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 1-9 of 9 results.

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

A242333 Numbers k such that k^2 + 5 is a semiprime.

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 14, 18, 21, 22, 24, 26, 27, 28, 30, 33, 42, 44, 51, 54, 57, 58, 62, 63, 64, 68, 69, 82, 84, 86, 90, 93, 98, 99, 102, 104, 108, 111, 118, 132, 134, 138, 144, 152, 154, 156, 166, 174, 177, 180, 183, 184, 186, 188, 189, 194, 208, 210, 212, 216
Offset: 1

Views

Author

Vincenzo Librandi, May 14 2014

Keywords

Comments

The semiprimes of this form are: 6, 9, 14, 21, 69, 86, 201, 329, 446, 489, 581, 681, 734, 789, 905, 1094, 1769, 1941, 2606, 2921, 3254, ...

Links

Crossrefs

Cf. A056905, A078402, A085722, A242330, A242331, A242332.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [0..300] | IsSemiprime(s) where s is n^2+5];
  • Mathematica
    Select[Range[0, 300], PrimeOmega[#^2 + 5] == 2 &]

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 &]

A253906 Numbers n such that n^2 + 3 and n^3 + 3 are semiprime.

Original entry on oeis.org

1, 6, 20, 34, 40, 44, 46, 56, 102, 116, 120, 170, 174, 196, 200, 204, 220, 226, 232, 234, 252, 260, 262, 294, 296, 320, 334, 336, 344, 346, 358, 360, 382, 386, 392, 412, 426, 464, 476, 482, 490, 494, 514, 520, 526, 536, 556, 564, 582, 586, 592, 646, 658, 716
Offset: 1

Views

Author

K. D. Bajpai, Jan 24 2015

Keywords

Comments

All terms in this sequence, except a(1), are even.

Examples

			a(2) = 6;
6^2 + 3 = 39 = 3 * 13;
6^3 + 3 = 219 = 3 * 73;
Both are semiprime.

Links

Crossrefs

Cf. A001358, A242331, A108868.

Programs

  • Mathematica
    Select[Range[10^3], k = 3; PrimeOmega[(#^2 + k)] == 2 && PrimeOmega[(#^3 + k)] == 2 &]
  • PARI
    issemiprime(q) = q>0 && bigomega(q)==2
select(n->issemiprime(n^2+3)&&issemiprime(n^3+3), vector(2000, n, n)) \\ Colin Barker, Jan 28 2015

A253907 Numbers n such that n^2 + 3, n^3 + 3, n^4 + 3, n^5 + 3, n^6 + 3 and n^7 + 3 are semiprime.

Original entry on oeis.org

1, 976, 5380, 16582, 17864, 22316, 27922, 34930, 44954, 50744, 61264, 72670, 107534, 147776, 193774, 195266, 240170, 260920, 265292, 281582, 314462, 337832, 342014, 367060, 379784, 383930, 384704, 392050, 421226, 455734, 463790, 498134, 499306, 510194, 538384
Offset: 1

Views

Author

K. D. Bajpai, Jan 24 2015

Keywords

Comments

All terms in this sequence, except a(1), are even.
Subsequence of A253906.

Examples

			a(2) = 976;
976^2 + 3 = 952579 = 43 * 22153;
976^3 + 3 = 929714179 = 1013 * 917783;
976^4 + 3 = 907401035779 = 7 * 129628719397;
976^5 + 3 = 885623410917379 = 2224441 * 398133019;
976^6 + 3 = 864368449055358979 = 97327 * 8881075642477;
976^7 + 3 = 843623606278030360579 = 16403765263 * 51428656333;
All six are semiprime.

Links

Crossrefs

Cf. A001358, A108868, A242331, A253906.

Programs

  • Mathematica
    Select[Range[10^5], k = 3; PrimeOmega[(#^2 + k)] == 2 && PrimeOmega[(#^3 + k)] == 2 && PrimeOmega[(#^4 + k)] == 2 && PrimeOmega[(#^5 + k)] == 2 &&  PrimeOmega[(#^6 + k)] == 2  && PrimeOmega[(#^7 + k)] == 2 &]
Select[Range[54*10^4],Union[PrimeOmega[#^Range[2,7]+3]]=={2}&] (* Harvey P. Dale, Jul 30 2022 *)

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] &]

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

Views

Author

Elmo R. Oliveira, Feb 18 2023

Keywords

Comments

A242331 gives the corresponding values of k.

Examples

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

Crossrefs

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

Programs

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

Formula

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

Views

Author

Elmo R. Oliveira, Feb 18 2023

Keywords

Comments

A242332 gives the corresponding values of k.
Except for 4, all terms == 5 (mod 8). - Robert Israel, Feb 18 2023

Examples

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

Crossrefs

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

Programs

  • Maple
    select(t -> numtheory:-bigomega(t)=2, [seq(i^2+4,i=0..1000)]); # Robert Israel, Feb 18 2023
  • Mathematica
    Select[Range[0, 200]^2 + 4, PrimeOmega[#] == 2 &] (* Amiram Eldar, Feb 18 2023 *)

Formula

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

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 1-9 of 9 results.

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