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 1-7 of 7 results.

A128653 Primes occurring in A084704 exactly twice.

Original entry on oeis.org

43, 53, 59, 79, 113, 151, 163, 167, 197, 229, 269, 313, 331, 359, 367, 397, 419, 421, 449, 541, 607, 617, 619, 683, 719, 739, 757, 857, 883, 887, 907, 911, 997, 1021, 1051, 1063, 1103, 1117, 1193, 1249, 1289, 1303, 1427, 1433, 1459, 1471, 1489, 1493, 1499
Offset: 1

Views

Author

Zak Seidov, Mar 18 2007

Keywords

Examples

			43 occurs as the 7th and 10th terms in A084704: (prime(8)+43)/2 = (19+43)/2 = 31 prime and (prime(11)+43)/2 = (31+43)/2 = 37 prime;
53 is 9th and 12th terms in A084704: (prime(10)+53)/2 = (29+53)/2 = 41 prime and (prime(13)+53)/2 = (41+53)/2 = 47 prime.

Crossrefs

For primes occurring in A084704 exactly 3 and 4 times see A128654, A128655. Cf. A084704 Smallest prime p > prime(n) such that (p+prime(n))/2 is prime.

A128654 Primes occurring in A084704 exactly thrice.

Original entry on oeis.org

263, 821, 953, 1049, 1597, 2143, 2273, 2677, 2683, 2699, 2749, 3251, 3389, 3709, 3739, 3761, 3929, 4013, 4243, 4409, 4519, 4603, 4621, 4639, 4657, 4943, 5009, 5387, 5449, 5483, 5783, 6173, 6311, 6373, 6703, 6737, 6869, 6983, 7517, 7603, 7717, 7993
Offset: 1

Views

Author

Zak Seidov, Mar 18 2007

Keywords

Examples

			263 occurs as the 42nd, 51st and 53rd terms: (prime(#)+263)/2/(AT){43,52,54}= {227,251,257} primes.

Crossrefs

Cf. A084704, A128653, A128655.

A128655 Primes occurring in A084704 exactly 4 times.

Original entry on oeis.org

1223, 1907, 2423, 4079, 7993, 11383, 14431, 20113, 20359, 22189, 23209, 26993, 28603, 29339, 30259, 31013, 33211, 33629, 35083, 35353, 39191, 40177, 42797, 44549, 47963, 49663, 51283, 52027, 52727, 53077, 54577, 58657, 58991, 60859
Offset: 1

Views

Author

Zak Seidov, Mar 18 2007

Keywords

Examples

			1223 occurs as the 165th, 184th, 189th and 191st terms in A084704:
(prime[ #+1]+1223)/(AT){165,184,189,191} = {1103,1163,1187,1193} primes.

Crossrefs

Cf. A084704, A128653, A128654.

Extensions

a(5)-a(34) from Donovan Johnson, Apr 17 2010

A126938 a(1) = 3, a(n) = the smallest prime p > a(n-1) such that (a(n-1)+p)/2 is prime.

Original entry on oeis.org

3, 7, 19, 43, 79, 127, 151, 163, 199, 223, 331, 367, 379, 439, 487, 607, 619, 643, 739, 883, 991, 1051, 1087, 1171, 1231, 1327, 1471, 1627, 1699, 1747, 1759, 1987, 1999, 2179, 2383, 2551, 2683, 2731, 2767, 3067, 3259, 3343, 3571, 3643, 3739, 3847, 3907
Offset: 1

Views

Author

Zak Seidov, Mar 18 2007

Keywords

Comments

Starting with a(2)=7 all terms are 7 mod 12. - Zak Seidov, Feb 26 2017

Examples

			(3+7)/2=5 prime, (7+19)/2=13 prime, (19+43)/2=31 prime, etc.

Links

Crossrefs

Cf. A084704, A128653, A128654, A128655.

Programs

  • Maple
    A[1]:= 3: A[2]:= 7:
for n from 3 to 100 do A[n]:= f(A[n-1]) od:
seq(A[i],i=1..100); # Robert Israel, Feb 27 2017
  • Mathematica
    s={3};pn=3;n=PrimePi[pn];Do[Do[p=Prime[i];If[PrimeQ[(pn+p)/2],AppendTo[s,p];pn=p;n=i;Break[]],{i,n+1,10000}],{112}];s
sp[n_]:=Module[{p=NextPrime[n]},While[!PrimeQ[(n+p)/2],p=NextPrime[p]];p]; NestList[sp,3,50] (* Harvey P. Dale, Apr 12 2013 *)
  • PARI
    step(q)=forprime(p=q+1,, if(isprime((p+q)/2), return(p)))
first(n)=my(v=vector(n)); v[1]=3; for(k=2,n, v[k]=step(v[k-1])); v \\ Charles R Greathouse IV, Feb 27 2017

A165138 Smallest prime p > prime(n) such that p+prime(n) is a semiprime.

Original entry on oeis.org

7, 7, 17, 19, 23, 61, 29, 43, 59, 53, 43, 97, 53, 79, 59, 89, 83, 73, 79, 107, 181, 127, 131, 113, 109, 113, 151, 167, 193, 149, 151, 167, 197, 163, 197, 163, 229, 199, 179, 281, 347, 241, 263, 229, 257, 223, 271, 331, 239, 313, 269, 263, 313, 263, 269, 359, 293
Offset: 1

Views

Author

Zak Seidov, Sep 04 2009

Keywords

Comments

Except for having an additional first term the sequence coincides with A084704.
For n>2, a(n)-prime(n) is a multiple of 12, e.g., 17-5, 19-7, 23-11, etc. - Zak Seidov, Oct 15 2015

Examples

			2+7=9=3*3 (semiprime), 3+7=10=2*5 (semiprime), 5+17=22=2*11 (semiprime).

Links

Crossrefs

Cf. A084704, A001358, A145471.

Programs

  • Mathematica
    sp[n_]:=Module[{np=NextPrime[n]},While[PrimeOmega[n+np]!=2,np= NextPrime[ np]]; np]; sp/@Prime[Range[60]] (* Harvey P. Dale, Apr 06 2016 *)
  • PARI
    a(n) = {q = prime(n); p = nextprime(q+1); while (bigomega(p+q)!=2, p = nextprime(p+1)); p;} \\ Michel Marcus, Oct 15 2015

Extensions

Prior Mathematica program deleted by Harvey P. Dale, Apr 06 2016

A224895 Let p = prime(n). Smallest odd number m > p such that m + p is semiprime.

Original entry on oeis.org

7, 7, 9, 15, 15, 21, 21, 27, 35, 33, 43, 45, 45, 51, 59, 65, 63, 73, 75, 75, 85, 87, 95, 105, 105, 105, 111, 111, 117, 141, 135, 143, 141, 159, 153, 163, 169, 171, 179, 185, 183, 201, 195, 201, 201, 223, 235, 231, 231, 237, 245, 243, 261, 263, 269, 275, 273
Offset: 1

Views

Author

Zak Seidov, Jul 24 2013

Keywords

Comments

Apparently a(n) = A210497(n) for n>1, which basically indicates that the search for the smallest even semiprime larger than 2*prime(n) produces 2*prime(n+1). - R. J. Mathar, Jul 27 2013
a(n) <= A165138(n); a(n) = A165138(n) when a(n) is prime, corresponding n's: 1, 2, 11, 15, 18, 36, 39, 46, 54, 55, 58, 73, 91,.. .
Also of interest is that sequence in not monotonic: e.g., a(10) - a(9) = 33 - 35 = -2, a(31) - a(30) = 135 - 141 = -6.

Examples

			2 + 7 = 9 = 3*3, 3 + 7 = 10 = 2*5, 5 + 9 = 14 = 2*7.

Crossrefs

Cf. A000040, A001358, A084704, A165138, A211280, A210497.

Programs

  • Maple
    A224895 := proc(n)
    local p,m ;
    p := ithprime(n) ;
    for m from p+1 do
        if type(m,'odd') and numtheory[bigomega](m+p) = 2 then
            return m ;
        end if;
    end do:
end proc: # R. J. Mathar, Jul 28 2013
  • Mathematica
    Reap[Sow[7];Do[p=Prime[n];k=p+2;While[!PrimeQ[r=(p+k)/2],k=k+2];Sow[k],{n,2,100}]][[2,1]]
son[n_]:=Module[{m=If[EvenQ[n],n+1,n+2]},While[PrimeOmega[n+m]!=2,m = m+2]; m]; Table[son[n],{n,Prime[Range[60]]}] (* Harvey P. Dale, Apr 24 2017 *)

A100742 Smallest prime p such that (p+prime(n))/2 is prime.

Original entry on oeis.org

2, 3, 5, 3, 3, 13, 5, 3, 3, 5, 3, 37, 5, 3, 11, 5, 3, 13, 7, 3, 13, 3, 3, 5, 37, 5, 3, 11, 13, 5, 7, 3, 5, 3, 17, 7, 37, 3, 11, 5, 23, 13, 3, 13, 5, 3, 3, 3, 47, 73, 29, 23, 13, 3, 5, 11, 5, 3, 37, 17, 19, 5, 7, 3, 13, 17, 3, 61, 11, 13, 5, 3, 19, 13, 3, 3, 5, 61, 53, 13, 3, 37, 23, 13, 7, 3, 5
Offset: 1

Views

Author

David Wasserman, Jan 03 2005

Keywords

Crossrefs

Cf. A084704.

Programs

  • PARI
    a(n) = {if (n==1, return (2)); p = 3; while(! isprime((prime(n)+p)/2), p = nextprime(p+1)); return (p);} \\ Michel Marcus, Jun 16 2013
Showing 1-7 of 7 results.

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