A273950 -id:A273950 - cp's OEIS frontend

A274514 a(n) is the only number m such that 12^(2^m) + 1 is divisible by A273950(n).

1, 0, 3, 1, 2, 3, 2, 7, 5, 6, 12, 15, 6, 4, 14, 3, 11, 11, 18, 8, 12, 19, 18, 6, 12, 18, 19, 11, 21, 10, 19, 29, 8, 26, 16, 4, 38, 21, 23, 39, 14, 42, 40, 30

Offset: 1

Arkadiusz Wesolowski, Jun 25 2016

Cf. A273950.

forstep(p=3, 10^15, 2, if(!Mod(p, 3)==0, if(isprime(p), o=znorder(Mod(12, p)); x=ispower(2*o); if(2^(x-1)==o, print1(x-2, ", ")))));

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, #]] &]

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]

A272949 Products of three distinct Fibonacci numbers > 1.

Original entry on oeis.org

30, 48, 78, 80, 120, 126, 130, 195, 204, 208, 210, 312, 315, 330, 336, 340, 504, 510, 520, 534, 544, 546, 550, 816, 819, 825, 840, 864, 880, 884, 890, 1320, 1326, 1335, 1360, 1365, 1398, 1424, 1428, 1430, 1440, 2136, 2142, 2145, 2160, 2184, 2200, 2210, 2262

Offset: 1

Clark Kimberling, May 13 2016

			a(1) = 30 = 2*3*5.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000045, A160009, A272947, A271354, A273950.

s = {1}; nn = 60; f = Fibonacci[2 + Range[nn]]; Do[s = Union[s, Select[s*f[[i]], # <= f[[nn]] &]], {i, nn}]; s =  Prepend[s, 0]; Take[s, 100]  (* A160009 *)
isFibonacciQ[n_] := Apply[Or, Map[IntegerQ, Sqrt[{# + 4, # - 4} &[5 n^2]]]];
ans = Join[{{0}}, {{1}}, Table[#[[Flatten[Position[Map[Apply[Times, #] &, #], s[[n]]]][[1]]]] &[Rest[Subsets[Rest[Map[#[[1]] &, Select[Map[{#, isFibonacciQ[#]} &, Divisors[s[[n]]]], #[[2]] &]]]]]], {n, 3, 500}]]
Map[Length, ans] (* A272947 *)
Flatten[Position[Map[Length, ans], 1]]  (* A272948 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 1 &]]  (* A000045 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 2 &]]  (* A271354 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 3 &]]  (* A272949 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 4 &]]  (* A272950 *)
(* Peter J. C. Moses, May 11 2016 *)
up=10^9; F=Fibonacci; i=3; Union[ Reap[ While[(a = F[i++]) < up, j=i; While[ (b = F[j++]*a) < up, h=j; While[ (c = F[h++]*b) < up, Sow@c ]]]][[2, 1]]] (* Giovanni Resta, May 14 2016 *)

A272947 Number of factors Fibonacci(i) > 1 of A160009(n+1).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 3, 1, 2, 2, 2, 3, 1, 2, 2, 2, 3, 3, 1, 2, 2, 2, 2, 3, 3, 3, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 4, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 1, 4, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 1, 4, 4, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Clark Kimberling, May 13 2016

			A160009(15) = 30 = 2*3*5, so that a(15) = 3.

MathOverflow, Distinctness of products of Fibonacci numbers

Cf. A000045, A160009, A272948, A271354, A272949, A273950.

s = {1}; nn = 60; f = Fibonacci[2 + Range[nn]]; Do[s = Union[s, Select[s*f[[i]], # <= f[[nn]] &]], {i, nn}]; s =  Prepend[s, 0]; Take[s, 100]  (* A160009 *)
isFibonacciQ[n_] := Apply[Or, Map[IntegerQ, Sqrt[{# + 4, # - 4} &[5 n^2]]]];
ans = Join[{{0}}, {{1}}, Table[#[[Flatten[Position[Map[Apply[Times, #] &, #], s[[n]]]][[1]]]] &[Rest[Subsets[Rest[Map[#[[1]] &, Select[Map[{#, isFibonacciQ[#]} &, Divisors[s[[n]]]], #[[2]] &]]]]]], {n, 3, 500}]]
Map[Length, ans] (* A272947 *)
Flatten[Position[Map[Length, ans], 1]]  (* A272948 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 1 &]]  (* A000045 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 2 &]]  (* A271354 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 3 &]]  (* A272949 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 4 &]]  (* A272950 *)
(* Peter J. C. Moses, May 11 2016 *)

A272948 Positions of Fibonacci numbers in ordered sequence A160009 of all products of Fibonacci numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 27, 35, 44, 56, 70, 87, 108, 133, 163, 199, 242, 292, 352, 421, 504, 599, 712, 841, 994, 1167, 1371, 1602, 1873, 2179, 2535, 2936, 3401, 3924, 4528, 5206, 5985, 6858, 7857, 8976, 10252, 11679, 13299, 15109, 17159, 19446, 22028

Offset: 1

Clark Kimberling, May 13 2016

nonn

			A160009 = (0,1,2,3,5,6,8,10,13,15,16,21,...), so that a = (1,2,3,4,5,7,9,12,...).

Rémy Sigrist, Table of n, a(n) for n = 1..121
Distinctness of Fibonacci products
Rémy Sigrist, PARI program for A272948

Cf. A000045, A160009, A272947, A271354, A272949, A273950.

s = {1}; nn = 60; f = Fibonacci[2 + Range[nn]]; Do[s = Union[s, Select[s*f[[i]], # <= f[[nn]] &]], {i, nn}]; s =  Prepend[s, 0]; Take[s, 100]  (* A160009 *)
isFibonacciQ[n_] := Apply[Or, Map[IntegerQ, Sqrt[{# + 4, # - 4} &[5 n^2]]]];
ans = Join[{{0}}, {{1}}, Table[#[[Flatten[Position[Map[Apply[Times, #] &, #], s[[n]]]][[1]]]] &[Rest[Subsets[Rest[Map[#[[1]] &, Select[Map[{#, isFibonacciQ[#]} &, Divisors[s[[n]]]], #[[2]] &]]]]]], {n, 3, 500}]]
Map[Length, ans] (* A272947 *)
Flatten[Position[Map[Length, ans], 1]]  (* A272948 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 1 &]]  (* A000045 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 2 &]]  (* A271354 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 3 &]]  (* A272949 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 4 &]]  (* A272950 *)
(* Peter J. C. Moses, May 11 2016 *)

More terms from Rémy Sigrist, Mar 17 2019

A275383 Number of prime factors (with multiplicity) of generalized Fermat number 12^(2^n) + 1.

Original entry on oeis.org

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

Offset: 0

Arkadiusz Wesolowski, Jul 25 2016

			b(n) = 12^(2^n) + 1.
Complete Factorizations
b(0) = 13
b(1) = 5*29
b(2) = 89*233
b(3) = 17*97*260753
b(4) = 153953*1200913648289
b(5) = 769*44450180997616192602560262634753
b(6) = 36097*81281*69619841*73389730593973249*P35
b(7) = 257*P136
b(8) = 8253953*295278642689*5763919006323142831065059613697*P96*P132

Arkadiusz Wesolowski, A 96-digit prime factor of b(8)

Cf. A152585, A273950.

Table[PrimeOmega[12^(2^n) + 1], {n, 0, 7}] (* Michael De Vlieger, Jul 26 2016 *)

PARI
```
a(n) = bigomega(factor(12^(2^n)+1))
```

a(n) = A001222(A152585(n)). - Felix Fröhlich, Jul 25 2016

a(8) was found in 2009 by Tom Womack

cp's OEIS Frontend

A274514 a(n) is the only number m such that 12^(2^m) + 1 is divisible by A273950(n).

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PARI

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

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Mathematica

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

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Mathematica

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

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Mathematica

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

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Maple

Mathematica

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

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Mathematica

A272949 Products of three distinct Fibonacci numbers > 1.

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Mathematica

A272947 Number of factors Fibonacci(i) > 1 of A160009(n+1).

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Mathematica

A272948 Positions of Fibonacci numbers in ordered sequence A160009 of all products of Fibonacci numbers.

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