A309906 -id:A309906 - cp's OEIS frontend

A341668 a(n) is the number of divisors of prime(n)^7 - 1.

2, 4, 6, 16, 16, 12, 10, 24, 16, 18, 16, 36, 32, 48, 16, 24, 64, 24, 32, 48, 24, 128, 16, 16, 96, 36, 64, 32, 96, 60, 144, 64, 32, 64, 12, 48, 48, 20, 16, 24, 16, 144, 128, 56, 96, 192, 96, 128, 32, 48, 64, 96, 80, 16, 72, 32, 192, 64, 96, 192, 32, 48, 48, 64

Offset: 1

Jon E. Schoenfield, Feb 26 2021

nonn

a(n) >= A309906(7) = 8 for n > 3.

			       p =                     factorization
  n  prime(n)   p^7 - 1          of p^7 - 1       a(n)
  -  --------  ----------  ---------------------  ----
  1      2            127  127                      2
  2      3           2186  2 * 1093                 4
  3      5          78124  2^2 * 19531              6
  4      7         823542  2 * 3 * 29 * 4733       16
  5     11       19487170  2 * 5 * 43 * 45319      16
  6     13       62748516  2^2 * 3 * 5229043       12
  7     17      410338672  2^4 * 25646167          10
  8     19      893871738  2 * 3^2 * 701 * 70841   24
  9     23     3404825446  2 * 11 * 29 * 5336717   16

Cf. A000005, A000040, A309906, A341669.

a[n_] := DivisorSigma[0, Prime[n]^7 - 1]; Array[a, 50] (* Amiram Eldar, Feb 27 2021 *)

a(n) = numdiv(prime(n)^7-1); \\ Michel Marcus, Feb 27 2021

a(n) = A000005(A000040(n)^7 - 1).

A341669 Primes p such that p^7 - 1 has 8 divisors.

Original entry on oeis.org

863, 1439, 2039, 3167, 3803, 4799, 10559, 11423, 14087, 14207, 15287, 15803, 16139, 18743, 20663, 21059, 21179, 22343, 25307, 25919, 26459, 29483, 29759, 30803, 32507, 32987, 33107, 34319, 34367, 35879, 43427, 45887, 46559, 46643, 46919, 54959, 57119, 57587

Offset: 1

Jon E. Schoenfield, Feb 26 2021

nonn

For each term p, p^7 - 1 = (p-1)*(p^6 + p^5 + p^4 + p^3 + p^2 + p + 1) is a number of the form 2*q*r (where q and r are distinct primes): p-1 = 2*q and p^6 + p^5 + p^4 + p^3 + p^2 + p + 1 = r.

Conjecture: sequence is infinite.

			      p =
  n   a(n)        factorization of p^7 - 1
  -  -----  ------------------------------------
  1    863  2 *  431 *        413588356833933793
  2   1439  2 *  719 *       8885189025331426081
  3   2039  2 * 1019 *      71897932302115976281
  4   3167  2 * 1583 *    1009312223899992366817
  5   3803  2 * 1901 *    3026022586778671180093
  6   4799  2 * 2399 *   12217856103420111345601
  7  10559  2 * 5279 * 1386046726502834819142721
  8  11423  2 * 5711 * 2221872233870122705845793
  9  14087  2 * 7043 * 7815232779386331437540137

Cf. A000005, A000040, A309906, A341668.

Select[Range[60000], PrimeQ[#] && DivisorSigma[0, #^7 - 1] == 8 &] (* Amiram Eldar, Feb 27 2021 *)

isok(p) = isprime(p) && (numdiv(p^7-1) == 8); \\ Michel Marcus, Feb 27 2021

A342065 Primes p such that p^9 - 1 has 16 divisors.

Original entry on oeis.org

383, 12227, 44519, 44687, 56003, 97523, 130259, 148727, 160739, 169007, 208799, 258887, 270563, 281783, 331883, 336143, 353099, 364979, 498119, 501707, 550679, 573107, 577667, 716747, 753023, 775367, 781007, 784727, 861299, 887543, 1084247, 1085159, 1099139

Offset: 1

Jon E. Schoenfield, Feb 27 2021

nonn

Conjecture: sequence is infinite.
The only primes p such that p^9 - 1 has fewer than A309906(9)=16 divisors are p=2 (2^9 - 1 = 511 = 7*73 has 4 divisors) and p=3 (3^9 - 1 = 19682 = 2*13*757 has 8 divisors).
For every term p, p^9 - 1 is of the form 2*q*r*s, where q = (p-1)/2, r = (p^2 + p + 1), and s = (p^6 + p^3 + 1) are primes (see Example section).
The Generalized Dickson's Conjecture implies there are infinitely many p such that p, (p-1)/2, p^2+p+1 and p^6+p^3+1 are prime. - Robert Israel, Feb 28 2021

			                       factorization of p^9 - 1
    p =   ===================================================
n   a(n)  2 * (p-1)/2 * (p^2+p+1) *      (p^6 + p^3 + 1)
-  -----  ---------------------------------------------------
1    383  2 *   191   *    147073 *          3156404483062657
2  12227  2 *  6113   * 149511757 * 3341330794198073514753973

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A000005, A000040, A309906, A341670.

R:= NULL: count:= 0: q:= 1:
while count < 100 do
  q:= nextprime(q);
  p:= 2*q+1;
  if isprime(p) and isprime(p^2+p+1) and isprime(p^6+p^3+1) then
    count:= count+1; R:= R, p;
  fi
od:
R; # Robert Israel, Feb 28 2021

A341663 a(n) is the number of divisors of prime(n)^3 - 1.

Original entry on oeis.org

2, 4, 6, 12, 16, 18, 10, 16, 16, 24, 24, 48, 16, 24, 16, 24, 8, 72, 72, 16, 32, 72, 16, 16, 36, 18, 24, 32, 60, 40, 32, 16, 64, 48, 48, 72, 36, 96, 8, 12, 16, 96, 96, 84, 36, 32, 192, 24, 16, 72, 32, 32, 60, 32, 36, 48, 48, 40, 144, 64, 48, 12, 64, 32, 72, 24

Offset: 1

Jon E. Schoenfield, Feb 26 2021

nonn

a(n) >= A309906(3) = 8 for n > 3.

			        p =                   factorization
   n  prime(n)  p^3 - 1         of p^3 - 1        a(n)
  --  --------  -------  -----------------------  ----
   1      2           7  7                          2
   2      3          26  2 * 13                     4
   3      5         124  2^2 * 31                   6
   4      7         342  2 * 3^2 * 19              12
   5     11        1330  2 * 5 * 7 * 19            16
   6     13        2196  2^2 * 3^2 * 61            18
   7     17        4912  2^4 * 307                 10
   8     19        6858  2 * 3^3 * 127             16
   9     23       12166  2 * 7 * 11 * 79           16
  10     29       24388  2^2 * 7 * 13 * 67         24
  11     31       29790  2 * 3^2 * 5 * 331         24
  12     37       50652  2^2 * 3^3 * 7 * 67        48
  13     41       68920  2^3 * 5 * 1723            16
  14     43       79506  2 * 3^2 * 7 * 631         24
  15     47      103822  2 * 23 * 37 * 61          16
  16     53      148876  2^2 * 7 * 13 * 409        24
  17     59      205378  2 * 29 * 3541              8
  18     61      226980  2^2 * 3^2 * 5 * 13 * 97   72
  19     67      300762  2 * 3^2 * 7^2 * 11 * 31   72
  20     71      357910  2 * 5 * 7 * 5113          16

Cf. A000005, A000040, A309906, A341659.

a[n_] := DivisorSigma[0, Prime[n]^3 - 1]; Array[a, 50] (* Amiram Eldar, Feb 26 2021 *)

a(n) = numdiv(prime(n)^3-1); \\ Michel Marcus, Feb 26 2021

a(n) = A000005(A000040(n)^3 - 1).

A342066 Primes p such that p^10 - 1 has 256 divisors.

Original entry on oeis.org

1187, 4723, 33037, 66973, 72797, 87973, 100523, 197123, 219683, 229693, 276293, 278827, 440653, 448997, 482837, 562963, 601333, 621443, 670493, 742723, 846877, 892357, 1033427, 1149307, 1166027, 1245067, 1256747, 1614413, 1679773, 1865693, 1950323, 1970467

Offset: 1

Jon E. Schoenfield, Feb 27 2021

nonn

Conjecture: sequence is infinite.
The only primes p such that p^10 - 1 has fewer than A309906(10)=256 divisors are 2, 3, 5, 7, 11, 13, and 43.
p^10 - 1 = (p-1)*(p+1)*(p^4 - p^3 + p^2 - p + 1)*(p^4 + p^3 + p^2 + p + 1). For every p > 11, one of these five factors is divisible by 11; one of p-1 and p+1 is divisible by 3; and p-1 and p+1 are consecutive even numbers, so one of them is divisible by 4 and their product is divisible by 8; thus, p^10 - 1 is divisible by 2^3 * 3 * 11.
For every term p with the exception of a(1)=1187, p^10 - 1 is of the form 2^3 * 3 * 11 * q * r * s * t, where q, r, s, and t are distinct primes > 11.

			For p = a(1) = 1187, p^10 - 1 = 2^3 * 3^3 * 11 * 593 * 1983522604541 * 1986867499321;
for p = a(2) = 4723, p^10 - 1 = 2^3 * 3 * 11 * 787 * 1181 * 45245048697451 * 497484826300381.

Cf. A000005, A000040, A309906, A341670.

A342067 Primes p such that p^11 - 1 has 8 divisors.

Original entry on oeis.org

3, 467, 2039, 4679, 5399, 5939, 6899, 8783, 12347, 16487, 18443, 23879, 25583, 33647, 35879, 36299, 44819, 47207, 53147, 57119, 67499, 74507, 90239, 93287, 96059, 119759, 125003, 133499, 135119, 136223, 157019, 159539, 164999, 165059, 168887, 178799, 188159

Offset: 1

Jon E. Schoenfield, Feb 28 2021

nonn

Conjecture: sequence is infinite.
The only primes p such that p^11 - 1 has fewer than A309906(11)=8 divisors are 2 and 5.
p^11 - 1 = (p-1)*(p^10 + p^9 + p^8 + p^7 + p^6 + p^5 + p^4 + p^3 + p^2 + p + 1).
For every term p, p^11 - 1 is of the form 2*q*r, where q and r are distinct odd primes. With the exception of p=a(1)=3, each term p is a number such that (p-1)/2 and (p^10 + p^9 + p^8 + ... + p^2 + p + 1) are primes.

			   p =
n  a(n)             factorization of p^11 - 1
-  ----  ------------------------------------------------
1     3  2 *   23 *                                  3851
2   467  2 *  233 *           494424256962371823779424877
3  2039  2 * 1019 *    1242754384106847037173120489949801
4  4679  2 * 2339 * 5030640462820574591105701447273296601

Cf. A000005, A000040, A309906, A341670.

isok(p) = isprime(p) && (numdiv(p^11-1) == 8); \\ Michel Marcus, Feb 28 2021

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A341668 a(n) is the number of divisors of prime(n)^7 - 1.

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A341669 Primes p such that p^7 - 1 has 8 divisors.

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A342065 Primes p such that p^9 - 1 has 16 divisors.

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A341663 a(n) is the number of divisors of prime(n)^3 - 1.

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A342066 Primes p such that p^10 - 1 has 256 divisors.

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A342067 Primes p such that p^11 - 1 has 8 divisors.

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