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a(n) is the only number m such that 12^(2^m) + 1 is divisible by A273950(n).
+20
0
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
PROG
(PARI) 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, ", ")))));
Odd prime factors of generalized Fermat numbers of the form 3^(2^m) + 1 with m >= 0.
+10
8
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
COMMENTS
Odd primes p such that the multiplicative order of 3 (mod p) is a power of 2.
MATHEMATICA
Select[Prime@Range[2, 10^5], IntegerQ@Log[2, MultiplicativeOrder[3, #]] &]
Odd prime factors of generalized Fermat numbers of the form 5^(2^m) + 1 with m >= 0.
+10
8
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
COMMENTS
Odd primes p such that the multiplicative order of 5 (mod p) is a power of 2.
REFERENCES
Hans Riesel, Common prime factors of the numbers A_n=a^(2^n)+1, BIT 9 (1969), pp. 264-269.
MATHEMATICA
Select[Prime@Range[2, 10^5], IntegerQ@Log[2, MultiplicativeOrder[5, #]] &]
Prime factors of generalized Fermat numbers of the form 6^(2^m) + 1 with m >= 0.
+10
8
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
COMMENTS
Primes p other than 5 such that the multiplicative order of 6 (mod p) is a power of 2.
REFERENCES
Hans Riesel, Common prime factors of the numbers A_n=a^(2^n)+1, BIT 9 (1969), pp. 264-269.
MATHEMATICA
Select[Prime@Range[4, 10^5], IntegerQ@Log[2, MultiplicativeOrder[6, #]] &]
Odd prime factors of generalized Fermat numbers of the form 7^(2^m) + 1 with m >= 0.
+10
8
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
COMMENTS
Odd primes p other than 3 such that the multiplicative order of 7 (mod p) is a power of 2.
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)
REFERENCES
Hans Riesel, Common prime factors of the numbers A_n=a^(2^n)+1, BIT 9 (1969), pp. 264-269.
MAPLE
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
MATHEMATICA
Select[Prime@Range[3, 10^5], IntegerQ@Log[2, MultiplicativeOrder[7, #]] &]
Odd prime factors of generalized Fermat numbers of the form 11^(2^m) + 1 with m >= 0.
+10
8
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
COMMENTS
Odd primes p other than 5 such that the multiplicative order of 11 (mod p) is a power of 2.
REFERENCES
Hans Riesel, Common prime factors of the numbers A_n=a^(2^n)+1, BIT 9 (1969), pp. 264-269.
MATHEMATICA
Delete[Select[Prime@Range[2, 10^5], IntegerQ@Log[2, MultiplicativeOrder[11, #]] &], 2]
Products of three distinct Fibonacci numbers > 1.
+10
7
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
MATHEMATICA
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}]]
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 *) 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 *)
Number of factors Fibonacci(i) > 1 of A160009(n+1).
+10
6
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
EXAMPLE
A160009(15) = 30 = 2*3*5, so that a(15) = 3.
MATHEMATICA
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}]]
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 *)
Positions of Fibonacci numbers in ordered sequence A160009 of all products of Fibonacci numbers.
+10
4
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
EXAMPLE
A160009 = (0,1,2,3,5,6,8,10,13,15,16,21,...), so that a = (1,2,3,4,5,7,9,12,...).
MATHEMATICA
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}]]
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 *)
Number of prime factors (with multiplicity) of generalized Fermat number 12^(2^n) + 1.
+10
3
1, 2, 2, 3, 2, 2, 5, 2, 5
EXAMPLE
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
PROG
(PARI) a(n) = bigomega(factor(12^(2^n)+1))
EXTENSIONS
a(8) was found in 2009 by Tom Womack
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