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 1-4 of 4 results.

A181602 Primes p such that p-1 is a semiprime and p+2 is prime or prime squared.

Original entry on oeis.org

5, 7, 11, 23, 47, 59, 107, 167, 179, 227, 347, 359, 839, 1019, 1319, 1367, 1487, 1619, 2027, 2207, 2999, 3119, 3167, 3467, 4127, 4259, 4547, 4787, 4799, 5099, 5639, 5879, 6659, 6779, 6827, 7559, 8819, 10007, 10607, 11699, 12107, 12539, 14387, 14867
Offset: 1

Views

Author

Giovanni Teofilatto, Nov 01 2010

Keywords

Comments

Except for the second term, a(n)+1 is divisible by 6.
[Proof: a(n)=p is a prime, with p-1=q*r and two primes q<=r by definition. Omitting the special case p=2, p is odd, p+1 is even, so p+1=q*r+2 = 2(1+q*r/2). To show that p+1 is divisible by 6 we show that it is divisible by 2 and by 3; divisibility by 2 has already been shown in the previous sentence. (1+q*r/2 must be integer, so q*r/2 must be integer, so the smaller prime q of the semiprime must be q=2, so p=2*r+1. This shows that p=a(n) are a subset of A005383.) First subcase of the definition is that p+2 is also prime. Then p is a smaller twin prime and by a comment in A003627, p+1 is divisible by 3. Second subcase of the definition is that p+2 = s^2 with s a prime. s can be 3*k+1 or 3*k+2 --p=7 is the exception-- which leads to s^2 = 9*k^2+6*k+1 or s^2=9*k^2+12*k+4, so p+1 = 9*k^2+6*k or 9*k^2+12*k+3, and in both cases p+1 is divisible by 3.]
In consequence, except for the first three terms, first differences a(n+1)-a(n) are also divisible by 6.

Links

Crossrefs

Cf. A001358 (semiprimes), A001248 (squares of primes).
Cf. A173176, A172487, A172240, A181669.

Programs

  • Magma
    [ p: p in PrimesInInterval(3,15000) | &+[ k[2]: k in Factorization(p-1) ] eq 2 and (IsPrime(p+2) or (q^2 eq p+2 and IsPrime(q) where q is Isqrt(p+2))) ]; // Klaus Brockhaus, Nov 03 2010
  • Mathematica
    semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger@n == 2; fQ[n_] := Block[{fi = FactorInteger@n}, Length@ fi == 1 && fi[[1, 2]] == 1 || fi[[1, 2]] == 2]; Select[ Prime@ Range@ 1293, semiPrimeQ[ # - 1] && fQ[ # + 2] &] (* Robert G. Wilson v, Nov 06 2010 *)
Select[Prime[Range[2000]],PrimeOmega[#-1]==2&&Or@@PrimeQ[{#+2, Sqrt[ #+2]}]&] (* Harvey P. Dale, Aug 12 2012 *)

Extensions

Corrected (29 removed) and extended by Klaus Brockhaus, Robert G. Wilson v and R. J. Mathar, Nov 03 2010

A172240 Odd primes not in A181669.

Original entry on oeis.org

3, 7, 13, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 173, 181, 191, 193, 197, 199, 211, 223, 229, 233, 239, 241, 251, 257, 263, 269, 271
Offset: 1

Views

Author

Giovanni Teofilatto, Nov 20 2010

Keywords

Comments

Except for term 5, the sequence contains all greater of twin primes

Crossrefs

Cf. A001359, A173176, A172487, A181669, A181602, A173641.

A172487 Lesser of twin primes in A172240.

Original entry on oeis.org

3, 17, 29, 41, 71, 101, 137, 149, 191, 197, 239, 269, 281, 311, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1427, 1451, 1481, 1607, 1667, 1697, 1721, 1787, 1871, 1877, 1931, 1949, 1997
Offset: 1

Views

Author

Giovanni Teofilatto, Nov 21 2010

Keywords

Comments

For a(n) > 3, the first differences of the sequence are divisible by 6. (Is this a conjecture or a theorem?)

Crossrefs

Cf. A173176, A172240, A181669, A181602.

Programs

  • Maple
    isA001359 := proc(p) isprime(p) and isprime(p+2) ; end proc:
isA000430 := proc(p) if isprime(p) then true; else if issqr(p) then isprime(sqrt(p)) ; else false; end if; end if; end proc:
isA181669 := proc(p) if isprime(p) and (p mod 6)= 5 then if numtheory[bigomega](p-1) =2 and  isA000430(p+2) then true; else false; end if;else false; end if ; end proc:
isA172240 := proc(n) isprime(n) and not isA181669(n) ; end proc:
isA172487 := proc(n) isA172240(n) and isA001359(n) ; end proc:
for n from 2 to 2000 do if isA172487(n) then printf("%d,",n) ; end if;end do:

Formula

A001359 INTERSECT A172240.

A173194 a(n) = -sin^2 (2*n*arccos n) = - sin^2 (2*n*arcsin n).

Original entry on oeis.org

0, 0, 9408, 384199200, 54471499791360, 20405558846592060000, 16793517249722147195701440, 26730228454204365035835498694848, 75019085697452515216001640927169855488, 346154755746154620929434271983392498083891520
Offset: 0

Views

Author

Artur Jasinski, Feb 12 2010

Keywords

Links

Crossrefs

Cf. A132592, A146311, A146312, A146313, A173115, A173116 A173121, A173127, A173128, A173129, A173130, A173131, A173133, A173134, A173148, A173151, A173170, A173171, A173174, A173175, A173176.

Programs

  • Maple
    A173194 := proc(n) ((n+sqrt(n^2-1))^(2*n)-(n-sqrt(n^2-1))^(2*n))^2 ; expand(%/4) ; simplify(%) ; end proc: # R. J. Mathar, Feb 26 2011
  • Mathematica
    Round[Table[ -N[Sin[2 n ArcSin[n]], 100]^2, {n, 0, 15}]] (* Artur Jasinski *)
Table[FullSimplify[(-1/2 (x - Sqrt[ -1 + x^2])^(2 x) + 1/2 (x + Sqrt[ -1 + x^2])^(2 x))^2], {x, 0, 7}] (* Artur Jasinski, Feb 17 2010 *)
Table[(n^2-1)*ChebyshevU[2*n-1, n]^2, {n, 0, 20}] (* Vaclav Kotesovec, Jan 05 2019 *)
  • PARI
    {a(n) = (n^2-1)*n^2*(sum(k=0, n-1, binomial(2*n, 2*k+1)*(n^2-1)^(n-1-k)*n^(2*k)))^2} \\ Seiichi Manyama, Jan 05 2019
  • PARI
    {a(n) = (n^2-1)*polchebyshev(2*n-1, 2, n)^2} \\ Seiichi Manyama, Jan 05 2019

Formula

4*a(n) = ( (n+sqrt(n^2-1))^(2*n) - (n-sqrt(n^2-1))^(2*n) )^2. - Artur Jasinski, Feb 17 2010
From Seiichi Manyama, Jan 05 2019: (Start)
a(n) = (n^2-1) * n^2 * (Sum_{k=0..n-1} binomial(2*n,2*k+1)*(n^2-1)^(n-1-k)*n^(2*k))^2.
For n > 0, a(n) = (n^2-1) * U_{2*n-1}(n)^2 where U_{n}(x) is a Chebyshev polynomial of the second kind. (End)
a(n) ~ 2^(4*n - 2) * n^(4*n). - Vaclav Kotesovec, Jan 05 2019

Extensions

a(9) from Seiichi Manyama, Jan 05 2019
Showing 1-4 of 4 results.

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