A164554 -id:A164554 - cp's OEIS frontend

A166574 If p, q are successive primes, and there is a number k with p < k <= q such that r = p+k is a prime, then r is in the sequence.

5, 7, 11, 17, 23, 29, 41, 47, 59, 67, 83, 89, 97, 107, 109, 127, 137, 149, 151, 167, 179, 181, 197, 227, 229, 233, 239, 257, 263, 281, 283, 307, 317, 337, 347, 349, 359, 367, 383, 389, 401, 409, 431, 433, 449, 461, 467, 479, 487, 491

Offset: 1

Vladimir Shevelev, Oct 17 2009

nonn

The old definition was: Primes p>=5 with the property: if Prime(k)

If A(x) is the counting function of a(n) not exceeding x, then, in view of the symmetry, it is natural to conjecture that A(x)~pi(x)/2.

			Taking p=2, q=3, k=3 we get r=2+3=5, the first term.
Taking p=3, q=5, k=4 we get r=3+4=7, the second term.
From p=89, q=97 we can take both k=90 and k=92, getting the terms 89+90=179 and 89+92=181. - _Art Baker_, Mar 16 2019

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A182365, A166307, A166252, A166251, A164368, A104272, A080359, A164333, A164288, A164294, A164554

Reap[Do[p=Prime[n]; k=PrimePi[p/2]; If[p<=Prime[k]+Prime[k+1], Sow[p]], {n,3,PrimePi[1000]}]][[2,1]]
Select[#[[1]]+Range[#[[1]]+1,#[[2]]],PrimeQ]&/@Partition[Prime[Range[60]],2,1]//Flatten (* Harvey P. Dale, Jul 02 2024 *)

Extended by T. D. Noe, Dec 01 2010

Edited with simpler definition based on a suggestion from Art Baker. -N. J. A. Sloane, Mar 16 2019

A182391 Numbers n for which A104272(n) = A080359(n).

Original entry on oeis.org

1, 9, 11, 18, 22, 23, 25, 30, 32, 34, 35, 37, 38, 41, 46, 47, 48, 49, 52, 53, 54, 63, 64, 66, 70, 75, 76, 79, 80, 82, 84, 94, 98, 99, 101, 102, 105, 108, 109, 110, 113, 114, 115, 124, 127, 128, 131, 135, 136, 139, 140, 148, 149, 150, 151, 154, 156, 158, 160

Offset: 1

Vladimir Shevelev, Apr 27 2012

nonn

Number m is in the sequence iff 1) there exists only composite number k such that 2*k-1 is prime and A060715(k)=m; 2) there is no prime p such that 2*p-1 is prime and A060715(p)=m-1.

Cf. A104272, A080359, A164554, A193507, A194184, A194186, A194217, A182366.

A194217(n)=0.

A166575 Primes p>=5 with the property: if Prime(k)
=Prime(k)+ Prime(k+1).

Original entry on oeis.org

5, 13, 19, 31, 37, 43, 53, 61, 71, 73, 79, 101, 103, 113, 131, 139, 157, 163, 173, 191, 193, 199, 211, 223, 241, 251, 269, 271, 277, 293, 311, 313, 331, 353, 373, 379, 397, 419, 421, 439, 443, 457, 463, 499, 509, 521, 523, 541, 577, 601, 607, 613, 619, 631, 653, 659, 661, 673, 691

Offset: 1

Vladimir Shevelev, Oct 17 2009

nonn

If A(x) is the counting function of a(n) not exceeding x, then, in view of the symmetry, it is natural to conjecture that A(x)~pi(x)/2.

			Let p=13. Then we have 5<13/2<7. Since 13>5+7, then 13 is in the sequence.

Cf. A166574, A182365, A166307, A166252, A166251, A164368, A104272, A080359, A164333, A164288, A164294, A164554

Reap[Do[p=Prime[n]; k=PrimePi[p/2]; If[p>=Prime[k]+Prime[k+1], Sow[p]], {n,3,PrimePi[1000]}]][[2,1]]

A182392 Numbers n for which there exists only composite number k such that A060715(k) = n and 2*k-1 is prime, but A104272(n) differs from A080359(n).

Original entry on oeis.org

3, 8, 36, 55, 58, 83, 129, 134, 143, 155, 186, 197, 207, 218, 269, 295, 309, 310, 361, 362, 380, 396, 412, 454, 466, 473, 505, 511, 514, 544, 549, 556, 563, 616, 631, 660, 666, 677, 683, 697, 771, 781, 788, 797, 812, 873, 874, 881, 883, 894, 906, 953

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Apr 27 2012

nonn

There exists a prime p=p(n) such that 2*p-1 is prime and A060715(p)=a(n)-1 (cf. comment in A182391).

Cf. A104272, A080359, A164554, A193507, A194184, A194186, A194217, A182366, A182391.

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cp's OEIS Frontend

A166574 If p, q are successive primes, and there is a number k with p < k <= q such that r = p+k is a prime, then r is in the sequence.

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A182391 Numbers n for which A104272(n) = A080359(n).

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A166575 Primes p>=5 with the property: if Prime(k)
=Prime(k)+ Prime(k+1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A182392 Numbers n for which there exists only composite number k such that A060715(k) = n and 2*k-1 is prime, but A104272(n) differs from A080359(n).

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Author

Keywords

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cp's OEIS Frontend

A166574 If p, q are successive primes, and there is a number k with p < k <= q such that r = p+k is a prime, then r is in the sequence.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A182391 Numbers n for which A104272(n) = A080359(n).

Views

Author

Keywords

Comments

Crossrefs

Formula

A166575 Primes p>=5 with the property: if Prime(k)=Prime(k)+ Prime(k+1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A182392 Numbers n for which there exists only composite number k such that A060715(k) = n and 2*k-1 is prime, but A104272(n) differs from A080359(n).

Views

Author

Keywords

Comments

Crossrefs

A166575 Primes p>=5 with the property: if Prime(k)
=Prime(k)+ Prime(k+1).