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 1-5 of 5 results.

A103533 Even semiprimes of the form prime(n)*prime(n+1) - 1.

Original entry on oeis.org

14, 34, 142, 898, 1762, 5182, 19042, 79522, 213442, 359998, 412162, 627238, 685582, 777922, 1192462, 1299478, 1695202, 2005006, 2585662, 2663398, 3849322, 4536898, 5143822, 5588446, 5673922, 6594502, 7225342, 8363638, 8538058, 12110278
Offset: 1

Views

Author

Giovanni Teofilatto, Mar 22 2005

Keywords

Comments

5 is the only odd number of the form prime(n)*prime(n+1) - 1. - Klaus Brockhaus, Mar 29 2005
2*A086870(n) is a subsequence of this sequence. They first differ when 313619 is not in A086870, but 2*313619 = 627238 = a(12). This is because 787 and 797 are the first such pair of consecutive primes that are not twins and (787*797-1)/2 is prime. - Jason Kimberley, Oct 22 2015

Examples

			a(1)=14 because prime(2)*prime(3)- 1=3*5-1=14=2*7;
a(2)=34 because prime(3)*prime(4)- 1=5*7-1=34=2*17;
a(3)=142 because prime(5)*prime(6)-1=11*13-1=142=2*71.

Links

Crossrefs

Cf. A001358, A006881, A086870, A103746, A269663.

Programs

  • Magma
    [a:n in[2..1000]|IsPrime(a div 2)where a is NthPrime(n)*NthPrime(n+1)-1]; // Jason Kimberley, Oct 22 2015
  • Mathematica
    fQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ Prime[ Range[490]]*Prime[ Range[2, 491]] - 1, fQ[ # ] &] (* Robert G. Wilson v, Mar 24 2005 *)
Select[Times@@#-1&/@Partition[Prime[Range[500]],2,1],EvenQ[#] && PrimeOmega[ #]==2&] (* Harvey P. Dale, Apr 24 2018 *)
  • PARI
    for(n=1,490,if(bigomega(k=prime(n)*prime(n+1)-1)==2,print1(k,","))) \\ Klaus Brockhaus, Mar 24 2005

Extensions

More terms from Robert G. Wilson v and Klaus Brockhaus, Mar 24 2005

A102770 (p*q - 1)/2 where p and q are consecutive odd primes.

Original entry on oeis.org

7, 17, 38, 71, 110, 161, 218, 333, 449, 573, 758, 881, 1010, 1245, 1563, 1799, 2043, 2378, 2591, 2883, 3278, 3693, 4316, 4898, 5201, 5510, 5831, 6158, 7175, 8318, 8973, 9521, 10355, 11249, 11853, 12795, 13610, 14445, 15483, 16199, 17285, 18431, 19010
Offset: 1

Views

Author

W. Neville Holmes, Feb 10 2005

Keywords

Comments

Primes in this sequence: 7, 17, 71, 449, 881, 2591, ... - Zak Seidov, Jan 14 2013

Examples

			a(1) = (3*5 - 1)/2 = 7.
a(2) = (5*7 - 1)/2 = 17.
a(3) = (7*11 - 1)/2 = 38.

Links

Crossrefs

Cf. A006094, A023515, A086870, A120876.

Programs

Formula

a(n) = (prime(n + 1)*prime(n + 2) - 1)/2.
a(n) ~ 0.5 n^2/log^2 n. - Charles R Greathouse IV, Jan 14 2013
a(n) = A023515(n+2)/2. - Jason Kimberley, Oct 23 2015

A120876 (Product of twin primes - 1)/2.

Original entry on oeis.org

7, 17, 71, 161, 449, 881, 1799, 2591, 5201, 5831, 9521, 11249, 16199, 18431, 19601, 25991, 28799, 36449, 39761, 48671, 60551, 88199, 93311, 106721, 136241, 162449, 179999, 190961, 206081, 217799, 328049, 337841, 342791, 368081, 388961, 520199, 532511, 551249, 563921
Offset: 1

Views

Author

Lekraj Beedassy, Jul 09 2006

Keywords

Comments

This sequence is a subsequence of A102770.

Links

Crossrefs

Cf. The subsequence A086870.

Programs

  • Mathematica
    (Times@@#-1)/2&/@Select[Partition[Prime[Range[200]], 2,1],Last[#]- First[#]== 2&] (* Harvey P. Dale, Jun 26 2011 *)
  • PARI
    for(n=1, 200, if(prime(n+1)-prime(n)==2, print1((prime(n)*prime(n+1)-1)/2", "))) \\ Altug Alkan, Oct 23 2015
  • PARI
    p=2; forprime(q=3, 1e3, if(q-p==2, print1(p*q\2", ")); p=q) \\ Charles R Greathouse IV, Apr 01 2016

Formula

a(n) = A120875(n)/2 = A075369(n)/2-1 = A075369(n)^2/2-1.
a(n) = 18*A002822(n-1)^2-1, for n>1.
a(n) = A102770(A107770(n)). - Jason Kimberley, Nov 10 2015

Extensions

Corrected by T. D. Noe, Oct 25 2006
Edited by Jason Kimberley, Oct 23 2015

A109945 Primes p such that [p,p+2] is a pair of twin primes and (p*(p+2)-1)/2 is prime.

Original entry on oeis.org

3, 5, 11, 29, 41, 71, 137, 281, 461, 599, 641, 827, 881, 1091, 1301, 1607, 2129, 2267, 2381, 2687, 3527, 3557, 3581, 4127, 4229, 4337, 4547, 5009, 5741, 6131, 6791, 6959, 7211, 7487, 7547, 8009, 8597, 8861, 9041, 9281, 10007, 10037, 10427, 10889, 11117
Offset: 1

Views

Author

Hugo Pfoertner, Jul 09 2005

Keywords

Examples

			3 is in the sequence because [3,5] is a pair of twin primes and (3*5 - 1)/2=7 is prime.

Links

Crossrefs

Cf. A086870 [corresponding primes], A093706 [primes p such that (p*nextprime(p)-1)/2 is prime], A061351 [number separating twin pair is squarefree].

Programs

  • Mathematica
    lst={}; d=2; Do[p1=Prime[n]; p2=Prime[n+1]; If[p2-p1==2&&PrimeQ[(p1*p2-1)/2], AppendTo[lst, p1]], {n, 10^3}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 08 2008 *)

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 1-5 of 5 results.

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