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.

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A166252 Primes which are not the smallest or largest prime in an interval of the form (2*prime(k),2*prime(k+1)).

Original entry on oeis.org

71, 101, 109, 151, 181, 191, 229, 233, 239, 241, 269, 283, 311, 349, 373, 409, 419, 433, 439, 491, 571, 593, 599, 601, 607, 643, 647, 653, 659, 683, 727, 823, 827, 857, 941, 947, 991, 1021, 1031, 1033, 1051, 1061, 1063, 1091, 1103, 1301, 1373, 1427, 1429
Offset: 1

Views

Author

Vladimir Shevelev, Oct 10 2009, Oct 14 2009

Keywords

Comments

Called "central primes" in A166251, not to be confused with the central polygonal primes A055469.
The primes tabulated in intervals (2*prime(k),2*prime(k+1)) are
5, k=1
7, k=2
11,13, k=3
17,19, k=4
23, k=5
29,31, k=6
37, k=7
41,43, k=8
47,53, k=9
59,61, k=10
67,71,73, k=11
79, k=12
83, k=13
89, k=14
97,101,103, k=15
and only rows with at least 3 primes contribute primes to the current sequence.
For n >= 2, these are numbers of A164368 which are in A194598. - Vladimir Shevelev, Apr 27 2012

Examples

			Since 2*31 < 71 < 2*37 and the interval (62, 74) contains prime 67 < 71 and prime 73 > 71, then 71 is in the sequence.

Links

Crossrefs

Cf. A166307, A182365, A166251, A164368, A104272, A080359, A164333, A164288, A164294, A164554, A194598.

Programs

  • Mathematica
    n = 0; t = {}; While[Length[t] < 100, n++; ps = Select[Range[2*Prime[n], 2*Prime[n+1]], PrimeQ]; If[Length[ps] > 2, t = Join[t, Rest[Most[ps]]]]]; t (* T. D. Noe, Apr 30 2012 *)

A166307 The smallest prime in some interval of the form (2*prime(k),2*prime(k+1)) if this interval contains at least 2 primes.

Original entry on oeis.org

11, 17, 29, 41, 47, 59, 67, 97, 107, 127, 137, 149, 167, 179, 197, 227, 263, 281, 307, 347, 367, 401, 431, 461, 487, 503, 521, 569, 587, 617, 641, 677, 719, 739, 751, 769, 809, 821, 853, 881, 907, 937, 967, 983, 1009, 1019, 1049, 1087, 1097, 1117, 1151, 1163, 1187, 1217, 1229, 1249, 1277
Offset: 1

Views

Author

Vladimir Shevelev, Oct 11 2009, Oct 17 2009

Keywords

Comments

These are called "right primes" in A166251.

Examples

			For p=29 we have: 2*13 < 29 < 2*17 and interval (26, 29) is free from primes while interval (29, 34) contains a prime. Therefore 29 is in the sequence for k=6.

Links

Crossrefs

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

Programs

  • Mathematica
    f[n_] := Block[{t = Select[ Table[i, {i, 2 Prime[n], 2 Prime[n + 1]}], PrimeQ]}, If[ Length@ t > 1, t[[1]], 0]]; Rest@ Union@ Array[f, 115] (* Robert G. Wilson v, May 08 2011 *)

A195271 1.5-gap primes: Prime p is a term iff there is no prime between 1.5*p and 1.5*q, where q is the next prime after p.

Original entry on oeis.org

2, 5, 17, 29, 41, 79, 101, 137, 149, 163, 191, 197, 227, 269, 281, 313, 349, 353, 461, 463, 521, 541, 569, 593, 599, 613, 617, 641, 757, 769, 809, 821, 827, 857, 881, 887, 941, 1009, 1049, 1061, 1087, 1093, 1097, 1117, 1151, 1223, 1229, 1277, 1279, 1289
Offset: 1

Views

Author

Vladimir Shevelev, Sep 14 2011

Keywords

Comments

For a real r>1, a prime p is called an r-gap prime, if there is no prime between r*p and r*q, where q is the next prime after p. In particular, 2-gap primes form A080192 and 3-gap primes form A195270.

Crossrefs

Cf. A080192, A195270, A193507, A194186, A164368, A194598, A194658, A194659, A194674, A164288, A164294.

Programs

  • Mathematica
    Select[Prime[Range[400]], PrimePi[3*NextPrime[#]/2] == PrimePi[3*#/2] &] (* T. D. Noe, Sep 14 2011 *)

A182365 The largest prime in some interval of the form (2*prime(k),2*prime(k+1)) if this interval contains at least 2 primes.

Original entry on oeis.org

13, 19, 31, 43, 53, 61, 73, 103, 113, 131, 139, 157, 173, 193, 199, 251, 271, 293, 313, 353, 379, 421, 443, 463, 499, 509, 523, 577, 613, 619, 661, 691, 733, 743, 757, 773, 811, 829, 859, 883, 911, 953, 971, 997, 1013, 1039, 1069, 1093, 1109, 1123, 1153
Offset: 1

Views

Author

Vladimir Shevelev, Apr 26 2012

Keywords

Comments

These are called "left primes" in A166251.

Examples

			For k=6 we have 2*13 < 29 < 31 < 2*17, and the interval contains two primes. Therefore 31 is in the sequence.

Crossrefs

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

Programs

  • Mathematica
    n = 0; t = {}; While[Length[t] < 100, n++; ps = Select[Range[2*Prime[n], 2*Prime[n + 1]], PrimeQ]; If[Length[ps] >= 2, AppendTo[t, ps[[-1]]]]]; t (* T. D. Noe, Apr 30 2012 *)

A194674 Positions of nonzero terms of A194659(n)-A194186(n+1), n>=1.

Original entry on oeis.org

20, 27, 73, 77, 85, 95, 106, 116, 117, 122, 125, 132, 137, 144, 145, 152, 162, 167, 168, 189, 191, 192, 193, 198, 201, 208, 213, 234, 235, 236, 243, 249, 258, 259, 265, 275, 279, 286, 287, 291, 318, 319, 321, 329, 330, 331, 340
Offset: 1

Views

Author

Vladimir Shevelev, Sep 01 2011

Keywords

Comments

The sequence (together with A194953) characterizes a right-left symmetry in the distribution of primes over intervals (2*p_n, 2*p_(n+1)), n=1,2,..., where p_n is the n-th prime.

Links

Crossrefs

Cf. A104272, A080359, A193507, A194186, A164368, A194598, A194658, A194659, A194953, A164288, A164294.

A195377 2.5-gap primes: Prime p is a term if there is no prime between 2.5*p and 2.5*q, where q is the next prime after p.

Original entry on oeis.org

127, 197, 281, 311, 347, 431, 613, 659, 673, 739, 877, 991, 1049, 1229, 1277, 1289, 1367, 1481, 1579, 1613, 1667, 1721, 1787, 1877, 1907, 2027, 2081, 2087, 2141, 2203, 2213, 2237, 2239, 2269, 2287, 2309, 2377, 2383, 2473, 2657, 2689, 2707, 2749, 2767, 2801
Offset: 1

Views

Author

Vladimir Shevelev, Sep 17 2011

Keywords

Crossrefs

Cf. A195270, A195271, A195325, A194658, A164294.

Programs

  • Mathematica
    Select[Prime[Range[500]], PrimePi[5*NextPrime[#]/2] == PrimePi[5*#/2] &] (* T. D. Noe, Sep 20 2011 *)

A164958 Primes p with the property that if p/3 is in the interval (p_m, p_(m+1)), where p_m>=3 and p_k is the k-th prime, then the interval (3p_m, p) contains a prime.

Original entry on oeis.org

2, 3, 5, 13, 19, 29, 31, 43, 47, 61, 67, 73, 79, 83, 101, 103, 107, 109, 137, 139, 151, 157, 167, 173, 181, 193, 197, 199, 229, 233, 241, 257, 263, 271, 277, 281, 283, 313, 317, 349, 353, 359, 367, 373, 379, 389, 401, 409, 431, 433, 439, 443, 461, 463, 467, 487, 499
Offset: 1

Views

Author

Vladimir Shevelev, Sep 02 2009

Keywords

Comments

For k>1 (not necessarily integer), we call a Labos k-prime L_n^(k) the prime a_k(n) which is the smallest number such that pi(a_k(n)) - pi(a_k(n)/k)= n. Note that, the sequence of all primes corresponds to the case of "k=oo". Let p be a k-Labos prime, such that p/k is in the interval (p_m, p_(m+1)), where p_m>=3 and p_n is the n-th prime. Then the interval (k*p_(m), p) contains a prime. Conjecture. For every k>1 there exist non-k-Labos primes, which possess the latter property. For example, for k=2, the smallest such prime is 131. Problem. For every k>1 to estimate the smallest non-k-Labos prime, which possess the latter property. [From Vladimir Shevelev, Sep 02 2009]
All 3-Labos primes are in this sequence.

Examples

			If p=61, the p/3 is in the interval (19, 23); we see that the interval (57, 61) contains a prime (59). Thus 61 is in the sequence.

Crossrefs

Cf. A104272, A080359, A164952, A164368, A164288, A164294

Programs

  • Mathematica
    nn=1000; t=Table[0, {nn+1}]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/3], s--]; If[s<=nn && t[[s+1]]==0, t[[s+1]]=k], {k, Prime[3*nn]}]; Rest[t]

Extensions

Extended by T. D. Noe, Nov 23 2010

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.

Original entry on oeis.org

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

Views

Author

Vladimir Shevelev, Oct 17 2009

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

A195329 Records of A195325.

Original entry on oeis.org

2, 59, 71, 149, 191, 641, 809, 3371, 5849, 9239, 20507, 20981, 32117, 48779, 176777, 191249, 204509, 211061, 223679, 245129, 358877, 654161, 2342771, 3053291, 4297961, 4755347, 6750221, 8019509, 9750371, 10196759, 11237981, 23367077, 34910219, 93929219, 186635747
Offset: 1

Views

Author

Vladimir Shevelev, Sep 15 2011

Keywords

Comments

The sequence is infinite. Conjecture. For n>=2, all terms are in A001359. This conjecture (weaker than the conjecture in comment to A195325) also implies the twin prime conjecture.

Crossrefs

Cf. A195325, A080192, A195270, A195271, A193507, A194186, A164368, A194598, A194658, A194659, A194674, A164288, A164294.

A195379 3.5-gap primes: Primes prime(k) such that there is no prime between 7*prime(k)/2 and 7*prime(k+1)/2.

Original entry on oeis.org

2, 137, 281, 521, 641, 883, 937, 1087, 1151, 1229, 1277, 1301, 1489, 1567, 1607, 1697, 2027, 2081, 2237, 2381, 2543, 2591, 2657, 2687, 2729, 2801, 2851, 2969, 3119, 3257, 3301, 3359, 3463, 3467, 3529, 3673, 3733, 3793, 3821, 3851, 4073, 4217, 4229, 4241, 4259, 4283, 4337, 4421, 4481
Offset: 1

Views

Author

Vladimir Shevelev, Sep 17 2011

Keywords

Crossrefs

Cf. A080192, A195270, A195271, A195377, A195325, A195329, A193507, A194186, A164368, A194598, A194658, A194659, A194674, A164288, A164294.

Programs

  • Mathematica
    Select[Prime[Range[1000]], PrimePi[7*NextPrime[#]/2] == PrimePi[7*#/2] &] (* T. D. Noe, Sep 20 2011 *)

Extensions

Corrected by R. J. Mathar, Sep 20 2011
Previous Showing 11-20 of 24 results. Next

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