A085722 -id:A085722 - cp's OEIS frontend

A360739 Semiprimes of the form k^2 + 2.

6, 38, 51, 123, 146, 291, 326, 731, 843, 1227, 1371, 1766, 1851, 2306, 2603, 2811, 2918, 3027, 3602, 4227, 4358, 4763, 5186, 5331, 5627, 6243, 6891, 7058, 7571, 8102, 8651, 9411, 13227, 14163, 15627, 17426, 17691, 18227, 18771, 19883, 20738, 22502, 23411, 24027

Offset: 1

Elmo R. Oliveira, Feb 18 2023

A242330 gives the corresponding values of k.

			123 is a term because 11^2 + 2 = 123 = 3*41.

Cf. A001358, A059100, A085722, A144255, A242330.

Select[Range[0, 200]^2 + 2, PrimeOmega[#] == 2 &] (* Amiram Eldar, Feb 18 2023 *)

a(n) = A242330(n)^2 + 2.

A360740 Semiprimes of the form k^2 + 3.

Original entry on oeis.org

4, 39, 259, 327, 403, 579, 679, 1027, 1159, 1299, 1603, 1939, 2119, 2307, 3139, 3603, 4359, 4627, 6087, 6403, 7747, 9607, 10003, 10407, 10819, 11667, 13459, 13927, 14403, 16387, 18499, 21907, 23107, 26899, 28903, 30279, 30979, 33127, 35347, 36103, 36867, 38419

Offset: 1

Elmo R. Oliveira, Feb 18 2023

A242331 gives the corresponding values of k.

			259 is a term because 16^2 + 3 = 259 = 7*37.

Cf. A001358, A085722, A117950, A144255, A242330, A242331.

Select[Range[0, 200]^2 + 3, PrimeOmega[#] == 2 &] (* Amiram Eldar, Feb 18 2023 *)

a(n) = A242331(n)^2 + 3.

A360741 Semiprimes of the form k^2 + 4.

Original entry on oeis.org

4, 85, 365, 445, 533, 629, 965, 1685, 1853, 2605, 2813, 3029, 3973, 4765, 5045, 5629, 5933, 6245, 6893, 8285, 8653, 11029, 11453, 11885, 12773, 14165, 15133, 16645, 17165, 17693, 20453, 21029, 22205, 22805, 23413, 24653, 27229, 29245, 29933, 30629, 32765, 34229

Offset: 1

Elmo R. Oliveira, Feb 18 2023

A242332 gives the corresponding values of k.

Except for 4, all terms == 5 (mod 8). - Robert Israel, Feb 18 2023

			85 is a term because 9^2 + 4 = 85 = 5*17.

Cf. A001358, A085722, A087475, A144255, A242330, A242331, A242332.

select(t -> numtheory:-bigomega(t)=2, [seq(i^2+4,i=0..1000)]); # Robert Israel, Feb 18 2023

Select[Range[0, 200]^2 + 4, PrimeOmega[#] == 2 &] (* Amiram Eldar, Feb 18 2023 *)

a(n) = A242332(n)^2 + 4.

A375391 a(n) is the greatest odd number k such that n^2 + j is a semiprime for all odd numbers j from 1 to k.

Original entry on oeis.org

-1, -1, 1, -1, 1, -1, -1, 1, 1, -1, 1, 1, -1, -1, 1, -1, -1, -1, 1, -1, -1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, -1, -1, 3, 1, -1, -1, -1, 1, -1, -1, 1, -1, 9, 1, 3, -1, 3, 1, 1, 1, 1, -1, -1, -1, -1, -1, 1, 1, 3, 1, 1, -1, 1, 1, -1, -1, -1, 1, -1, 1, -1, -1, -1, -1, 1, -1, 3, 1, 3, -1, -1, -1, -1

Offset: 1

Robert Israel, Aug 15 2024

sign

a(n) = -1 if n^2 + 1 is not a semiprime.
a(n) <= 1 if n is odd, since n^2 + 3 is divisible by 4.
a(n) <= 15 since one of n^2 + 1, n^2 + 3, ..., n^2 + 17 is divisible by 9.
First occurrences of values: a(3) = 1, a(34) = 3, a(152) = 5, a(102) = 7, a(44) = 9, a(824264) = 11, a(21394) = 13, a(121364) = 15.

			a(44) = 9 since 44^2 + 1 = 1937 = 13 * 149, 44^2 + 3 = 1939 = 7 * 277, 44^2 + 5 = 1941 = 3 * 647, 44^2 + 7 = 1943 = 29 * 67 and 44^2 + 9 = 1945 = 5 * 389 are all semiprimes but 44^2 + 11 = 1947 = 3 * 11 * 59 is not a semiprime.

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

Cf. A001358, A085722, A242331, A242333, A375390.

f:= proc(n) local i;
  for i from 1 by 2 while numtheory:-bigomega(n^2+i) = 2 do od:
  i-2
end proc:
map(f, [$1..100]);

A115398 Numbers k such that both k^2+1 and 2^k + 1 are semiprimes.

Original entry on oeis.org

3, 5, 11, 12, 19, 28, 61, 64, 79, 92, 101, 104, 199, 356, 596, 692, 1709, 3539, 3824

Offset: 1

Zak Seidov, Mar 08 2006

Intersection of A085722 and A092559.

			11 is a term because 11^2 + 1 = 122 = 2*61 (semiprime) and 2^11 + 1 = 2049 = 3*683 (semiprime).

Cf. A085722, A092559.

IsSemiprime:=func; [n: n in [2..700] | IsSemiprime(n^2+1) and IsSemiprime(2^n+1)]; // Vincenzo Librandi, Oct 10 2013

Select[Range[700],PrimeOmega[#^2+1]==PrimeOmega[2^#+1]==2&] (* Harvey P. Dale, Apr 14 2019 *)

isok(n) = (bigomega(n^2+1) == 2) && (bigomega(2^n+1) == 2); \\ Michel Marcus, Oct 10 2013

a(17)-a(19) from Robert Israel, Nov 27 2023

A258780 a(n) is the least k such that k^2 + 1 is a semiprime p*q, p < q, and (q - p)/2^n is prime.

Original entry on oeis.org

8, 12, 140, 64, 2236, 196, 1300, 1600, 6256, 5084, 248756, 246196, 484400, 36680, 887884, 821836, 1559116, 104120, 126072244, 9586736, 4156840, 542759984, 1017981724, 2744780140, 405793096, 148647496, 1671024916

Offset: 2

Michel Lagneau, Jun 10 2015

The corresponding primes are 2, 3, 71, 7, 1069, 7, 5, 5, 59, 2, 368471, 180463, 12421, 2, 29, 125683, 226169, 5, 369704891, 197, 5, 263, 7444559, 239621423, 594271, 2, 474359, ...

All terms are even, in order for k^2+1 to be odd. Otherwise, with k^2+1 being even, p-q would be odd and hence not a multiple of 2^n. - Michel Marcus, Apr 13 2019

			a(2)=8 because 8^2+1 = 5*13 and (13-5)/2^2 = 2 is prime. The number 8 is the first term of the sequence 8, 22, 34, 46, 50, 58, ...
a(3)=12 because 12^2+1 = 5*29 and (29-5)/2^3 = 3 is prime. The number 12 is the first term of the sequence 12, 28, 44, 52, 76, 80, ...
a(4)=140 because 140^2+1 = 17*1153 and (1153-17)/2^4 = 71 is prime. The number 140 is the first term of the sequence 140, 296, 404, 604, ...

Cf. A085722, A144255.

lst={};Do[k=2;While[!(Plus@@Last/@FactorInteger[k^2+1]==2&&PrimeQ[(FactorInteger[k^2+1][[-1,1]]-FactorInteger[k^2+1][[1,1]])/2^n]),k=k+2];Print[n," ",k],{n,2,19}];lst

isok(k, n) = my(kk=k^2+1, f=factor(kk)[,1]~); (bigomega(kk) == 2) && (#f == 2) && (p=f[1]) && (q=f[2]) && (qq=(q-p)/2^n) && !frac(qq) && isprime(qq);
a(n) = my(k=2); while (!isok(k,n), k+=2); k; \\ Michel Marcus, Apr 13 2019

Name edited by Jon E. Schoenfield, Sep 12 2017
a(20)-a(22) from Daniel Suteu, Apr 13 2019
a(23)-a(28) from Daniel Suteu, Nov 09 2019

A321519 Let d(n,i), i = 1..k be the k divisors of n^2 + 1 (the number 1 is not counted). a(n) is the number of ordered pairs d(n,i) < d(n,j) such that gcd(d(n,i), d(n,j)) = 1.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 6, 0, 1, 0, 6, 2, 1, 0, 6, 1, 6, 0, 1, 0, 6, 1, 1, 1, 6, 2, 6, 1, 1, 0, 6, 2, 1, 0, 2, 1, 11, 1, 1, 1, 25, 1, 1, 1, 1, 1, 6, 0, 6, 0, 16, 1, 1, 1, 1, 1, 6, 1, 1, 0, 6, 3, 1, 2, 1, 6, 25, 0, 6, 1, 6, 1, 1, 1, 6, 2, 25, 0, 1, 1

Offset: 1

Michel Lagneau, Nov 12 2018

nonn

Terms only depends on prime signature of n^2+1. - David A. Corneth, Nov 14 2018
We observe an interesting statistic for n <= 10^5: the four values of a(n) = 0, 1, 6, 25 represent more than 82% (see the table below).
a(A005574(n)) = 0, a(A085722(n)) = 1, a(A272078(n)) = 6, a(A316351(n)) = 25.
In the general case, a(k) = m if k^2+1 = p*q^m, m = 1, 2, 3, ... with p, q primes.
+--------------+-----------------------+------------+
|              | number of occurrences |            |
|     a(n)     |     for n <= 10^5     | percentage |
+--------------+-----------------------+------------+
|       0      |          6656         |    6.656%  |
|       1      |         23255         |   23.255%  |
|       6      |         31947         |   31.947%  |
|      25      |         20461         |   20.461%  |
| other values |         17681         |   17.681%  |
+--------------+-----------------------+------------+

			a(13) = 6 because the divisors {d(i)} of 13^2 + 1 = 170 (without the number 1)  are  {2, 5, 10, 17, 34, 85, 170}, and gcd(d(i), d(j)) = 1 for the 6 following pairs of elements of {d(i)}: (2, 5), (2, 17), (2, 85), (5, 17), (5, 34) and (10, 17).

Cf. A005574, A002522, A085722, A089233, A193432, A272078, A316351.

with(numtheory):nn:=10^3:
for n from 1 to nn do:
  it:=0:d:=divisors(n^2+1):n0:=nops(d):
   for k from 2 to n0-1 do:
    for l from k+1 to n0 do:
     if gcd(d[k],d[l])= 1
      then
      it:=it+1
      else
     fi:
   od:
  od:
  printf(`%d, `,it):
od:

f[n_] := (DivisorSigma[0, n^2] - 1)/2 - DivisorSigma[0, n] + 1; Map[f, Range[0,100]^2+1] (* Amiram Eldar, Nov 14 2018 after Robert G. Wilson v at A089233 *)

a(n) = {my(d=divisors(n^2+1)); sum(k=2, #d, sum(j=2, k-1, gcd(d[k], d[j]) == 1));} \\ Michel Marcus, Nov 12 2018

a(n) = A089233(n^2+1). - Michel Marcus, Nov 13 2018

A321985 Numbers m such that m^2+1 is semiprime with (m-1)^2+1 and (m+1)^2+1 primes.

Original entry on oeis.org

3, 5, 15, 25, 205, 715, 1095, 1315, 1615, 2055, 2405, 2925, 3755, 4615, 4795, 5015, 5055, 5475, 6785, 7855, 8115, 8175, 9425, 9475, 10415, 10845, 11025, 11245, 12335, 12765, 15225, 16225, 16395, 16405, 18145, 18175, 18275, 21345, 21915, 22905, 23165, 23815

Offset: 1

Michel Lagneau, Nov 23 2018

nonn

Subsequence of A085722.
For n>1, a(n) == 5 (mod 10).
The corresponding pairs of primes (p, q) = ((m-1)^2+1, (m+1)^2+1) are congruent to 7 (mod 10), and the semiprimes are of the form m^2+1 = 2r where r is congruent to 3 (mod 10). So, a(n) = (q - 2r - 1)/2 = (2r - p + 1)/2 = (q - p)/4.

			15 is in the sequence because 15^2 + 1 = 2*113 is semiprime, and 14^2 + 1 = 197, 16^2 + 1 = 257 are prime numbers.

Cf. A005574, A085722, A321795.

Select[Range[50000],PrimeQ[(#-1)^2+1]&&PrimeOmega [#^2+1]==2&&PrimeQ[(#+1)^2+1]&]

isok(m) = (bigomega(m^2+1) == 2) && isprime((m-1)^2+1) && isprime((m+1)^2+1); \\ Michel Marcus, Nov 23 2018

A348594 Numbers m such that m^2 + 1 = p*q with p, q primes and m = (p + q)/2 - 1.

Original entry on oeis.org

8, 50, 1250, 1800, 2450, 9800, 14450, 20000, 24200, 101250, 105800, 135200, 162450, 168200, 204800, 304200, 336200, 451250, 480200, 490050, 530450, 696200, 924800, 966050, 1008200, 1125000, 1155200, 1428050, 1805000, 2332800, 2420000, 2576450, 2761250, 2832200

Offset: 1

Michel Lagneau, Jan 26 2022

nonn

Subsequence of A085722.
The corresponding pairs (p, q) of the sequence are (5, 13), (41, 61), (1201, 1301), (1741, 1861), (2381, 2521), (9661, 9941), (14281, 14621), (19801, 20201), (23981, 24421), (100801, 101701), ...
Property:
a(n) = 2* A109306(n)^2 and a(n) == 0 (mod 50) for n > 1. Proof:
From the relations:
(1)        m^2 + 1 = p*q
(2)        (p + q)/2 = m + 1
We obtain:
(3)        p = m + 1 - sqrt(8*m)/2
(4)        q = m + 1 + sqrt(8*m)/2
with m = 2*k^2 we obtain:
(5)        p = k^2 + (k-1)^2
(6)        q = k^2 + (k+1)^2
For n > 1, A109306(n) == 0 (mod 5) => 2*A109306(n)^2 == 0 (mod 50).

			50 = 2*5^2 is in the sequence because 50^2 + 1 = 41*61 with 50 = (41 + 61)/2 - 1.

Cf. A014442, A085722, A089120, A109306, A144255.

with(numtheory):nn:=250:printf(`%d, `,8):
for k from 0 to nn do:
n:=50*k^2:d:=factorset(n^2+1):
  if bigomega(n^2+1)=2 and (d[1]+d[2])/2 - 1 = n
   then
    printf(`%d, `,n):
    else
  fi:
od:

q[n_] := Module[{f = FactorInteger[n^2 + 1]}, f[[;; , 2]] == {1, 1} && f[[1, 1]] + f[[2, 1]] == 2*n + 2]; Select[Range[3*10^5], q] (* Amiram Eldar, Jan 26 2022 *)

isok(m) = my(x); if (bigomega(x=m^2+1)==2, my(f=factor(x)); (f[1,1]+f[2,1] == 2*(m+1))); \\ Michel Marcus, Jan 26 2022

cp's OEIS Frontend

A360739 Semiprimes of the form k^2 + 2.

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A360740 Semiprimes of the form k^2 + 3.

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A360741 Semiprimes of the form k^2 + 4.

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A375391 a(n) is the greatest odd number k such that n^2 + j is a semiprime for all odd numbers j from 1 to k.

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A115398 Numbers k such that both k^2+1 and 2^k + 1 are semiprimes.

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A258780 a(n) is the least k such that k^2 + 1 is a semiprime p*q, p < q, and (q - p)/2^n is prime.

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A321519 Let d(n,i), i = 1..k be the k divisors of n^2 + 1 (the number 1 is not counted). a(n) is the number of ordered pairs d(n,i) < d(n,j) such that gcd(d(n,i), d(n,j)) = 1.

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A321985 Numbers m such that m^2+1 is semiprime with (m-1)^2+1 and (m+1)^2+1 primes.

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