A341656 -id:A341656 - cp's OEIS frontend

A341670 a(n) is conjecturally the largest prime p such that, for every prime q > p, q^n - 1 has more divisors than does p^n - 1, or -1 if no such prime p exists.

5, 73, 5, 101, 3, 167, 5, 71, 3, 43, 5, 167, 5, 73, 3, 19, 2, 17, 2, 19, 3, 23, 2, 71, 2, 7, 2, 29, -1, 13, 2, 11, 2, 7, 2, 13, 2, 3, 2, 11, 2, 7, -1, 23, 2, 17, 5, 17, 5, 7, -1, 7, -1, 11, -1, 5, 2, 7, 2, 11, 2, 2, 3, 3, 2, 5, 2, 2, 2, 3, 3, 11, -1, 2, -1, 7

Offset: 1

Jon E. Schoenfield, Feb 26 2021

sign

Conjecture: a(n) is also the largest prime p for which there exists no prime q > p such that p^n - 1 and q^n - 1 have the same number of divisors.

			It is conjectured that there are infinitely many primes q such that q-1 has exactly A309906(1)=4 divisors (see also A005385), and p=5 is the largest prime p such that p-1 has fewer than 4 divisors, so a(1)=5.
Similarly, there appear to be infinitely many primes q such that q^2 - 1 has exactly A309906(2)=32 divisors (e.g., primes q such that, of the two factors q-1 and q+1 of q^2 - 1, one is twice a prime > 3 and the other is 12 times a prime > 3), and p=73 is the largest prime p such that p^2 - 1 has fewer than 32 divisors (see A341655), so a(2)=73.
a(4)=101 is the largest prime p such that p^4 - 1 has fewer than 160 divisors, and there are conjecturally infinitely many primes q such that q^4 - 1 has exactly A309906(4)=160 divisors.
a(29)=-1 because there are conjecturally infinitely many primes q such that q^29 - 1 has exactly A309906(29)=8 divisors, and there exists no prime p such that p^29 - 1 has fewer than 8 divisors.

Cf. A000005, A000040, A005385, A309906, A341655, A341656, A341657, A341658, A341659.

A341657 a(n) is the number of divisors of prime(n)^6 - 1.

Original entry on oeis.org

6, 16, 48, 60, 192, 96, 192, 256, 360, 384, 504, 512, 240, 384, 576, 320, 384, 768, 576, 320, 320, 864, 384, 640, 504, 1152, 960, 1280, 1280, 576, 576, 768, 960, 768, 1152, 720, 384, 768, 240, 768, 2048, 2048, 2304, 384, 1536, 1920, 3072, 672, 1152, 1536, 1280

Offset: 1

Jon E. Schoenfield, Feb 25 2021

nonn

a(n) >= A309906(6) = 384 for n > 39.
p^6 - 1 = A*B*C*D where A=(p-1), B=(p+1), C=(p^2 - p + 1), and D=(p^2 + p + 1), and A, B, C, and D are pairwise coprime except that 2 may divide both A and B and that 3 may divide both A and D or both B and C. For prime p > 7, A and B are consecutive even numbers (so one of them is divisible by 4), so 8|AB; 3 divides both A and D or both B and C, so 9|ABCD; and 7 divides exactly one of A, B, C, and D. Thus, 8*9*7 = 2^3 * 3^2 * 7^1 = 504|ABCD = p^6 - 1. Generally, for sufficiently large primes p, the factors of ABCD, counted with multiplicity, include at least three 2's, two 3's, one 7, and at least four distinct larger primes, so tau(ABCD) = A000005(ABCD) >= (3+1)*(2+1)*(1+1)*(1+1)^4 = 384. (For sufficiently large primes p such that one of A, B, C, or D has no prime factors other than 2, 3, or 7, ABCD will still have at least four distinct prime factors > 7 unless the other three of A, B, C, and D have only one such larger prime factor each; in every such case where p > 167 (e.g., at p = 193, 383, 1373, and 6047), even though ABCD has only 3 distinct prime factors > 7, the multiplicities of 2, 3, and 7 in ABCD are collectively large enough that ABCD nevertheless has at least 384 divisors.)
The largest prime p at which tau(p^6 - 1) < 384 is p = prime(39) = 167: the prime factorizations of A, B, C, and D are A = 166 = 2 * 83, B = 168 = 2^3 * 3 * 7, C = 27723 = 3 * 9241, and D = 28057, so p^6 - 1 = ABCD = 2^4 * 3^2 * 7 * 83 * 9241 * 28057, and thus tau(p^6 - 1) = (4+1)*(2+1)*(1+1)*(1+1)*(1+1)*(1+1) = 5*3*2*2*2*2 = 240. (Note that the prime factorization of 167^6 - 1 contains four 2's, two 3's, one 7, and only 3 distinct primes > 7; B = 168 is 7-smooth.)

			   n  prime(n)    factorization of prime(n)^6 - 1      a(n)
  --  --------  -----------------------------------    ----
   1      2           3^2     * 7                         6
   2      3     2^3           * 7   * 13                 16
   3      5     2^3 * 3^2     * 7   * 31                 48
   4      7     2^4 * 3^2           * 19*43              60
   5     11     2^3 * 3^2 * 5 * 7   * 19*37             192
   6     13     2^3 * 3^2     * 7   * 61*157             96
   7     17     2^5 * 3^3     * 7   * 13*307            192
   8     19     2^3 * 3^3 * 5 * 7^3 * 127               256
   9     23     2^4 * 3^2     * 7   * 11*13^2*79        360
  10     29     2^3 * 3^2 * 5 * 7   * 13*67*271         384
  11     31     2^6 * 3^2 * 5 * 7^2 * 19*331            504
  12     37     2^3 * 3^3     * 7   * 19*31*43*67       512
  13     41     2^4 * 3^2 * 5 * 7   * 547*1723          240
  14     43     2^3 * 3^2     * 7   * 11*13*139*631     384
  15     47     2^5 * 3^2     * 7   * 23*37*61*103      576
  16     53     2^3 * 3^4     * 7   * 13*409*919        320
  17     59     2^3 * 3^2 * 5 * 7   * 29*163*3541       384
  18     61     2^3 * 3^2 * 5 * 7   * 13*31*97*523      768
  19     67     2^3 * 3^2     * 7^2 * 11*17*31*4423     576
  20     71     2^4 * 3^3 * 5 * 7   * 1657*5113         320
  21     73     2^4 * 3^3     * 7   * 37*751*1801       320
  ...
  39    167     2^4 * 3^2     * 7   * 83*9241*28057     240

Cf. A000005, A000040, A309906, A341655, A341656.

a[n_] := DivisorSigma[0, Prime[n]^6 - 1]; Array[a, 50] (* Amiram Eldar, Feb 26 2021 *)

a(n) = numdiv(prime(n)^6-1); \\ Michel Marcus, Feb 26 2021

a(n) = A000005(A000040(n)^6 - 1).

A341661 Primes p such that p^4 - 1 has fewer than 160 divisors.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 59, 61, 71, 79, 101

Offset: 1

Jon E. Schoenfield, Feb 26 2021

For all primes p > 101, p^4 - 1 has at least A309906(4)=160 divisors.

			      p =
   n  a(n)   p^4 - 1   factorization of p^4 - 1  tau(p^4 - 1)
  --  ----  ---------  ------------------------- ------------
   1    2          15  3 * 5                           4
   2    3          80  2^4 * 5                        10
   3    5         624  2^4 * 3 * 13                   20
   4    7        2400  2^5 * 3 * 5^2                  36
   5   11       14640  2^4 * 3 * 5 * 61               40
   6   13       28560  2^4 * 3 * 5 * 7 * 17           80
   7   17       83520  2^6 * 3^2 * 5 * 29             84
   8   19      130320  2^4 * 3^2 * 5 * 181            60
   9   23      279840  2^5 * 3 * 5 * 11 * 53          96
  10   29      707280  2^4 * 3 * 5 * 7 * 421          80
  11   31      923520  2^7 * 3 * 5 * 13 * 37         128
  12   37     1874160  2^4 * 3^2 * 5 * 19 * 137      120
  13   41     2825760  2^5 * 3 * 5 * 7 * 29^2        144
  14   59    12117360  2^4 * 3 * 5 * 29 * 1741        80
  15   61    13845840  2^4 * 3 * 5 * 31 * 1861        80
  16   71    25411680  2^5 * 3^2 * 5 * 7 * 2521      144
  17   79    38950080  2^6 * 3 * 5 * 13 * 3121       112
  18  101   104060400  2^4 * 3 * 5^2 * 17 * 5101     120

Cf. A000005, A000040, A309906, A341656.

Select[Range[101], PrimeQ[#] && DivisorSigma[0, #^4 - 1] < 160 &] (* Amiram Eldar, Feb 26 2021 *)

A341662 Primes p such that p^4 - 1 has 160 divisors.

Original entry on oeis.org

53, 67, 131, 139, 227, 277, 283, 347, 383, 641, 653, 661, 821, 877, 997, 1069, 1181, 1213, 1811, 2083, 2389, 2459, 2819, 3803, 4021, 4253, 4723, 6619, 6829, 7213, 7933, 8069, 9013, 9187, 10589, 11261, 16139, 17827, 18133, 18587, 19309, 19541, 20477, 20947

Offset: 1

Jon E. Schoenfield, Feb 26 2021

nonn

Conjecture: sequence is infinite.

For every term p, p^4 - 1 is of the form 2^4 * 3 * 5 * q * r * s, where q, r, and s are distinct primes > 5, with three exceptions: p = 53, 383, and 641 (see Example section).

			      p =
   n  a(n)     p^4 - 1    factorization of p^4 - 1
  --  ----  ------------  -------------------------------
   1    53       7890480  2^4 * 3^3 * 5 * 13 * 281
   2    67      20151120  2^4 * 3 * 5 * 11 * 17 * 449
   3   131     294499920  2^4 * 3 * 5 * 11 * 13 * 8581
   4   139     373301040  2^4 * 3 * 5 * 7 * 23 * 9661
   5   227    2655237840  2^4 * 3 * 5 * 19 * 113 * 5153
   6   277    5887339440  2^4 * 3 * 5 * 23 * 139 * 7673
   7   283    6414247920  2^4 * 3 * 5 * 47 * 71 * 8009
   8   347   14498327280  2^4 * 3 * 5 * 29 * 173 * 12041
   9   383   21517662720  2^9 * 3 * 5 * 191 * 14669
  10   641  168823196160  2^9 * 3 * 5 * 107 * 205441
  11   653  181824635280  2^4 * 3 * 5 * 109 * 163 * 42641

Cf. A000005, A000040, A309906, A341656, A341661.

Select[Range[21000], PrimeQ[#] && DivisorSigma[0, #^4 - 1] == 160 &] (* Amiram Eldar, Feb 26 2021 *)

isok(p) = isprime(p) && (numdiv(p^4-1) == 160); \\ Michel Marcus, Feb 26 2021

cp's OEIS Frontend

A341670 a(n) is conjecturally the largest prime p such that, for every prime q > p, q^n - 1 has more divisors than does p^n - 1, or -1 if no such prime p exists.

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A341657 a(n) is the number of divisors of prime(n)^6 - 1.

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A341661 Primes p such that p^4 - 1 has fewer than 160 divisors.

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A341662 Primes p such that p^4 - 1 has 160 divisors.

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