cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A242331 Numbers k such that k^2 + 3 is a semiprime.

Original entry on oeis.org

1, 6, 16, 18, 20, 24, 26, 32, 34, 36, 40, 44, 46, 48, 56, 60, 66, 68, 78, 80, 88, 98, 100, 102, 104, 108, 116, 118, 120, 128, 136, 148, 152, 164, 170, 174, 176, 182, 188, 190, 192, 196, 200, 204, 212, 220, 226, 232, 234, 238, 246, 250, 252, 258, 260, 262, 266
Offset: 1

Views

Author

Vincenzo Librandi, May 14 2014

Keywords

Comments

The semiprimes of this form are: 4, 39, 259, 327, 403, 579, 679, 1027, 1159, 1299, 1603, 1939, 2119, 2307, 3139, 3603, 4359, 4627, ...

Links

Crossrefs

Cf. A049422, A049423, A085722, A242330, A242332, A242333.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [0..300] | IsSemiprime(s) where s is n^2+3];
  • Mathematica
    Select[Range[300], PrimeOmega[#^2 + 3] == 2 &]

A242330 Numbers k such that k^2 + 2 is a semiprime.

Original entry on oeis.org

2, 6, 7, 11, 12, 17, 18, 27, 29, 35, 37, 42, 43, 48, 51, 53, 54, 55, 60, 65, 66, 69, 72, 73, 75, 79, 83, 84, 87, 90, 93, 97, 115, 119, 125, 132, 133, 135, 137, 141, 144, 150, 153, 155, 159, 161, 165, 169, 174, 183, 186, 187, 189, 191, 192, 195, 198
Offset: 1

Views

Author

Vincenzo Librandi, May 14 2014

Keywords

Comments

The semiprimes of this form are: 6, 38, 51, 123, 146, 291, 326, 731, 843, 1227, 1371, 1766, 1851, 2306, 2603, 2811, 2918, 3027, 3602, ....
There are no four consecutive terms in this sequence, that is, a(n) > a(n-3) + 3 (check mod 6). Probably sieve theory can show that this sequence has density 0. - Charles R Greathouse IV, Feb 24 2023

Links

Crossrefs

Cf. A056899, A067201, A085722, A242331, A242332, A242333.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..200] | IsSemiprime(s) where s is n^2+2];
  • Mathematica
    Select[Range[300], PrimeOmega[#^2 + 2] == 2 &]
  • PARI
    issemi(n)=forprime(p=2,997,if(n%p==0, return(isprime(n/p)))); bigomega(n)==2
is(n)=issemi(n^2+2) \\ Charles R Greathouse IV, Feb 24 2023

Formula

a(n) > 2n for n > 1. - Charles R Greathouse IV, Feb 24 2023

A242332 Numbers k such that k^2 + 4 is a semiprime.

Original entry on oeis.org

0, 9, 19, 21, 23, 25, 31, 41, 43, 51, 53, 55, 63, 69, 71, 75, 77, 79, 83, 91, 93, 105, 107, 109, 113, 119, 123, 129, 131, 133, 143, 145, 149, 151, 153, 157, 165, 171, 173, 175, 181, 185, 187, 191, 195, 197, 201, 209, 221, 223, 225, 227, 241, 249, 251, 257, 259
Offset: 1

Views

Author

Vincenzo Librandi, May 14 2014

Keywords

Comments

The semiprimes of this form are: 4, 85, 365, 445, 533, 629, 965, 1685, 1853, 2605, 2813, 3029, 3973, 4765, 5045, 5629, 5933, 6245, ...

Links

Crossrefs

Cf. A005473, A007591, A085722, A242330, A242331, A242333.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [0..300] | IsSemiprime(s) where s is n^2+4];
  • Mathematica
    Select[Range[0, 300], PrimeOmega[#^2 + 4] == 2 &]

A375390 Numbers k such that k^2 + 1, k^2 + 3 and k^2 + 5 are semiprimes.

Original entry on oeis.org

44, 102, 104, 108, 152, 188, 226, 234, 296, 328, 426, 526, 586, 692, 720, 842, 846, 856, 926, 994, 1076, 1278, 1284, 1386, 1426, 1484, 1498, 1574, 1704, 1746, 1764, 1822, 1826, 1848, 1952, 2058, 2114, 2128, 2142, 2148, 2164, 2186, 2386, 2416, 2442, 2484, 2640, 2704, 2904, 2948, 3108, 3142, 3164
Offset: 1

Views

Author

Zak Seidov and Robert Israel, Aug 15 2024

Keywords

Comments

All terms are even.
a(n)^2 + 3 or a(n)^2 + 5 is 3 times a prime. In the first case, a(n)/3 is in A111051.

Examples

			a(3) = 104 is a term because 104^2 + 1 = 10817 = 29 * 373, 104^2 + 3 = 10819 = 31 * 349 and 104^2 + 5 = 10821 = 3 * 3607 are all semiprimes.

Links

Crossrefs

Cf. A001358, A111051. Intersection of A085722, A242331 and A242333.

Programs

  • Maple
    select(t -> andmap(s -> numtheory:-bigomega(t^2+s)=2, [1,3,5]), 2*[$1..2000]);
  • Mathematica
    Select[Range[3000], 2 == PrimeOmega[1 + #^2] == PrimeOmega[3 +
#^2] ==   PrimeOmega [5 + #^2] &]

A361696 Semiprimes of the form k^2 + 5.

Original entry on oeis.org

6, 9, 14, 21, 69, 86, 201, 329, 446, 489, 581, 681, 734, 789, 905, 1094, 1769, 1941, 2606, 2921, 3254, 3369, 3849, 3974, 4101, 4629, 4766, 6729, 7061, 7401, 8105, 8654, 9609, 9806, 10409, 10821, 11669, 12326, 13929, 17429, 17961, 19049, 20741, 23109, 23721, 24341, 27561, 30281, 31334, 32405
Offset: 1

Views

Author

Elmo R. Oliveira, Mar 20 2023

Keywords

Examples

			69 is a term because 8^2 + 5 = 69 = 3*23.

Links

Crossrefs

Cf. A144255, A242333, A360739, A360740, A360741.
Intersection of A117951 and A001358.

Programs

  • Mathematica
    Select[Range[200]^2 + 5, PrimeOmega[#] == 2 &] (* Paolo Xausa, Aug 21 2025 *)
  • PARI
    isok(k) = issquare(k-5) && (bigomega(k)==2); \\ Michel Marcus, Mar 27 2023

Formula

a(n) = A242333(n)^2 + 5.

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

Views

Author

Robert Israel, Aug 15 2024

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • Maple
    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]);
Showing 1-6 of 6 results.

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