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.

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.

A172079 First differences of A181669.

Original entry on oeis.org

6, 12, 24, 12, 48, 60, 12, 48, 120, 12, 480, 180, 300, 48, 120, 132, 408, 180, 792, 120, 48, 300, 660, 132, 288, 240, 12, 300, 540, 240, 780, 120, 48, 732, 1260, 1188, 600, 1092, 408, 432, 1848, 480
Offset: 1

Views

Author

Giovanni Teofilatto, Nov 19 2010

Keywords

Crossrefs

Cf. A181669.

Programs

  • Mathematica
    Differences[Select[6*Range[3000]-1,PrimeQ[#]&&PrimeOmega[#-1]==2&&AnyTrue[ {#+2,Sqrt[#+2]},PrimeQ]&]] (* Harvey P. Dale, Jul 01 2022 *)

Extensions

a(21) ff. corrected by Georg Fischer, Aug 31 2020

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

A173176 Greater twin primes in A172240.

Original entry on oeis.org

7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609, 1621, 1669, 1699, 1723, 1789, 1873, 1879, 1933, 1951, 1999
Offset: 1

Views

Author

Giovanni Teofilatto, Nov 22 2010

Keywords

Comments

For a(n) > 5, first difference of the sequence is divisible by 6. (Conjectured or proved?)
Also for a(n)>5, a(n)-1 is divisible by 6, if a(n)-2 is prime p such that p+1 is divisible by 6.

Crossrefs

Cf. A172487, A181669, A181602.

Programs

  • Maple
    isA006512 := 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:
isA181602 := proc(p) if isprime(p) then if numtheory[bigomega](p-1) =2 and  isA000430(p+2) then true; else false; end if; else false;   end if ; end proc:
isA181669 := proc(p) isA181602(p) and (p mod 6)= 5 ; end proc:
isA172240 := proc(n) isprime(n) and not isA181669(n) ; end proc:
isA173176 := proc(n) isA172240(n) and isA006512(n) ; end proc:
for n from 2 to 2000 do if isA173176(n) then printf("%d,",n) ; end if; end do:

Formula

A172240 INTERSECT A006512.

Extensions

Corrected by R. J. Mathar, Dec 01 2010

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.

A256386 Numbers m such that m-2, m-1, m+1, m+2 cannot all be represented in the form x*y + x + y for values x, y with x >= y > 1.

Original entry on oeis.org

2, 3, 4, 5, 8, 11, 59, 1319, 1619, 4259, 5099, 6659, 6779, 11699, 12539, 21059, 66359, 83219, 88259, 107099, 110879, 114659, 127679, 130199, 140759, 141959, 144539, 148199, 149519, 157559, 161339, 163859, 175079, 186479, 204599, 230939, 249539, 267959, 273899, 312839
Offset: 1

Views

Author

Alex Ratushnyak, Mar 31 2015

Keywords

Comments

Indices of terms surrounded by pairs of zeros in A255361.
Conjectures:
1. A255361(a(n)) > 0 for n > 4.
2. All terms > 8 are primes.
3. All terms > 8 are terms of these supersequences: A118072, A171667, A176821, A181602, A181669.
From Lamine Ngom, Feb 12 2022: (Start)
For n > 4, a(n) is not a term of A254636. This means that a(n)-2, a(n)-1, a(n)+1 and a(n)+2 are adjacent terms in A254636.
Number of terms < 10^k: 5, 7, 7, 13, 19, 96, 441, 2552, ...
Conjecture 2 would follow if we establish the equivalence "t is in sequence" <=> "t is a term of b(n): lesser of twin primes pair p and q such that (p - 1)/2 and (q + 1)/2 are also a pair of twin primes (A077800)".
It appears that b(n) = a(n) for n > 5. Verified for all terms < 10^9. (End)

Examples

			9, 10, 12, 13 cannot be represented as x*y + x + y, where x >= y > 1. Therefore 11 is in the sequence.

Crossrefs

Cf. A255361, A254636.
Cf. A118072, A171667, A176821, A181602, A181669.
Cf. A001097, A001359, A006512, A077800, A158870.

Formula

a(n) = A158870(n-5) - 2, n > 5 (conjectured). - Lamine Ngom, Feb 12 2022
Showing 1-6 of 6 results.

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