A046801 -id:A046801 - cp's OEIS frontend

A177855 Divisors of 2^1092 - 1.

1, 3, 5, 7, 9, 13, 15, 21, 29, 35, 39, 43, 45, 49, 53, 63, 65, 79, 87, 91, 105, 113, 117, 127, 129, 145, 147, 157, 159, 169, 195, 203, 215, 237, 245, 261, 265, 273, 301, 313, 315, 337, 339, 371, 377, 381, 387, 395, 435, 441, 455, 471, 477, 507, 547, 553, 559, 565

Offset: 1

Reinhard Zumkeller, May 14 2010

The sequence is finite with A000005(2^1092-1) = 178120883699712 terms.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (Terms for n = 1..5000 from Reinhard Zumkeller)
Dario A. Alpern, Factorization using the Elliptic Curve Method
Index entries for sequences related to divisors of numbers

Cf. A000005, A172290, A172291, A046801.

divisors(2^1092-1) \\ Charles R Greathouse IV, Jun 21 2017

A177855_list = [i for i in range(1,10**6) if (2**1092-1) % i == 0] # Chai Wah Wu, Jun 02 2021

A051464 Number of divisors of 4*(2^n-1) + 1.

Original entry on oeis.org

2, 2, 2, 2, 4, 4, 2, 2, 4, 2, 4, 2, 4, 6, 4, 4, 16, 2, 4, 2, 4, 2, 4, 8, 12, 8, 2, 4, 8, 4, 4, 4, 4, 4, 4, 8, 8, 4, 8, 16, 8, 4, 8, 8, 6, 16, 8, 8, 8, 16, 8, 4, 32, 32, 8, 4, 8, 4, 4, 8, 16, 8, 8, 16, 48, 16, 16, 8, 4, 16, 4, 16, 16, 8, 8, 8, 16, 16, 8, 16, 32

Offset: 1

Edwin D. Evans, eevans2(AT)pacbell.net

nonn

Create a table with tau(2^n-1) as the first row (A046801) and tau(m) as the first column (A000005). The second column is tau(A004760) and so on. Rows 2, 3 and 4 are easily described in terms of row 1. This sequence is row 5.

Sean A. Irvine, Table of n, a(n) for n = 1..591

Cf. A000005, A036563.

Array[DivisorSigma[0, 4*(2^# - 1) + 1] &, 81] (* Michael De Vlieger, Sep 15 2021 *)

a(n) = numdiv(4*(2^n-1) + 1); \\ Michel Marcus, Sep 16 2021

a(n) = tau(4*(2^n -1)+1), where d(n) = A000005(n).

a(81) corrected by Sean A. Irvine, Sep 15 2021

A086256 Number of base-2 Fermat pseudoprimes that divide 2^n-1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 2, 1, 4, 1, 2, 1, 1, 0, 13, 4, 5, 0, 2, 2, 1, 1, 13, 1, 1, 4, 7, 1, 11, 4, 14, 9, 4, 4, 28, 0, 12, 11, 12, 4, 2, 5, 28, 4, 26, 1, 63, 0, 1, 5, 12, 1, 29, 1, 12, 2, 44, 4, 101, 4, 11, 27, 12, 1, 26, 4, 15, 4, 11, 1, 75, 1, 11, 14, 36, 0, 40, 11

Offset: 1

T. D. Noe, Jul 14 2003

A base-2 Fermat pseudoprime is a composite number x such that 2^x = 2 mod x.

Amiram Eldar, Table of n, a(n) for n = 1..419
Eric Weisstein's World of Mathematics, Pseudoprime

Cf. A001567 (base-2 pseudoprimes), A046801, A086249.

Table[d=Divisors[2^n-1]; cnt=0; Do[m=d[[i]]; If[ !PrimeQ[m]&&PowerMod[2, m, m]==2, cnt++ ], {i, Length[d]}]; cnt, {n, 100}]

a(n) = Sum{d|n} A086249(d), the Mobius transform of A086249.

A283930 Numbers k such that tau(2^k - 1) = tau(2^k + 1).

Original entry on oeis.org

2, 11, 14, 21, 23, 29, 45, 47, 53, 71, 73, 74, 82, 86, 95, 99, 101, 105, 113, 115, 121, 142, 167, 169, 179, 181, 199, 203, 209, 233, 235, 277, 307, 311, 317, 335, 337, 343, 347, 349, 353, 355, 358, 361, 382, 434, 449, 465, 494, 509, 515, 518, 529, 535, 547, 549, 570, 583, 585, 599

Offset: 1

Jaroslav Krizek, Mar 18 2017

nonn

tau(k) is the number of divisors of k (A000005).
Numbers k such that A046801(k) = A046798(k).
Numbers k such that A000005(A000225(k)) = A000005(A000051(k)).
Corresponding values of tau(2^k +- 1): 2, 4, 8, 12, 4, 8, 64, 8, 8, 8, 8, 32, 32, 32, 32, 256, 4, 1536, ...
Corresponding pairs of numbers (2^k - 1, 2^k + 1): (3, 5); (2047, 2049); (16383, 16385); (2097151, 2097153); (8388607, 8388609); ...

			For n = 11; tau(2047) = tau(2049) = 4.

Cf. A000005, A000051, A000225, A046798, A046801, A283931.

[n: n in [1..500] | NumberOfDivisors(2^n - 1) eq NumberOfDivisors(2^n + 1)]

Select[Range@ 200, Function[n, Equal @@ Map[DivisorSigma[0, 2^n + #] &, {-1, 1}]]] (* Michael De Vlieger, Mar 18 2017 *)

for(n=1, 600, if(numdiv(2^n - 1) == numdiv(2^n + 1), print1(n,", "))) \\ Indranil Ghosh, Mar 18 2017

from sympy import divisor_count
print([n for n in range(1, 601) if divisor_count(2**n + 1) == divisor_count(2**n - 1)]) # Indranil Ghosh, Mar 18 2017

A336104 Number of permutations of the prime indices of A000225(n) = 2^n - 1 with at least one non-singleton run.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 96, 0, 120, 6, 0, 0, 720, 0, 0, 0, 0, 0, 720, 0, 0, 0, 0, 0, 322560, 0, 0, 0, 5040, 0, 4320, 0, 0, 0, 0, 0, 362880, 0, 0

Offset: 1

Gus Wiseman, Sep 03 2020

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(21) = 6 permutations of {4, 4, 31, 68}:
  (4,4,31,68)
  (4,4,68,31)
  (31,4,4,68)
  (31,68,4,4)
  (68,4,4,31)
  (68,31,4,4)

A335432 is the anti-run version.
A335459 is the version for factorial numbers.
A336105 counts all permutations of this multiset.
A336107 is not restricted to predecessors of powers of 2.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A008480 counts permutations of prime indices.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A333489 ranks anti-run compositions.
A335433 lists numbers whose prime indices have an anti-run permutation.
A335448 lists numbers whose prime indices have no anti-run permutation.
A335452 counts anti-run permutations of prime indices.
A335489 counts strict permutations of prime indices.
Cf. A056239, A106351, A112798, A114938, A292884, A336102.
The numbers 2^n - 1: A000225, A001265, A001348, A046051, A046800, A046801, A049093, A325610, A325611, A325612, A325625.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[2^n-1]],MatchQ[#,{_,x_,x_,_}]&]],{n,30}]

a(n) = A336107(2^n - 1).

a(n) = A336105(n) - A335432(n).

A336105 Number of permutations of the prime indices of 2^n - 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 6, 2, 6, 2, 60, 1, 6, 6, 24, 1, 120, 1, 360, 12, 24, 2, 2520, 6, 6, 6, 720, 6, 2520, 1, 120, 24, 6, 24, 604800, 2, 6, 24, 20160, 2, 10080, 6, 5040, 720, 24, 6, 1814400, 2, 5040, 120, 5040, 6, 15120, 720, 40320, 24, 720, 2

Offset: 1

Gus Wiseman, Sep 03 2020

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(n) permutations for n = 2, 4, 6, 8, 21:
  (2)  (2,3)  (2,2,4)  (2,3,7)  (31,4,4,68)
       (3,2)  (2,4,2)  (2,7,3)  (31,4,68,4)
              (4,2,2)  (3,2,7)  (31,68,4,4)
                       (3,7,2)  (4,31,4,68)
                       (7,2,3)  (4,31,68,4)
                       (7,3,2)  (4,4,31,68)
                                (4,4,68,31)
                                (4,68,31,4)
                                (4,68,4,31)
                                (68,31,4,4)
                                (68,4,31,4)
                                (68,4,4,31)

A008480 is not restricted to predecessors of powers of 2.
A325617 is the version for factorial numbers.
A335489 counts strict permutations of prime indices.
Cf. A056239, A112798, A335432, A336104.
The numbers 2^n - 1: A000225, A046051, A046800, A046801, A049093, A325610, A325611, A325612, A325625.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Permutations[primeMS[2^n-1]]],{n,30}]

a(n) = A008480(2^n - 1).

a(n) = A336104(n) + A335432(n).

A111303 Numbers n such that 2^tau(n) = n + 1 (where tau(n) = number of divisors of n).

Original entry on oeis.org

1, 3, 15, 63, 255, 65535, 4294967295

Offset: 1

Joseph L. Pe, Nov 02 2005

It is clear that n+1 must be a power of 2. Hence n=2^k-1 for some k. Found k=1, 2, 4, 6, 8, 16, 32. No other k<150. - T. D. Noe, Nov 04 2005

Cf. A046801 (number of divisors of 2^n-1), A019434.

Select[Range[10^6], 2^DivisorSigma[0, # ] == # + 1 &]
2^Select[Range[150], DivisorSigma[0, 2^#-1]==#&] - 1 (Noe)

from sympy import divisor_count as tau
def afind(klimit, kstart=1):
    for k in range(kstart, klimit+1):
        m = 2**k - 1
        if 2**tau(m) == m + 1: print(m, end=", ")
afind(klimit=100) # Michael S. Branicky, Dec 16 2021

Note that this is different from the sequence A019434 - 2.

One more term from T. D. Noe, Nov 04 2005

A350142 Numbers m of the form 2^k + 1 such that tau(m-2) = tau(m-1) - 1.

Original entry on oeis.org

3, 5, 17, 65, 257, 65537, 4294967297

Offset: 1

Jaroslav Krizek, Dec 16 2021

Corresponding pairs of values [tau(m-2), tau(m-1)]: [1, 2], [2, 3], [4, 5], [6, 7], [8, 9], [16, 17], [32, 33], ...
There are no other terms <= 2^1206 + 1 (from A046801 data).
The first 5 known Fermat primes from A019434 are in this sequence. Corresponding values of tau(A019434(n - 2)): 1, 2, 4, 8, 16, ...
Conjecture 1: Also numbers m of the form 2^k + 1 such that tau(m - 2) = k.
Conjecture 2: If 6th Fermat prime F_p6 exists, then tau(F_p6 - 2) is a power of 2 and tau(F_p6 - 1) = tau(F_p6 - 2) + 1.
Conjecture 3: Sequence is finite with 7 terms; supersequence of A262534.

			For number 257 holds: tau(255) = 8, tau(256) = 9.

Cf. A000005, A019434, A262534, A343144, A347078.

Intersection of (A055927+2) and A000051.

[2^k + 1: k in [1..50] | #Divisors(2^k) - #Divisors(2^k-1) eq 1];

cp's OEIS Frontend

A177855 Divisors of 2^1092 - 1.

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A051464 Number of divisors of 4*(2^n-1) + 1.

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A086256 Number of base-2 Fermat pseudoprimes that divide 2^n-1.

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A283930 Numbers k such that tau(2^k - 1) = tau(2^k + 1).

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A336104 Number of permutations of the prime indices of A000225(n) = 2^n - 1 with at least one non-singleton run.

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A336105 Number of permutations of the prime indices of 2^n - 1.

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A111303 Numbers n such that 2^tau(n) = n + 1 (where tau(n) = number of divisors of n).

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A350142 Numbers m of the form 2^k + 1 such that tau(m-2) = tau(m-1) - 1.

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