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.

A241493 Primes p such that p + 4, p + 16, p + 64, p + 256 and p + 1024 are all semiprimes.

Original entry on oeis.org

1627, 2917, 3583, 4603, 5581, 6367, 6379, 8263, 9697, 12517, 12763, 13339, 14197, 15289, 16339, 16993, 17539, 17737, 18199, 19267, 19531, 20023, 28057, 28879, 29587, 32647, 33427, 34033, 34537, 35353, 35617, 37039, 37087, 37657, 37663, 42337, 43093, 47533, 48049
Offset: 1

Views

Author

K. D. Bajpai, Apr 24 2014

Keywords

Comments

The constants in the definition (4, 16, 64, 256 and 1024 ) are in geometric progression.

Examples

			1627 is prime and appears in the sequence because 1627+4 = 1631 = 7*233, 1627+16 = 1643 = 31*53, 1627+64 = 1691 = 19*89, 1627+256 = 1883 = 7*269 and 1627+1024 = 2651 = 11*241, which are all semiprime.

Links

Crossrefs

Cf. A072381, A082919, A241483, A241484.

Programs

  • Maple
    with(numtheory): KD:= proc() local a,b,d,e,f,k; k:=ithprime(n); a:=bigomega(k+4); b:=bigomega(k+16); d:=bigomega(k+64); e:=bigomega(k+256); f:=bigomega(k+1024); if a=2 and  b=2 and d=2 and  e=2 and f=2 then RETURN (k); fi; end: seq(KD(), n=1..10000);
  • Mathematica
    KD = {}; Do[t = Prime[n]; If[PrimeOmega[t + 4] == 2 && PrimeOmega[t + 16] == 2 && PrimeOmega[t + 64] == 2 && PrimeOmega[t + 256] == 2 && PrimeOmega[t + 1024] == 2, AppendTo[KD, t]], {n, 10000}]; KD
(* For the b-file *) c = 0; Do[t = Prime[n]; If[PrimeOmega[t + 4] == 2 && PrimeOmega[t + 16] == 2 && PrimeOmega[t + 64] == 2 && PrimeOmega[t + 256] == 2 && PrimeOmega[t + 1024] == 2, c++; Print[c, "  ", t]], {n, 1,5*10^6}];
Select[Prime[Range[5000]],Union[PrimeOmega[#+{4,16,64,256,1024}]] == {2}&] (* Harvey P. Dale, Nov 28 2017 *)

A241659 Primes p such that p^3 + 2 is semiprime.

Original entry on oeis.org

2, 11, 13, 17, 19, 23, 31, 41, 53, 59, 89, 101, 131, 137, 149, 193, 211, 223, 227, 229, 233, 239, 251, 271, 293, 317, 331, 359, 401, 449, 461, 557, 563, 571, 593, 599, 619, 641, 659, 677, 691, 719, 739, 751, 809, 821, 853, 929, 971, 991, 1009, 1013, 1039, 1051
Offset: 1

Views

Author

K. D. Bajpai, Apr 26 2014

Keywords

Examples

			11 is prime and appears in the sequence because 11^3 + 2 = 1333 = 31 * 43, which is a semiprime.
17 is prime and appears in the sequence because 17^3 + 2 = 4915 =  5 * 983, which is a semiprime.
37 is prime but does not appear in the sequence because 37^3 + 2 = 50655 =  3 * 5 * 11 * 983, which is not a semiprime.

Links

Crossrefs

Cf. A001358, A063637, A063638, A228183, A237627, A241483, A241493.

Programs

  • Maple
    with(numtheory): KD:= proc() local a, b, k; k:=ithprime(n); a:=bigomega(k^3+2); if a=2 then RETURN (k); fi; end: seq(KD(), n=1..500);
  • Mathematica
    A241659 = {}; Do[t = Prime[n]; If[PrimeOmega[t^3 + 2] == 2, AppendTo[A241659, t]], {n, 500}]; A241659
(*For the b-file*) c = 0; Do[t = Prime[n]; If[PrimeOmega[t^3 + 2] == 2, c++; Print[c, "  ", t]], {n, 1,6*10^4}];
Select[Prime[Range[200]],PrimeOmega[#^3+2]==2&] (* Harvey P. Dale, Feb 05 2025 *)
  • PARI
    s=[]; forprime(p=2, 1200, if(bigomega(p^3+2)==2, s=concat(s, p))); s \\ Colin Barker, Apr 27 2014

A241716 Primes p such that p^3 - 2 is semiprime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 41, 43, 47, 61, 79, 89, 101, 107, 139, 157, 181, 199, 239, 271, 307, 311, 331, 337, 347, 349, 379, 397, 409, 421, 431, 479, 487, 499, 521, 523, 541, 571, 607, 613, 641, 643, 661, 673, 701, 719, 761, 769, 811, 823, 829, 839, 877, 881, 883
Offset: 1

Views

Author

K. D. Bajpai, Apr 27 2014

Keywords

Examples

			11 is prime and appears in the sequence because 11^3 - 2 = 1329 = 3 * 443, which is a semiprime.
17 is prime and appears in the sequence because 17^3 - 2 = 4911 = 3 * 1637, which is a semiprime.
23 is prime but does not appear in the sequence because 23^3 - 2 = 12165 =  3 * 5 * 811, which is not a semiprime.

Links

Crossrefs

Cf. A001358, A063637, A063638, A072381, A082919, A145292, A228183, A237627, A241483, A241493, A241659.

Programs

  • Maple
    with(numtheory):A241716:= proc() local k; k:=ithprime(x); if bigomega(k^3-2)=2 then RETURN (k); fi; end: seq(A241716(), x=1..500);
  • Mathematica
    A241716 = {}; Do[t = Prime[n]; If[PrimeOmega[t^3 - 2] == 2, AppendTo[A241716, t]], {n, 500}]; A241716
Select[Prime[Range[200]],PrimeOmega[#^3-2]==2&] (* Harvey P. Dale, Dec 09 2018 *)

A241732 Primes p such that p^3 + 2 and p^3 - 2 are semiprime.

Original entry on oeis.org

2, 11, 13, 17, 41, 89, 101, 239, 271, 331, 571, 641, 719, 1051, 1231, 1321, 1549, 1559, 1721, 1741, 1831, 1993, 1999, 2029, 2311, 2459, 2749, 2837, 2861, 2939, 3389, 3467, 3671, 4049, 4111, 4273, 4787, 4919, 4969, 5657, 5689, 5861, 6221, 6679, 6691, 6829, 7109
Offset: 1

Views

Author

K. D. Bajpai, Apr 27 2014

Keywords

Examples

			11 is prime and appears in the sequence because 11^3 + 2 = 1333 = 31 * 43 and 11^3 - 2 = 1329 = 3 * 443, both are semiprime.
41 is prime and appears in the sequence because 41^3 + 2 = 68923 = 157 * 439 and 41^3 - 2 = 68919 = 3 * 22973, both are semiprime.

Links

Crossrefs

Cf. A001358, A063637, A063638, A072381, A082919, A145292, A228183, A237627, A241483, A241493, A241659.

Programs

  • Maple
    with(numtheory): KD:= proc() local k; k:=ithprime(n); if bigomega(k^3+2)=2 and bigomega(k^3-2)=2 then k; fi; end: seq(KD(), n=1..2000);
  • Mathematica
    A241732 = {}; Do[t = Prime[n]; If[PrimeOmega[t^3 + 2] == 2 && PrimeOmega[t^3 - 2] == 2, AppendTo[A241732, t]], {n, 500}]; A241732
Select[Prime[Range[1000]],PrimeOmega[#^3+2]==PrimeOmega[#^3-2]==2&] (* Harvey P. Dale, Jun 20 2019 *)

A268862 Primes p such that p+2, p+4, p+6, p+8, p+10, p+12 and p+14 are all semiprime.

Original entry on oeis.org

3089, 182747, 209477, 239087, 313409, 1619507, 2425799, 4113353, 4705049, 6058379, 6870089, 10395083, 10716077, 12818297, 14678057, 16173929, 16369337, 17694587, 28526699, 30318437, 31361699, 31772207, 32025107, 34132349, 37031609, 38112797, 48926477
Offset: 1

Views

Author

Zak Seidov, Feb 15 2016

Keywords

Comments

All terms are == 11 (mod 18).

Links

Crossrefs

Subsequence of A241483.
Primes in A268578.

Programs

  • Mathematica
    Select[Prime[Range[400000]], Union[PrimeOmega[# + {2, 4, 6, 8, 10, 12, 14}]] == {2} &] (* Vincenzo Librandi, Feb 17 2016 *)

A241959 Primes p such that p+2, p+4, p+6, p+8, p+10 are semiprimes.

Original entry on oeis.org

211, 1381, 3089, 5087, 10399, 18803, 26903, 27031, 31583, 41161, 47189, 49081, 53759, 62939, 63949, 76801, 87383, 93739, 98491, 107509, 109397, 113341, 128099, 143093, 158699, 182747, 186889, 193727, 197507, 201413, 204331, 209477, 239087, 252949, 255989, 256079
Offset: 1

Views

Author

K. D. Bajpai, May 03 2014

Keywords

Comments

Each term in the sequence is prime p which yields 5 semiprimes in arithmetic progression with common difference of 2.

Examples

			a(1) = 211 is prime: 213, 215, 217, 219 and 221 are semiprimes.
a(2) = 1381 is prime: 1383, 1385, 1387, 1389 and 1391 are semiprimes.

Links

Crossrefs

Cf. A001358, A082919, A092128, A241483.

Programs

  • Maple
    with(numtheory): A241959:= proc() local p;p:=ithprime(x);if  bigomega(p+2)=2 and bigomega(p+4)=2 and bigomega(p+6)=2 and bigomega(p+8)=2 and bigomega(p+10)=2 then RETURN (p); fi; end: seq(A241959 (), x=1..100000);
Showing 1-6 of 6 results.

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