A069062 -id:A069062 - cp's OEIS frontend

A108278 Numbers k such that k^2-1 and k^2+1 are semiprimes.

12, 30, 42, 60, 102, 108, 198, 312, 462, 522, 600, 810, 828, 1020, 1050, 1062, 1278, 1452, 1488, 1872, 1950, 2028, 2130, 2142, 2340, 2790, 2802, 2970, 3000, 3120, 3252, 3300, 3330, 3672, 3930, 4020, 4092, 4230, 4548, 4800, 5280, 5640, 5652, 5658, 6198

Offset: 1

Hugo Pfoertner, May 30 2005

nonn

Subsequence of A069062. - Michel Marcus, Jan 22 2016

Subsequence of A014574. - Robert Israel, Jan 24 2016

			a(1)=12 because 12^2-1=143=11*13 and 12^2+1=145=5*29 are both semiprimes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358 (semiprimes), A069062 (k^2-1 and k^2+1 have the same number of divisors), A014574 (average of twin prime pairs).

IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [4..7000] | IsSemiprime(n^2+1) and IsSemiprime(n^2-1) ]; // Vincenzo Librandi, Jan 22 2016

filter:= n -> isprime(n+1) and isprime(n-1) and numtheory:-bigomega(n^2+1)=2:
select(filter, [seq(i,i=2..1000, 2)]); # Robert Israel, Jan 24 2016

Select[Range[7000], PrimeOmega[#^2 - 1] == PrimeOmega[#^2 + 1]== 2 &] (* Vincenzo Librandi, Jan 22 2016 *)

isok(n) = (bigomega(n^2-1) == 2) && (bigomega(n^2+1) == 2); \\ Michel Marcus, Jan 22 2016

A373209 Numbers k such that k^2 - 1 and k^2 + 1 have 8 divisors each.

Original entry on oeis.org

68, 112, 128, 162, 200, 212, 252, 294, 318, 336, 338, 372, 448, 450, 498, 502, 542, 578, 592, 598, 612, 648, 672, 678, 708, 752, 762, 808, 812, 852, 878, 888, 938, 952, 992, 996, 1012, 1038, 1098, 1102, 1116, 1122, 1188, 1202, 1212, 1248, 1258, 1328, 1362, 1380

Offset: 1

Jon E. Schoenfield, Jun 21 2024

nonn

Among the first 10000 terms (from a(1) = 68 through a(10000) = 697578), k^2 - 1 and k^2 + 1 are each the product of three distinct primes, except for
     125 terms for which k^2 + 1 =  5^3 times a prime
       6 terms for which k^2 + 1 = 13^3 times a prime
       1 terms for which k^2 + 1 = 17^3 times a prime
       1 terms for which k^2 + 1 = 29^3 times a prime, and
       4 terms for which k^2 - 1 =  p^3 * (p^3 +/- 2) (with p = 19, 29, 37, 83, respectively).
The first term for which both k^2 - 1 and k^2 + 1 are of the form p^3 * q is k = 41457661182: k^2 - 1 = 3461^3 * 41457661183, while k^2 + 1 = 5^3 * 13749901365452077097.

			68 is a term: both 68^2 - 1 = 4623 = 3 * 23 * 67 and 68^2 + 1 = 4625 = 5^3 * 37 have 8 divisors.

Cf. A000005, A002522, A005563, A069062, A108278, A193432, A347191.

{ k : tau(k^2 - 1) = tau(k^2 + 1) = 8}, where tau() is the number of divisors function, A000005.

A373903 Numbers k such that k^2 - 1 has fewer divisors than k^2 + 1.

Original entry on oeis.org

18, 72, 132, 138, 182, 192, 228, 242, 268, 278, 282, 327, 348, 360, 378, 382, 408, 418, 432, 438, 618, 632, 642, 660, 682, 684, 693, 718, 772, 788, 798, 822, 843, 858, 882, 918, 948, 957, 1032, 1048, 1068, 1092, 1113, 1118, 1143, 1152, 1227, 1228, 1230, 1282, 1292

Offset: 1

Amiram Eldar, Jun 22 2024

Numbers k such that A347191(k) < A193432(k).

			18 is a term since 18^2 - 1 = 323 has 4 divisors (1, 17, 19 and 323) while 18^2 + 1 = 325 has 6 divisors (1, 5, 13, 25, 65 and 325).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A002522, A005563, A069062, A188629, A193432, A347191.

Select[Range[2, 1300], DivisorSigma[0, #^2 - 1] < DivisorSigma[0, #^2 + 1] &]

is(k) = k > 1 && numdiv(k^2 - 1) < numdiv(k^2 + 1);

A373213 Numbers k such that k^2 - 1 and k^2 + 1 have 6 divisors each.

Original entry on oeis.org

168, 1368, 97968, 10374840, 16104168, 44049768, 68674368, 100741368, 281803368, 486775968, 1177381968, 1262878368, 1336852968, 2321986968, 2404627368, 3476635368, 4374102768, 5102102040, 5142754368, 5182128168, 5385651768, 6035269968, 9218496168, 10657878168

Offset: 1

Jon E. Schoenfield, Jun 21 2024

nonn

Each term is a number of the form k = sqrt(p^2 * q + 1) such that q = p^2 - 2 and k^2 + 1 = r^2 * s, where p, q, r, and s are distinct primes.

			168 is a term: both 168^2 - 1 = 28223 = 13^2 * 167 and 168^2 + 1 = 28225 = 5^2 * 1129 have 6 divisors.

Cf. A000005, A002522, A005563, A069062, A108278, A193432, A347191, A373209.

{ k : tau(k^2 - 1) = tau(k^2 + 1) = 6}, where tau() is the number of divisors function, A000005.

A373756 Table read by antidiagonals: T(n,k) is the smallest m > 1 such that m^2 - 1 and m^2 + 1 have 2n and 2k divisors, respectively, or -1 if no such m exists.

Original entry on oeis.org

2, 4, -1, 10, 3, -1, 14, 8, 18, -1, 28560, 5, 168, 72, -1, 26, 9, 32, 360, 16068, -1, 25071688922457240, 15, 7, 68, 369465818568, 1620, -1, 56, 728, 332, 28398240, 182, 744768, 1407318, -1, 170, 11, 161245807967271241368, 98, 248872305817685706212070112080, 132, 4175536688568, 642, -1

Offset: 1

Jon E. Schoenfield, Jun 16 2024

m=1 is excluded because m^2 - 1 would be 0.
For all m > 1, both m^2 - 1 and m^2 + 1 are nonsquares, so each has an even number of divisors.
For k=1, m^2 + 1 is a prime, so T(n,1) == 0 (mod 2) for all n.
For n=1, m^2 - 1 = (m-1)*(m+1) is a prime, which occurs only at m=2; 2^2 + 1 = 5 is also a prime, so T(1,1) = 2 and T(1,k) = -1 for k > 1.
For n=2, m^2 - 1 = (m-1)*(m+1) has 4 divisors, so (except for T(2,2) = 3) T(2,k) is the average of a twin prime pair (A014574).
Is T(n,k) > 0 for all n > 1?

			T(5,1) is the smallest integer m > 1 such that m^2 - 1 and m^2 + 1 have 10 and 2 divisors, respectively; since m^2 - 1 cannot be the 9th power of a prime, this requires that p^4 * q + 1 = m^2 = r - 1, where p, q, and r are distinct primes. The smallest such m is 28560, which gives a solution with p = 13, q = 28559, r = 815673601.
T(5,5) is the smallest integer m > 1 such that m^2 - 1 and m^2 + 1 each have 10 divisors; since neither m^2 - 1 nor m^2 + 1 can be the 9th power of a prime, this is the smallest m such that p^4 * q + 1 = m^2 = r^4 * s - 1, where p, q, r, and s are distinct primes: 22335421^4 * 248872305817685706212070112079 + 1 = 248872305817685706212070112080^2 = 13^4 * 2168601400616633822685176617536070987718973054081571441 - 1.
The first eight antidiagonals of the table are shown below.
.
  n\k|                 1   2   3        4            5      6       7  8
  ---+------------------------------------------------------------------
   1 |                 2  -1  -1       -1           -1     -1      -1 -1
   2 |                 4   3  18       72        16068   1620 1407318
   3 |                10   8 168      360 369465818568 744768
   4 |                14   5  32       68          182
   5 |             28560   9   7 28398240
   6 |                26  15 332
   7 | 25071688922457240 728
   8 |                56

Cf. A000005, A002522, A005563, A014574, A069062, A193432, A347191.

Define f(m) = tau(m^2 - 1) and g(m) = tau(m^2 + 1), where tau is the number of divisors function (A000005). Then
    T(n,k) = min_{ m : f(m) = 2n and g(m) = 2k },
             or -1 if no such m exists.

cp's OEIS Frontend

A108278 Numbers k such that k^2-1 and k^2+1 are semiprimes.

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A373209 Numbers k such that k^2 - 1 and k^2 + 1 have 8 divisors each.

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A373903 Numbers k such that k^2 - 1 has fewer divisors than k^2 + 1.

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A373213 Numbers k such that k^2 - 1 and k^2 + 1 have 6 divisors each.

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A373756 Table read by antidiagonals: T(n,k) is the smallest m > 1 such that m^2 - 1 and m^2 + 1 have 2n and 2k divisors, respectively, or -1 if no such m exists.

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