A003524 -id:A003524 - cp's OEIS frontend

A245027 Divisors of 7^12 - 1.

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 16, 18, 19, 20, 24, 25, 26, 30, 32, 36, 38, 39, 40, 43, 45, 48, 50, 52, 57, 60, 65, 72, 75, 76, 78, 80, 86, 90, 95, 96, 100, 104, 114, 117, 120, 129, 130, 144, 150, 152, 156, 160, 171, 172, 180, 181, 190, 195, 200, 208

Offset: 1

Bruno Berselli, Jul 10 2014

Number of divisors of k^12-1 for k = 2..20: 24 (2), 80 (3), 96 (4), 240 (5), 128 (6), 864 (7), 512 (8), 384 (9), 256 (10), 1920 (11), 256 (12), 960 (13), 384 (14), 448 (15), 768 (16), 1792 (17), 768 (18), 3840 (19), 384 (20).

The following triangular numbers belong to this sequence: 1, 3, 6, 10, 15, 36, 45, 78, 120, 171, 190, 300, 325, 741, 780, 2080, 2850, 4560, 8385, 14706, 16290, 5915080, 1730160900.

			13841287200 = 2^5 * 3^2 * 5^2 * 13 * 19 * 43 * 181.

Bruno Berselli, Table of n, a(n) for n = 1..864
Index entries for sequences related to divisors of numbers

Cf. Divisors of k^12-1: A003524 (k=2); A003532 (k=4); A003543 (k=8), A027902 (k=9), A027897 (k=10), A245028 (k=11).

Magma
```
Divisors(7^12-1);
```
Mathematica
```
Divisors[7^12 - 1]
```
Maxima
```
divisors(7^12-1);
```
PARI
```
divisors(7^12-1)
```
Sage
```
divisors(7^12-1)
```

A245028 Divisors of 11^12 - 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 24, 26, 28, 30, 35, 36, 37, 38, 39, 40, 42, 45, 48, 52, 56, 57, 60, 61, 63, 65, 70, 72, 74, 76, 78, 80, 84, 90, 91, 95, 104, 105, 111, 112, 114, 117, 120, 122, 126, 130, 133, 140, 144, 148, 152

Offset: 1

Bruno Berselli, Jul 10 2014

See Comments section in A245027.
The following 36 triangular numbers belong to this sequence: 1, 3, 6, 10, 15, 21, 28, 36, 45, 78, 91, 105, 120, 171, 190, 210, 630, 666, 703, 741, 780, 1596, 1830, 4095, 4560, 5460, 6216, 16653, 33670, 46360, 103740, 115440, 221445, 274170, 365085, 392303547090.
The following terms of A001082 (without 1, 21 and 120, which appear in the previous list) are in sequence: 5, 8, 16, 40, 56, 65, 133, 208, 280, 456, 481, 560, 936, 1008, 1281, 1365, 1680, 1776, 1976, 4880, 5985, 10920, 11285, 44408, 47880, 590520, 658008, 731120, 973560, 1046142792240.
Also, 4/5 of the members are divisible by 3 and 2/3 of them are even.

			3138428376720 = 2^4 * 3^2 * 5 * 7 * 13 * 19 * 37 * 61 * 1117.

Bruno Berselli, Table of n, a(n) for n = 1..1920
Index entries for sequences related to divisors of numbers

Cf. Divisors of k^12-1: A003524 (k=2); A003532 (k=4); A245027 (k=7), A003543 (k=8), A027902 (k=9), A027897 (k=10).

Magma
```
Divisors(11^12-1);
```
Mathematica
```
Divisors[11^12 - 1]
```
Maxima
```
divisors(11^12-1);
```
PARI
```
divisors(11^12-1)
```
Sage
```
divisors(11^12-1)
```

A334133 Numbers k > 2 such that gpf(A111076(k)^lambda(k) - 1) = gpf(lambda(k) + 1); where gpf is the greatest prime factor (A006530), and lambda = A002322 is the Carmichael function.

Original entry on oeis.org

3, 5, 6, 9, 10, 12, 13, 15, 16, 20, 21, 24, 30, 35, 39, 40, 45, 60, 63, 65, 80, 91, 105, 117, 120, 195, 240, 273, 315, 455, 585, 819, 1365, 4095

Offset: 1

Thomas Ordowski, Apr 15 2020

Prime numbers in this sequence are 3, 5, and 13. These are primes p with primitive root 2 (A001122) such that gpf(2^(p-1)-1) = p.
The set of all numbers of this sequence is probably also finite and complete (all terms are on the list).
The odd terms of this sequence up to 4095 = 2^12-1 are exactly the divisors of this number (A003524) except 1 and 7. [Edited by M. F. Hasler, Apr 17 2020]
Conjecture: all odd terms {3, 5, 9, 13, 15, 21, 35, 39, 45, 63, 65, 91, 105, 117, 195, 273, 315, 455, 585, 819, 1365, 4095} are odd numbers k such that gpf(2^m-1) = gpf(m+1), where m = A002326((k-1)/2) is the multiplicative order of 2 mod 2k+1. - Amiram Eldar, Apr 15 2020
No further terms below 10^5. - M. F. Hasler, Apr 17 2020

Cf. A001122, A002322, A002326, A003524, A006530, A111076.

gpf[n_] := FactorInteger[n][[-1, 1]]; gpfQ[n_, p_] := Module[{ps = Select[Range[p], PrimeQ], np, m, k}, np = Length[ps]; m = n; If[Divisible[n, p], Do[m /= (ps[[k]]^IntegerExponent[m, ps[[k]]]), {k, 1, np}]; m == 1, False]]; f[n_] := Module[{k = 1, lam = CarmichaelLambda[n]}, While[! CoprimeQ[n, k] || MultiplicativeOrder[k, n] != lam, k++]; k]; Select[Range[2, 2^12], gpfQ[f[#]^(c = CarmichaelLambda[#]) - 1, gpf[c + 1]] &] (* Amiram Eldar, Apr 15 2020 *)

select( is_A334133(n)={n>2||return; my(o=lcm(znstar(n)[2]),k=1); while(gcd(k++,n)>1 || znorder(Mod(k,n))M. F. Hasler, Apr 17 2020

More terms from Amiram Eldar, Apr 15 2020

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A245027 Divisors of 7^12 - 1.

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A245028 Divisors of 11^12 - 1.

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A334133 Numbers k > 2 such that gpf(A111076(k)^lambda(k) - 1) = gpf(lambda(k) + 1); where gpf is the greatest prime factor (A006530), and lambda = A002322 is the Carmichael function.

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