A046801 -id:A046801 - cp's OEIS frontend

A366661 Number of divisors of 9^n-1.

4, 10, 16, 24, 24, 80, 16, 112, 128, 180, 64, 384, 16, 160, 768, 256, 128, 1280, 64, 864, 768, 640, 32, 14336, 384, 160, 4096, 1536, 256, 23040, 128, 576, 2048, 1280, 768, 12288, 128, 640, 12288, 16128, 128, 61440, 32, 12288, 196608, 320, 512, 131072, 2048

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

			a(2)=10 because 9^2-1 has divisors {1, 2, 4, 5, 8, 10, 16, 20, 40, 80}.

Max Alekseyev, Table of n, a(n) for n = 1..690

Cf. A024101, A000005, A057952, A366660, A366662, A366663.

Cf. A046801, A366575, A366602, A366612, A366621, A366633, A366652, A070528, A366683, A366709.

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

Mathematica
```
DivisorSigma[0, 9^Range[100]-1]
```
PARI
```
a(n) = numdiv(9^n-1);
```

a(n) = sigma0(9^n-1) = A000005(A024101(n)).

a(n) = A366575(2*n) = A366575(n) * A366577(n) * (4 + A007814(n)) / (2 * (3 + A007814(n))). - Max Alekseyev, Jan 07 2024

A086251 Number of primitive prime factors of 2^n - 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 1, 2, 3, 3, 3, 1, 3, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 1, 1, 3, 1, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 2, 3, 2, 2, 1, 3, 3, 2, 3, 2, 2, 3

Offset: 1

T. D. Noe, Jul 14 2003

A prime factor of 2^n - 1 is called primitive if it does not divide 2^r - 1 for any r < n. Equivalently, p is a primitive prime factor of 2^n - 1 if ord(2,p) = n. Zsigmondy's theorem says that there is at least one primitive prime factor for n > 1, except for n=6. See A086252 for those n that have a record number of primitive prime factors.
Number of odd primes p such that A002326((p-1)/2) = n. Number of occurrences of number n in A014664. - Thomas Ordowski, Sep 12 2017
The prime factors are not counted with multiplicity, which matters for a(364)=4 and a(1755)=6. - Jeppe Stig Nielsen, Sep 01 2020

			a(11) = 2 because 2^11 - 1 = 23*89 and both 23 and 89 have order 11.

Table of n, a(n) for n = 1..1206
Eric Weisstein's World of Mathematics, Zsigmondy Theorem

Cf. A046800, A046051 (number of prime factors, with repetition, of 2^n-1), A086252, A002588, A005420, A002184, A046801, A049093, A049094, A059499, A085021, A097406, A112927, A237043.

Join[{0}, Table[cnt=0; f=Transpose[FactorInteger[2^n-1]][[1]]; Do[If[MultiplicativeOrder[2, f[[i]]]==n, cnt++ ], {i, Length[f]}]; cnt, {n, 2, 200}]]

a(n) = sumdiv(n, d, moebius(n/d)*omega(2^d-1)); \\ Michel Marcus, Sep 12 2017

a(n) = my(m=polcyclo(n, 2)); omega(m/gcd(m,n)) \\ Jeppe Stig Nielsen, Sep 01 2020

a(n) = Sum{d|n} mu(n/d) A046800(d), inverse Mobius transform of A046800.
a(n) <= A182590(n). - Thomas Ordowski, Sep 14 2017
a(n) = A001221(A064078(n)). - Thomas Ordowski, Oct 26 2017

Terms to a(500) in b-file from T. D. Noe, Nov 11 2010
Terms a(501)-a(1200) in b-file from Charles R Greathouse IV, Sep 14 2017
Terms a(1201)-a(1206) in b-file from Max Alekseyev, Sep 11 2022

A366602 Number of divisors of 4^n-1.

Original entry on oeis.org

2, 4, 6, 8, 8, 24, 8, 16, 32, 48, 16, 96, 8, 64, 96, 32, 8, 512, 8, 192, 144, 128, 16, 768, 128, 128, 160, 256, 64, 4608, 8, 128, 384, 128, 512, 8192, 32, 128, 192, 768, 32, 9216, 32, 1024, 4096, 512, 64, 6144, 32, 8192, 1536, 1024, 64, 10240, 3072, 2048, 384

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(4)=8 because 4^4-1 has divisors {1, 3, 5, 15, 17, 51, 85, 255}.

Max Alekseyev, Table of n, a(n) for n = 1..1128

Cf. A024036, A000005, A057957, A295501, A366603, A366604.

Cf. A046798, A046801, A366575, A366612, A366621, A366633, A366652, A366661, A070528, A366683, A366709.

a:=n->numtheory[tau](4^n-1):
seq(a(n), n=1..100);

DivisorSigma[0,4^Range[100]-1] (* Paolo Xausa, Oct 14 2023 *)

PARI
```
a(n) = numdiv(4^n-1);
```

a(n) = sigma0(4^n-1) = A000005(A024036(n)).

a(n) = A046801(2*n) = A046798(n) * A046801(n). - Max Alekseyev, Jan 07 2024

A366633 Number of divisors of 7^n-1.

Original entry on oeis.org

4, 10, 12, 36, 8, 60, 16, 84, 64, 80, 16, 864, 8, 160, 96, 384, 16, 640, 16, 1536, 96, 160, 32, 16128, 32, 80, 1280, 1152, 32, 3840, 32, 1728, 384, 80, 128, 18432, 32, 160, 192, 14336, 32, 7680, 16, 4608, 2048, 160, 16, 147456, 256, 640, 768, 1152, 32, 25600

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(5)=8 because 7^5-1 has divisors {1, 2, 3, 6, 2801, 5602, 8403, 168061}.

Max Alekseyev, Table of n, a(n) for n = 1..388

Cf. A024075, A000005, A057954, A059889, A074249, A218358, A366632, A366634, A366635.

Cf. A046801, A366575, A366602, A366612, A366621, A366652, A366661, A070528, A366683, A366709.

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

Mathematica
```
DivisorSigma[0, 7^Range[100]-1]
```
PARI
```
a(n) = numdiv(7^n-1);
```

a(n) = sigma0(7^n-1) = A000005(A024075(n)).

A366612 Number of divisors of 5^n-1.

Original entry on oeis.org

3, 8, 6, 20, 12, 48, 6, 48, 24, 64, 6, 240, 6, 64, 96, 224, 12, 512, 24, 640, 48, 128, 12, 1152, 192, 64, 384, 320, 24, 6144, 12, 1024, 48, 128, 384, 10240, 24, 512, 48, 6144, 12, 18432, 12, 1280, 3072, 128, 6, 10752, 12, 4096, 192, 960, 24, 81920, 576, 1536

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(3)=6 because 5^3-1 has divisors {1, 2, 4, 31, 62, 124}.

Max Alekseyev, Table of n, a(n) for n = 1..502

Cf. A024049, A000005, A057956, A059887, A074479, A143665, A295502, A218357, A366611, A366613.

Cf. A046801, A366575, A366602, A366621, A366633, A366652, A366661, A070528, A366683, A366709.

a:=n->numtheory[tau](5^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[0, 5^Range[100]-1]
```
PARI
```
a(n) = numdiv(5^n-1);
```

a(n) = sigma0(5^n-1) = A000005(A024049(n)).

A366652 Number of divisors of 8^n-1.

Original entry on oeis.org

2, 6, 4, 24, 8, 32, 12, 96, 8, 96, 16, 512, 16, 144, 64, 768, 32, 160, 16, 4608, 96, 384, 16, 8192, 128, 192, 64, 9216, 64, 4096, 8, 6144, 256, 1536, 1536, 10240, 64, 384, 512, 73728, 32, 6144, 32, 24576, 1024, 384, 64, 262144, 64, 12288, 256, 147456, 256

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

			a(5)=8 because 8^5-1 has divisors {1, 7, 31, 151, 217, 1057, 4681, 32767}.

Max Alekseyev, Table of n, a(n) for n = 1..500

Cf. A024088, A000005, A057953, A059890, A366651, A366653, A366654.

Cf. A046801, A366575, A366602, A366612, A366621, A366633, A366661, A070528, A366683, A366709.

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

Mathematica
```
DivisorSigma[0, 8^Range[100]-1]
```
PARI
```
a(n) = numdiv(8^n-1);
```

a(n) = sigma0(8^n-1) = A000005(A024088(n)).

a(n) = A046801(3*n). - Max Alekseyev, Jan 09 2024

A366575 Number of divisors of 3^n - 1.

Original entry on oeis.org

2, 4, 4, 10, 6, 16, 4, 24, 8, 24, 8, 80, 4, 16, 24, 112, 8, 128, 8, 180, 16, 64, 8, 384, 24, 16, 64, 160, 16, 768, 16, 256, 32, 128, 48, 1280, 8, 64, 96, 864, 16, 768, 8, 640, 384, 32, 32, 14336, 128, 384, 64, 160, 16, 4096, 128, 1536, 128, 256, 8, 23040, 8

Offset: 1

Sean A. Irvine, Oct 13 2023

nonn

			a(4)=10 because 3^4-1 has divisors {1, 2, 4, 5, 8, 10, 16, 20, 40, 80}.

Max Alekseyev, Table of n, a(n) for n = 1..690

Cf. A024023, A000005, A046801, A133801, A295500, A366576.

Cf. A046801, A366602, A366612, A366621, A366633, A366652, A366661, A070528, A366683, A366709.

a:=n->numtheory[tau](3^n-1):
seq(a(n), n=1..100);

DivisorSigma[0,3^Range[100]-1] (* Paolo Xausa, Oct 15 2023 *)

a(n) = sigma0(3^n-1) = A000005(A024023).

A359082 Indices of records in A246600.

Original entry on oeis.org

1, 3, 15, 63, 255, 495, 4095, 96255, 98175, 130815, 203775, 1048575, 5810175, 6455295, 16777215, 67096575, 88062975, 389656575, 553517055, 850917375, 1157349375, 9141354495, 12826279935, 22828220415, 26818379775, 31684427775, 68719476735, 242870910975, 1168231038975

Offset: 1

Amiram Eldar, Dec 15 2022

Numbers k with a record number of divisors d such that the bitwise OR of k and d is equal to k (or equivalently, the bitwise AND of k and d is equal to d).
All the terms are odd since A246600(2*k) = A246600(k).
This sequence is infinite since A246600(2^m-1) = A000005(2^m-1) = A046801(m), and A046801 is unbounded (A046801(2^(m+1)) > A046801(2^m) for all m >= 0).
The corresponding record values are 1, 2, 4, 6, 8, 11, 24, 25, 28, 32, 35, 48, 56, 89, 96, 105, 121, 127, 148, 162, 216, 243, 245, 256, 319, 358, 512, 633, 768, ... .
2*10^11 < a(28) <= 2^48 - 1.

Cf. A046801, A246600, A359080, A359081, A359083.

s[n_] := DivisorSum[n, 1 &, BitAnd[n, #] == # &]; seq={}; sm = 0; Do[If[(sn = s[n]) > sm, sm = sn; AppendTo[seq, n]], {n, 1, 10^6}]; seq

lista(nmax) = {my(list = List(), ndmax = 0, d, s); for(n = 1, nmax, nd = sumdiv(n, d, bitand(d, n)==d); if(nd > ndmax, ndmax = nd; listput(list, n))); Vec(list)};

a(28)-a(29) from Martin Ehrenstein, Dec 19 2022

A359081 a(n) is the least number k such that A246600(k) = n, and -1 if no such k exists.

Original entry on oeis.org

1, 3, 39, 15, 175, 63, 1275, 255, 1215, 891, 495, 6975, 14175, 26367, 13311, 8127, 20475, 42735, 95931, 69615, 36855, 24255, 404415, 4095, 96255, 423423, 253935, 98175, 913275, 165375, 507375, 130815, 3198975, 1576575, 203775, 2154495, 4398975, 1616895, 1556415

Offset: 1

Amiram Eldar, Dec 15 2022

All the terms are odd since A246600(2*k) = A246600(k).

Cf. A000005, A046801, A246600, A359080, A359082, A359083.

seq[nmax_, kmax_] := Module[{s = Table[0, {nmax}], c = 0, k = 1, i}, While[c < nmax && k < kmax, i = DivisorSum[k, 1 &, BitOr[#, k] == k &]; If[i <= nmax && s[[i]] == 0, c++; s[[i]] = k]; k++]; s]; seq[20, 5*10^6]

lista(nmax, kmax=oo) = {my(s = vector(nmax), c = 0, k = 1, i); while(c < nmax && k < kmax, i = sumdiv(k, d, bitor(d, k) == k); if(i <= nmax && s[i] == 0, c++; s[i] = k); k++); s};

A335432 Number of anti-run permutations of the prime indices of Mersenne numbers A000225(n) = 2^n - 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 6, 2, 6, 2, 36, 1, 6, 6, 24, 1, 24, 1, 240, 6, 24, 2, 1800, 6, 6, 6, 720, 6, 1800, 1, 120, 24, 6, 24, 282240, 2, 6, 24, 15120, 2, 5760, 6, 5040, 720, 24, 6, 1451520, 2, 5040, 120, 5040, 6, 1800, 720, 40320, 24, 720, 2, 1117670400, 1, 6, 1800, 5040, 6

Offset: 1

Gus Wiseman, Jul 02 2020

nonn

An anti-run is a sequence with no adjacent equal parts.

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(1) = 1 through a(10) = 6 permutations:
  ()  (2)  (4)  (2,3)  (11)  (2,4,2)  (31)  (2,3,7)  (21,4)  (11,2,5)
                (3,2)                       (2,7,3)  (4,21)  (11,5,2)
                                            (3,2,7)          (2,11,5)
                                            (3,7,2)          (2,5,11)
                                            (7,2,3)          (5,11,2)
                                            (7,3,2)          (5,2,11)

Andrew Howroyd, Table of n, a(n) for n = 1..200

The version for factorial numbers is A335407.
Anti-run compositions are A003242.
Anti-run patterns are A005649.
Permutations of prime indices are A008480.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Strict permutations of prime indices are A335489.
Cf. A056239, A106351, A112798, A114938, A278990, A292884, A325535, A335125.
Mersenne numbers: A000225, A046051, A046800, A046801, A049093, A059305, A159611, 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,0,30}]

\\ See A335452 for count.
a(n) = {count(factor(2^n-1)[,2])} \\ Andrew Howroyd, Feb 03 2021

a(n) = A335452(A000225(n)).

Terms a(51) and beyond from Andrew Howroyd, Feb 03 2021

cp's OEIS Frontend

A366661 Number of divisors of 9^n-1.

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A086251 Number of primitive prime factors of 2^n - 1.

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A366602 Number of divisors of 4^n-1.

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A366633 Number of divisors of 7^n-1.

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A366612 Number of divisors of 5^n-1.

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A366652 Number of divisors of 8^n-1.

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A366575 Number of divisors of 3^n - 1.

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