A273947 - cp's OEIS frontend

A274510 a(n) is the only number m such that 6^(2^m) + 1 is divisible by A273947(n).

0, 3, 1, 7, 4, 2, 4, 5, 8, 15, 9, 3, 5, 7, 9, 4, 19, 7, 10, 25, 7, 6, 27, 25, 10, 25, 35, 23, 36, 21, 22, 11, 40, 33

Offset: 1

Arkadiusz Wesolowski, Jun 25 2016

forstep(p=3, 10^15, 2, if(!Mod(p, 3)==0, if(isprime(p), o=znorder(Mod(6, 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, #]] &]

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, #]] &]

A273949 Odd prime factors of generalized Fermat numbers of the form 11^(2^m) + 1 with m >= 0.

Original entry on oeis.org

3, 17, 61, 193, 257, 7321, 15361, 51329, 65537, 163841, 6304673, 15190529, 70254593, 1691123713, 1760464897, 3221225473, 3489660929, 4696846849, 6874464257, 53401878529, 111489577217, 149300051969, 184683593729, 206158430209, 447600088289, 1819992391681

Offset: 1

Arkadiusz Wesolowski, Jun 05 2016

nonn

Odd primes p other than 5 such that the multiplicative order of 11 (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..31
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, A199592, A273945 (base 3), A273946 (base 5), A273947 (base 6), A273948 (base 7), A273950 (base 12).

Delete[Select[Prime@Range[2, 10^5], IntegerQ@Log[2, MultiplicativeOrder[11, #]] &], 2]

A275379 Number of prime factors (with multiplicity) of generalized Fermat number 6^(2^n) + 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 3, 7, 3, 5

Offset: 0

Arkadiusz Wesolowski, Jul 25 2016

			b(n) = 6^(2^n) + 1.
Complete Factorizations
b(0) = 7
b(1) = 37
b(2) = 1297
b(3) = 17*98801
b(4) = 353*1697*4709377
b(5) = 2753*145601*19854979505843329
b(6) = 4926056449*447183309836853377*28753787197056661026689
b(7) = 257*763649*50307329*3191106049*2339340566463317436161*
       2983028405608735541756929*18247770097021321924017185281
b(8) = 18433*
       69615986569139423375849495295909549956813828853888948633601*P137
b(9) = 80897*3360769*12581314681802812884728041373153281*
       3513902440204553274892072241244613302018049*P311

Cf. A078303, A273947.

Table[PrimeOmega[6^(2^n) + 1], {n, 0, 6}] (* Michael De Vlieger, Jul 26 2016 *)

PARI
```
a(n) = bigomega(factor(6^(2^n)+1))
```

a(n) = A001222(A078303(n)). - Felix Fröhlich, Jul 25 2016

a(8) was found in 2001 by Robert Silverman

a(9) was found in 2007 by Nestor de Araújo Melo

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A274510 a(n) is the only number m such that 6^(2^m) + 1 is divisible by A273947(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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A273948 Odd prime factors of generalized Fermat numbers of the form 7^(2^m) + 1 with m >= 0.

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A273949 Odd prime factors of generalized Fermat numbers of the form 11^(2^m) + 1 with m >= 0.

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A275379 Number of prime factors (with multiplicity) of generalized Fermat number 6^(2^n) + 1.

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