A046798 -id:A046798 - cp's OEIS frontend

A366628 Number of divisors of 6^n+1.

2, 2, 2, 4, 2, 8, 8, 12, 4, 8, 8, 4, 4, 16, 8, 32, 8, 8, 64, 8, 8, 48, 16, 8, 16, 16, 16, 32, 32, 16, 512, 4, 8, 64, 8, 1536, 32, 16, 8, 512, 32, 16, 128, 4, 8, 128, 32, 4, 128, 64, 64, 256, 16, 32, 1024, 192, 64, 128, 8, 4, 64, 8, 4, 768, 8, 256, 2048, 32, 32

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

			a(3)=4 because 6^3+1 has divisors {1, 7, 31, 217}.

Max Alekseyev, Table of n, a(n) for n = 0..430

Cf. A062394, A000005, A057938, A366621, A366627, A366629, A366630, A366670.

Cf. A046798, A366577, A366606, A366616, A366637, A366656, A366665, A344897, A366688, A366714.

a:=n->numtheory[tau](6^n+1):
seq(a(n), n=0..100);

DivisorSigma[0, 6^Range[0, 70] + 1] (* Paolo Xausa, Apr 19 2025 *)

PARI
```
a(n) = numdiv(6^n+1);
```

a(n) = sigma0(6^n+1) = A000005(A062394(n)).

A366637 Number of divisors of 7^n+1.

Original entry on oeis.org

2, 4, 6, 8, 4, 16, 24, 16, 8, 16, 32, 16, 32, 16, 12, 64, 8, 8, 48, 16, 16, 128, 48, 8, 16, 32, 24, 32, 64, 8, 512, 32, 16, 128, 48, 1024, 256, 16, 12, 256, 64, 64, 96, 512, 32, 2048, 96, 8, 64, 2048, 640, 128, 32, 64, 384, 3072, 256, 256, 96, 64, 512, 8, 48

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

			a(4)=4 because 7^4+1 has divisors {1, 2, 1201, 2402}.

Max Alekseyev, Table of n, a(n) for n = 0..387

Cf. A034491, A000005, A057937, A227575, A366633, A366636, A366638, A366639.

Cf. A046798, A366577, A366606, A366616, A366628, A366656, A366665, A344897, A366688, A366714.

a:=n->numtheory[tau](7^n+1):
seq(a(n), n=0..100);

DivisorSigma[0, 7^Range[0, 62] + 1] (* Paul F. Marrero Romero, Oct 16 2023 *)

PARI
```
a(n) = numdiv(7^n+1);
```

a(n) = sigma0(7^n+1) = A000005(A034491(n)).

A366656 Number of divisors of 8^n+1.

Original entry on oeis.org

2, 3, 4, 8, 4, 12, 16, 12, 8, 20, 48, 24, 16, 12, 64, 64, 8, 48, 64, 24, 16, 64, 64, 24, 32, 96, 768, 192, 32, 24, 1536, 24, 8, 256, 512, 1536, 64, 96, 256, 64, 64, 96, 1024, 48, 128, 1280, 256, 96, 128, 96, 8192, 1024, 32, 48, 1024, 2304, 256, 192, 256, 192

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

			a(4)=4 because 8^4+1 has divisors {1, 17, 241, 4097}.

Max Alekseyev, Table of n, a(n) for n = 0..502

Cf. A062395, A000005, A057936, A274905, A366653, A366638, A366655, A366657, A366658.

Cf. A046798, A366577, A366606, A366616, A366628, A366637, A366665, A344897, A366688, A366714.

a:=n->numtheory[tau](8^n+1):
seq(a(n), n=0..100);

DivisorSigma[0, 8^Range[0,59] + 1] (* Paul F. Marrero Romero, Nov 12 2023 *)

PARI
```
a(n) = numdiv(8^n+1);
```

a(n) = sigma0(8^n+1) = A000005(A062395(n)).

a(n) = A046798(3*n). - Max Alekseyev, Jan 09 2024

A366665 Number of divisors of 9^n+1.

Original entry on oeis.org

2, 4, 4, 8, 8, 12, 8, 16, 4, 16, 8, 16, 64, 16, 16, 48, 4, 16, 16, 16, 32, 128, 32, 16, 16, 128, 16, 32, 64, 16, 128, 32, 4, 64, 32, 384, 256, 32, 64, 128, 32, 32, 1024, 128, 64, 384, 16, 16, 64, 512, 64, 256, 128, 64, 512, 192, 512, 512, 32, 8, 2048, 64, 16

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

			a(2)=4 because 9^2+1 has divisors {1, 2, 41, 82}.

Max Alekseyev, Table of n, a(n) for n = 0..345

Cf. A062396, A000005, A057935, A366661, A366664, A366666, A366667.

Cf. A046798, A366577, A366606, A366616, A366628, A366637, A366656, A344897, A366688, A366714.

a:=n->numtheory[tau](9^n+1):
seq(a(n), n=0..100);

DivisorSigma[0, 9^Range[0,62] + 1] (* Paul F. Marrero Romero, Nov 13 2023 *)

PARI
```
a(n) = numdiv(9^n+1);
```

a(n) = sigma0(9^n+1) = A000005(A062396(n)).

a(n) = A366577(2*n). - Max Alekseyev, Jan 08 2024

A073936 Numbers k such that 2^k + 1 is the product of two distinct primes.

Original entry on oeis.org

5, 6, 7, 11, 12, 13, 17, 19, 20, 23, 28, 31, 32, 40, 43, 61, 64, 79, 92, 101, 104, 127, 128, 148, 167, 191, 199, 256, 313, 347, 356, 596, 692, 701, 1004, 1228, 1268, 1709, 2617, 3539, 3824, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239

Offset: 1

Labos Elemer, Aug 13 2002

nonn

Original name: "2^n + 1 is squarefree and has exactly 2 prime factors."
From Giuseppe Coppoletta, May 08 2017: (Start)
As 3 divides 2^a(n) + 1 for any odd term a(n), all odd terms are prime and exactly the Wagstaff primes (A000978), at the exclusion of 3 (which gives 2^3 + 1 = 3^2 not squarefree).
For the even terms, let a(n) = d * 2^j with d odd integer and j > 0. If d > 1, as (2^2^j)^q  + 1 divides 2^a(n) + 1 for any odd prime q dividing d, then d must be prime.
So the even terms are all given by the following two class:
a) (d = 1) a(n) = 2^j such that Fj is a semiprime Fermat number. Up to now, only j = 5, 6, 7, 8 are known to give a Fermat semiprime, giving the even terms 32, 64, 128 and 256. We are also assured that 2^j is not a term for j = 9..19 because Fj is not a semiprime for those value of j (see Wagstaf's link). F20 is the first composite Fermat number which could give another even term (it would be 2^20 = 1048576). However, it seems highly unlikely that other Fermat semiprimes could exist.
b) (d = p odd prime) a(n) = p * 2^j with j such that Fj is a Fermat prime and p a prime verifying ((Fj - 1)^p + 1)/Fj is a prime.
Exemplifying that, we have:
for j = 1 this gives only the even term a(2) = 2 * 3 = 6  (see Jack Brennen's result in ref),
for j = 2 we have all the terms of type 2^2 * A057182.
for j = 3 the even terms are of type 2^3 * A127317.
For j = 4 at least up to 200000, there is only the term a(41) = 2^4 * 239 = 3824 (see comment in A127317).
All terms after a(50) refer to probabilistic primality tests for 2^a(n) + 1  (see Caldwell's link for the list of the largest certified Wagstaff primes).
After a(56), from the above, the primes 267017, 269987, 374321, 986191, 4031399 and the even value 4101572 are also terms, but still remains the (remote) possibility of some gaps in between. In addition, 13347311 and 13372531 are also terms, but possibly much further in the numbering (see comments in A000978).
(End)
Intersection of A092559 and A066263. - Eric Chen, Jun 13 2018

			11 is a member because 1 + 2^11 = 2049 = 3 * 683.
9 is not a term because 1 + 2^9 = 513 = 3^3 * 19

Giuseppe Coppoletta, Table of n, a(n) for n = 1..56
AMS Books Online, Factorizations of b^n = +-1, b=2,3,5,6,7,10,11,12 Up to High Powers, Third Edition.
Arjen Bot, Factors for 2^n-1 and 2^n+1 for 1200 < n < 10000.
Jack Brennen, Primes of the form (4^p+1)/5^t, Seqfan (Mar 15 2017).
C. Caldwell's The Top Twenty Wagstaff primes.
Mersennewiki, Factorizations Of Cunningham Numbers C+(2,n) (tables).
Samuel S. Wagstaff, The Cunningham Project.

Cf. A000005, A000051, A046798, A092559, A000978. Different from A066263.

Do[ If[ Length[ Divisors[1 + 2^n]] == 4, Print[n]], {n, 1, 200}]
(* Second program: *)
Select[Range@ 200, DivisorSigma[0, 2^# + 1] == 4 &] (* Michael De Vlieger, May 09 2017 *)

[n for n in xsrange(3,200) if sigma(2^n+1,0)==4]
# Second program (faster):

v=[]; N=2000
for n in xsrange(4,N):
    j=valuation(n,2)
    if j<5:
        Fj=2^2^j+1; p=ZZ(n/2^j); q=ZZ((2^n+1)/Fj)
        if p.is_prime() and q.is_prime(proof=false): v.append(n)
    elif j<9 and n.is_power_of(2): v.append(n)
print(v) # Giuseppe Coppoletta, May 11 2017

Solutions to A000005[A000051(x)]=4 or A046798[x]=4

Edited by Robert G. Wilson v, Aug 19 2002
a(28)-a(51) by Giuseppe Coppoletta, May 02 2017
Name reworded by Jon E. Schoenfield, Jun 15 2018

A374237 Irregular triangle read by rows where row n lists, in increasing order, the divisors of 2^n + 1.

Original entry on oeis.org

1, 2, 1, 3, 1, 5, 1, 3, 9, 1, 17, 1, 3, 11, 33, 1, 5, 13, 65, 1, 3, 43, 129, 1, 257, 1, 3, 9, 19, 27, 57, 171, 513, 1, 5, 25, 41, 205, 1025, 1, 3, 683, 2049, 1, 17, 241, 4097, 1, 3, 2731, 8193, 1, 5, 29, 113, 145, 565, 3277, 16385, 1, 3, 9, 11, 33, 99, 331, 993, 2979, 3641, 10923, 32769

Offset: 0

Paolo Xausa, Jul 02 2024

			Triangle begins:
   [0]  1,   2;
   [1]  1,   3;
   [2]  1,   5;
   [3]  1,   3,  9;
   [4]  1,  17;
   [5]  1,   3, 11,  33;
   [6]  1,   5, 13,  65;
   [7]  1,   3, 43, 129;
   [8]  1, 257;
   [9]  1,   3,  9,  19,  27,   57, 171, 513;
  [10]  1,   5, 25,  41, 205, 1025;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11484 (rows 0..135 of the triangle, flattened).
Index entries for sequences related to divisors of numbers

Subsequence of A027750.
Cf. A000051, A002586 (2nd column), A046798 (row lengths), A069060 (row products), A069061 (row sums).
Cf. A361438 (analogous for 2^n - 1).

T:= n-> sort([numtheory[divisors](2^n+1)[]])[]:
seq(T(n), n=0..15);  # Alois P. Heinz, Oct 20 2024

Mathematica
```
Divisors[2^Range[0, 20] + 1]
```

A074696 Numbers k such that 2^k+1 has more than k divisors (k such that A000005(2^k+1) > k).

Original entry on oeis.org

0, 1, 30, 42, 45, 50, 54, 63, 70, 75, 78, 81, 90, 99, 102, 105, 114, 118, 126, 130, 135, 138, 150, 153, 154, 162, 165, 168, 170, 171, 174, 175, 177, 180, 182, 186, 189, 190, 194, 195, 196, 198, 204, 210, 216, 220, 222, 225, 228, 230, 231, 234, 238

Offset: 1

Benoit Cloitre, Sep 03 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..441

Cf. A000005, A046798.

Select[Range[0, 150], DivisorSigma[0, 2^#+1] > # &] (* Amiram Eldar, May 10 2022 *)

isok(n) = numdiv(2^n+1) > n; \\ Michel Marcus, Nov 29 2013

More terms from Michel Marcus, Nov 29 2013

a(1) = 0 inserted by Amiram Eldar, May 10 2022

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

A283931 Numbers m such that tau(2^m) = tau(2^m + 1).

Original entry on oeis.org

1, 63, 511

Offset: 1

Jaroslav Krizek, Mar 18 2017

tau(m) is the number of divisors of m (A000005).
Numbers m such that A046798(m) = m + 1.
Numbers m such that A000005(A000079(m)) = A000005(A000051(m)).
Corresponding values of tau(2^m): 2, 64, 512, ...
Corresponding pairs of numbers (2^m, 2^m + 1): (2, 3); (9223372036854775808, 9223372036854775809); ...
All terms in the sequence are odd. (If m were an even number m = 2k, then tau(2^m) = m + 1 = 2k + 1 would be odd, but tau(2^m + 1) = tau(2^(2k) + 1) = tau(4^k + 1) would be even, since 4^k + 1 can never be a square.) - Jon E. Schoenfield, Mar 26 2017
Each of the known terms a(1), a(2), and a(3) is of the form 2^j - 1 (so tau(2^m) = 2^j); the corresponding values of j are 1, 6, and 9. Are there additional terms of this form? The sequence does not include 2^10 - 1 = 1023; even without completely factoring 2^1023 + 1, it can be seen that 3 must appear in that factorization with a multiplicity of 2, so tau(2^1023 + 1) is divisible by 3, and thus cannot be 1024. Since 2^2047 + 1 is 3 * 179 * 2796203 * 53484017 * 62020897 * 18584774046020617 times a 40-digit prime times a 537-digit composite (coprime to each of the known 7 prime factors), 2047 will be a term iff that 537-digit composite has exactly 2048/2^7 = 16 divisors. - Jon E. Schoenfield, Mar 30 2017; updated by Max Alekseyev, Feb 16 2023
Candidate next terms: 1151?, 1187?, 1231?, 1259?, 1319?, 1343?, 1399?, 1471?, 1535?, 1567?, 1583?, 1663?, 1759?, 1847?, 1913?, 1919?, 2047?. In particular, a(4) >= 1151.  - Max Alekseyev, Feb 16 2023

			For m = 1, tau(2) = tau(3) = 2.

Cf. A000005, A000051, A000079, A046798, A283930.

[n: n in [0..500] | NumberOfDivisors(2^n) eq NumberOfDivisors(2^n + 1)]

A348177 a(n) is the number of pair of positive integers (x, y) with 1 <= x <= y such that sum s = x + y and product p = x * y satisfy s + p = 2^n, with n > 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 3, 2, 1, 1, 1, 3, 5, 0, 1, 7, 1, 1, 5, 3, 1, 3, 7, 7, 9, 1, 3, 23, 1, 1, 11, 7, 15, 7, 3, 7, 5, 1, 3, 31, 1, 3, 31, 15, 3, 3, 3, 31, 23, 3, 3, 31, 23, 3, 11, 3, 7, 7, 1, 15, 31, 1, 31, 31, 3, 5, 11, 47, 3, 15, 3, 15, 47, 7, 31, 383, 1, 3

Offset: 1

Bernard Schott, Oct 05 2021

nonn

That is a generalization of a problem proposed by French site Diophante in link.
Some results:
x and y satisfy (x+1)*(y+1) = 2^n + 1.
x and y are both even, so 2 <= x <= y < 2^n.
There is only one case such that x = y, it is for n = 3 with x = y = 2 (Examples).
a(n) = 0 iff 2^n+1 is Fermat prime (A019434), hence iff n = 1, 2, 4, 8, 16.
a(n) = 1 iff 2^n+1 is semiprime (n is in A092559).

			For n = 3, only (x=y=2) satisfy s = 2+2 = 4, p = 2*2 = 4 and s+p = 8 = 2^3, hence a(3) = 1.
For n = 6, only (x=4, y=12) satisfy s = 4+12 = 16, p = 4*12 = 48 and s+p = 64 = 2^6 hence a(6) = 1.
For n = 9, (2,170), (8,56), (18,26) are the 3 solutions, with 172+340=512=2^9, 64+448=512, 44+468=512, hence a(9) = 3.
For n = 10, (4, 204) and (24, 40) are the 2 solutions, with 208+816=1024=2^10 and 64+960=1024, hence a(10) = 2.

Diophante, A4921 - Zéro, une, mille et plus (in French).

Cf. A046798, A092559, A019434.

with(numtheory):
M := seq(ceil((tau(2^n+1)-2)/2), n=1..100);

a[3] = 1; a[n_] := DivisorSigma[0, 2^n + 1]/2 - 1; Array[a, 80] (* Amiram Eldar, Oct 05 2021 *)

a(n) = ceil((numdiv(2^n+1) - 2)/2); \\ Michel Marcus, Oct 11 2021

For n<>3, the number of positive pairs solution (x,y) is a(n) = (tau(2^n+1) - 2)/2.
For n = 3, there is only one pair solution and a(3) = (tau(2^3+1) - 1)/2 = 1, with (x, y) = (2, 2).
a(n) = ceiling((tau(2^n+1) - 2)/2) = ceiling((A046798(n)-2)/2) is the general formula.

cp's OEIS Frontend

A366628 Number of divisors of 6^n+1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A366637 Number of divisors of 7^n+1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A366656 Number of divisors of 8^n+1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A366665 Number of divisors of 9^n+1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A073936 Numbers k such that 2^k + 1 is the product of two distinct primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Sage

Formula

Extensions

A374237 Irregular triangle read by rows where row n lists, in increasing order, the divisors of 2^n + 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A074696 Numbers k such that 2^k+1 has more than k divisors (k such that A000005(2^k+1) > k).

Views

Author

Keywords

Links

Crossrefs

Programs