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.

A103746 Numbers n such that prime(n)*prime(n+1) - 1 is semiprime.

Original entry on oeis.org

2, 3, 5, 10, 13, 20, 33, 60, 89, 109, 116, 138, 144, 152, 182, 189, 212, 223, 253, 258, 297, 320, 336, 350, 353, 375, 390, 418, 422, 487, 492, 498, 501, 549, 567, 579, 592, 616, 654, 671, 704, 755, 799, 800, 812, 826, 874, 893, 917, 921, 948, 951, 957, 967
Offset: 1

Views

Author

Klaus Brockhaus, Mar 29 2005

Keywords

Examples

			prime(10)*prime(11) - 1 = 29*31 - 1 = 898 = 2*449, hence 10 is a term.

Links

Crossrefs

Cf. A001358, A006881, A103533.

Programs

  • PARI
    for(n=1,1000,if(bigomega(prime(n)*prime(n+1)-1)==2,print1(n,",")))

A086870 Primes equal to a product of twin primes minus 1 divided by 2.

Original entry on oeis.org

7, 17, 71, 449, 881, 2591, 9521, 39761, 106721, 179999, 206081, 342791, 388961, 596231, 847601, 1292831, 2268449, 2571911, 2836961, 3612671, 6223391, 6329681, 6415361, 8520191, 8946449, 9409121, 10342151, 12550049, 16485281, 18800711
Offset: 1

Views

Author

Cino Hilliard, Aug 20 2003

Keywords

Comments

From Jason Kimberley, Oct 22 2015 (Start)
Prime elements of A120876.
For each p in this list, A001221(2p) = A001222(2p) = A001221(2p+1) = A001222(2p+1) = 2.
2*a(n) is a subsequence of A103533. They first differ when 313619 is not in this sequence, but 2*313619 = 627238 = A103533(12).
(End)

Examples

			t1 = 71,t2 = 73, (71*73-1)/2 = 5182/2 = 2591 = prime.

Links

Crossrefs

Cf. A103533, A120876.

Programs

  • Mathematica
    Select[(Times[#, # + 2] - 1)/2 &@ Select[Prime@ Range@ 1000, PrimeQ[# + 2] &], PrimeQ] (* Michael De Vlieger, Nov 06 2015 *)
  • PARI
    for(n=1, 1e3, if(prime(n+1)-prime(n)==2 && isprime(k=(prime(n)*prime(n+1)-1)/2), print1(k", "))) \\ Altug Alkan, Nov 06 2015

Formula

Primes of the form (t1*t2-1)/2, where t1, t2 are twin primes.

A103614 Semiprimes of the form prime(n)*prime(n+1)*prime(n+2) - 1.

Original entry on oeis.org

4198, 33262, 1564258, 6672202, 7566178, 18181978, 20193022, 178433278, 187466722, 229580146, 293158126, 467821918, 1125878062, 1341880018, 4317369778, 5198554618, 8493529942, 10138087306, 10594343758, 20940647698
Offset: 1

Views

Author

Jonathan Vos Post, Mar 24 2005

Keywords

Comments

This is the three-consecutive-prime minus one equivalent of A103533, which is Giovanni Teofilatto's two-consecutive-prime minus one sequence.

Examples

			n: prime(n) * prime(n+1) * prime(n+2) - 1
6: 13 *17 *19 - 1 = 4198 = 2 * 2099
10: 29 * 31 * 37 - 1 = 33262 = 2 * 16631
29: 109 * 113 * 127 - 1 = 1564258 = 2 * 782129
42: 181 * 191 * 193 -1 = 6672202 = 2 * 3336101
44: 193 * 197 * 199 -1 = 7566178 = 2 * 3783089
55: 257 * 263 * 269 -1 = 18181978 = 2 * 9090989
57: 269 * 271 * 277 -1 = 20193022 = 2 * 10096511
102: 557 * 563 * 569 -1 = 178433278 = 2 * 89216639

Links

Crossrefs

Cf. A000040, A001358, A006881, A103533, A103746, A104874, A104875.

Programs

  • Mathematica
    Bigomega[n_]:=Plus@@Last/@FactorInteger[n]; SemiprimeQ[n_]:=Bigomega[n]==2; Select[Table[Prime[n]*Prime[n+1]*Prime[n+2]-1, {n, 1000}], SemiprimeQ] (* Ray Chandler, Mar 29 2005 *)
Select[Times@@#-1&/@Partition[Prime[Range[500]],3,1],PrimeOmega[#]==2&] (* Harvey P. Dale, Mar 03 2025 *)
  • PARI
    for(n=1,420,if(bigomega(k=prime(n)*prime(n+1)*prime(n+2)-1)==2,print1(k,","))) (Brockhaus)

Extensions

Extended by Ray Chandler and Klaus Brockhaus, Mar 29 2005

A104874 Semiprimes of the form prime(n)*prime(n+1)*prime(n+2)*prime(n+3) - 1.

Original entry on oeis.org

209, 1154, 645328246, 2445956098, 2337448622686, 19317973275826, 22894376863198, 32220239865718, 51087435019342, 78382834887262, 163068083613646, 176031800345938, 622751201209726, 1292966939911018
Offset: 1

Views

Author

Jonathan Vos Post, Mar 29 2005

Keywords

Comments

This is the four-consecutive-prime minus one equivalent of A103533.

Examples

			n: prime(n) * prime(n+1) * prime(n+2) * prime(n+3) - 1
1: 2 * 3 * 5 * 7 - 1 = 209 = 11 * 19
2: 3 * 5 * 7 * 11 - 1 = 1154 = 2 * 577
36: 151 * 157 * 163 * 167 - 1 = 645328246 = 2 * 322664123
47: 211 * 223 * 227 * 229 - 1 = 2445956098 = 2 * 1222978049
201: 1229 * 1231 * 1237 * 1249 - 1 = 2337448622686 = 2 * 1168724311343.

Links

Crossrefs

Cf. A000040, A001358, A006881, A103533, A103614, A103746, A104875.

Programs

  • Mathematica
    Bigomega[n_]:=Plus@@Last/@FactorInteger[n]; SemiprimeQ[n_]:=Bigomega[n]==2; Select[Table[Prime[n]*Prime[n+1]*Prime[n+2]*Prime[n+3]-1, {n, 1000}], SemiprimeQ] (* Ray Chandler, Mar 29 2005 *)

Extensions

Extended by Ray Chandler, Mar 29 2005

A104875 Semiprimes of the form prime(n)*prime(n+1)*prime(n+2)*prime(n+3)*prime(n+4) - 1.

Original entry on oeis.org

15014, 1062346, 600662302, 2224636919002, 118335570521086, 168652154886862, 3790374062238502, 6290838589498366, 127018534712243098, 131125107904515418, 190740905520325018, 2057351971883521282, 3151949824862998762
Offset: 1

Views

Author

Jonathan Vos Post, Mar 29 2005

Keywords

Comments

This is the five-consecutive-prime minus one equivalent of A103533.

Examples

			n prime(n) * prime(n+1) * prime(n+2) * prime(n+3) * prime(n+4) - 1
1: 2 * 3 * 5 * 7 * 11 - 1 = 2309 is prime; examples hereafter are semiprime
2: 3 * 5 * 7 * 11 * 13 - 1 = 15014 = 2 * 7507
5: 11 * 13 * 17 * 19 * 23 - 1 = 1062346 = 2 * 531173
15: 47 * 53 * 59 * 61 * 67 - 1 = 600662302 = 2 * 300331151
60: 281 * 283 * 293 * 307 * 311 - 1 = 2224636919002 = 2 * 1112318459501
117: 643 * 647 * 653 * 659 * 661 - 1 = 118335570521086 = 2 * 59167785260543

Links

Crossrefs

Cf. A000040, A001358, A006881, A103533, A103614, A103746, A104874.

Programs

  • Mathematica
    Bigomega[n_]:=Plus@@Last/@FactorInteger[n]; SemiprimeQ[n_]:=Bigomega[n]==2; Select[Table[Prime[n]*Prime[n+1]*Prime[n+2]*Prime[n+3]*Prime[n+4]-1, {n, 1000}], SemiprimeQ] (* Ray Chandler, Mar 29 2005 *)

Extensions

Extended by Ray Chandler, Mar 29 2005

A269663 Semiprimes which are the product of a twin prime pair minus one.

Original entry on oeis.org

14, 34, 142, 898, 1762, 5182, 19042, 79522, 213442, 359998, 412162, 685582, 777922, 1192462, 1695202, 2585662, 4536898, 5143822, 5673922, 7225342, 12446782, 12659362, 12830722, 17040382, 17892898, 18818242, 20684302, 25100098, 32970562, 37601422, 46131262, 48441598
Offset: 1

Views

Author

K. D. Bajpai, Mar 02 2016

Keywords

Comments

Subsequence of A103533 and A001358.
All the terms in this sequence, except a(1), are congruent to 1 (mod 3).

Examples

			a(1) = 14 = 2 * 7 that is semiprime. Also, 3 * 5 - 1 = 14 where {3,5} is a twin prime pair.
a(2) = 34 = 2 * 17 that is semiprime. Also, 5 * 7 - 1 = 34 where {5,7} is a twin prime pair.

Links

Crossrefs

Cf. A001097, A001358, A001359, A006512, A086870, A103533, A118071.

Programs

  • Magma
    IsP2:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ s: n in [1..1000] | IsPrime(n) and IsPrime(n+2) and IsP2(s) where s is (n * (n+2) - 1)];
  • Maple
    A269663:= proc() local a, b, d; a:= ithprime(n); b:=a+2; d:=(a*b)-1; if isprime(b)and bigomega(d)=2 then return (d): fi; end: seq(A269663 (n), n=1..1000);
  • Mathematica
    A269663= {}; Do[a = Prime[n]; b = a + 2; c = a*b - 1; If[PrimeQ[b] && PrimeOmega[c] == 2, AppendTo[A269663, c]], {n, 1000}]; A269663
Select[Times @@ # - 1 & /@ Transpose@{#, 2 + #} &@ Select[Prime@ Range@ 900, NextPrime@ # == # + 2 &], PrimeOmega@ # == 2 &] (* Michael De Vlieger, Apr 01 2016 *)
Select[Times@@@Select[Partition[Prime[Range[1000]],2,1],#[[2]]-#[[1]]==2&]-1,PrimeOmega[ #]==2&] (* Harvey P. Dale, Mar 14 2023 *)
  • PARI
    for(n = 1, 1000, p = prime(n); q = p + 2; c=(p*q) - 1; if(isprime(q) && bigomega(c)==2, print1(c, ", ")));

Formula

a(n) = 2*A086870(n). - Ray Chandler, Apr 04 2016
Showing 1-6 of 6 results.

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