A072982 -id:A072982 - cp's OEIS frontend

A197224 Exponent in the power of 2 which is the period of the decimal fraction 1/A072982(n).

0, 1, 4, 3, 2, 3, 8, 5, 5, 5, 5, 9, 8, 6, 16, 5, 8, 6, 10, 16, 4, 6, 9, 17, 20, 21, 18, 10, 13, 19, 7, 27, 9, 30, 9, 12, 20, 30, 36, 20, 41, 23, 42, 41, 6

Offset: 1

T. D. Noe, Oct 21 2011

This means that the fraction 1/52613349377 repeats every 2^30 terms!

Every term with value up to 6 is accounted for.

Hans Riesel, Some factors of the numbers Gn = 6^2^n+1 and Hn = 10^2^n+1, Math. Comp. 23 (1969), pp. 413-415. With errata reported in Math. Comp. 24 (1970), p. 243.

Cf. A072982, A274512.

Table[Log[2, MultiplicativeOrder[10, p]], {p, A072982}] (* Ray Chandler, Oct 25 2019 *)

a(n+1) = A274512(n) + 1. - Ray Chandler, Oct 25 2019

a(37)-a(39) from Ray Chandler, Nov 03 2011

a(40)-a(45) from Ray Chandler, Oct 25 2019

A274512 a(n) is the only number m such that 10^(2^m) + 1 is divisible by A072982(n+1).

Original entry on oeis.org

0, 3, 2, 1, 2, 7, 4, 4, 4, 4, 8, 7, 5, 15, 4, 7, 5, 9, 15, 3, 5, 8, 16, 19, 20, 17, 9, 12, 18, 6, 26, 8, 29, 8, 11, 19, 29, 35, 19, 40, 22, 41, 40, 5

Offset: 1

Arkadiusz Wesolowski, Jun 25 2016

Cf. A072982, A197224.

forstep(p=3, 10^15, 2, if(!Mod(p, 5)==0, if(isprime(p), o=znorder(Mod(10, p)); x=ispower(2*o); if(2^(x-1)==o, print1(x-2, ", ")))));

a(n) = A197224(n+1) - 1. - Ray Chandler, Oct 25 2019

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, #]] &]

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]

A128948 Primes p for which the period length of 1/p is a perfect power, A001597.

Original entry on oeis.org

3, 17, 73, 101, 137, 163, 257, 353, 449, 577, 641, 751, 757, 883, 1297, 1409, 1801, 3137, 3529, 5477, 7057, 7351, 8929, 9397, 10753, 11831, 12101, 13457, 13553, 14401, 15361, 15377, 15973, 18523, 19841, 20809, 21401, 21601, 23549, 24001, 24337

Offset: 1

Robert G. Wilson v, May 05 2007

Number of primes p < 10^n whose period length of 1/p is a perfect power: 1,3,14,24,78,173,461,1190,3235,8933,....

The primes modulo any integer do not seem to be equally distributed.

			The prime 73 has a period of 8 = 2^3 which is a member of A001597, hence is a member of this sequence.

Ray Chandler & Robert G. Wilson v, Table of n, a(n) for n = 1..30000
Index entries for sequences related to decimal expansion of 1/n

Cf. A001597, A072859, A072982.

lst = {3}; p = 1; While[p < 10^8, p = NextPrime@p; If[GCD @@ Last /@ FactorInteger@ MultiplicativeOrder[10, p] > 1, AppendTo[lst, p]; Print@p]]; lst (* Ray Chandler, May 11 2007 *)

Correction (3 is a member of the sequence) from Ray Chandler, May 11 2007

B-file corrected by Ray Chandler, Oct 23 2011

A197225 Primes p with the period of the decimal fraction 1/p a prime power, A000961.

Original entry on oeis.org

3, 11, 17, 37, 41, 53, 73, 79, 83, 101, 107, 137, 163, 173, 227, 239, 257, 271, 317, 347, 353, 359, 449, 467, 479, 563, 587, 641, 643, 719, 733, 751, 757, 773, 797, 839, 907, 1031, 1187, 1231, 1283, 1307, 1319, 1409, 1439, 1493, 1523, 1627, 1637, 1879, 1907

Offset: 1

T. D. Noe, Oct 22 2011

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A072859 (period is prime).
Cf. A072982 (period is a power of 2).
Cf. A128948 (period is perfect power).
Cf. A197226 (the periods of this sequence).
Cf. A129727 (period is a semiprime).

myPerfectPowerQ[n_] := Length[FactorInteger[n]] == 1; Select[Prime[Range[500]], Mod[10,#] > 0 && myPerfectPowerQ[Length[RealDigits[1/#, 10][[1,1]]]] &]

cp's OEIS Frontend

A197224 Exponent in the power of 2 which is the period of the decimal fraction 1/A072982(n).

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A274512 a(n) is the only number m such that 10^(2^m) + 1 is divisible by A072982(n+1).

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

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