A164288 -id:A164288 - cp's OEIS frontend

A164952 Primes p with the property: 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 (p, 3p_(m+1)) contains a prime.

2, 3, 11, 17, 23, 29, 41, 43, 59, 61, 71, 73, 79, 97, 101, 103, 107, 131, 137, 149, 151, 163, 167, 179, 191, 193, 223, 227, 229, 239, 251, 257, 269, 271, 277, 281, 311, 331, 347, 349, 353, 359, 367, 373, 383, 397, 419, 421, 431, 433, 439, 457, 461, 463, 479, 491

Offset: 1

Vladimir Shevelev, Sep 01 2009

nonn

For k>1 (not necessarily integer), we call a Ramanujan k-prime R_n^(k) the prime a_k(n) which is the smallest number such that if x >= a_k(n), then pi(x)- pi(x/k) >= n. Note that, the sequence of all primes corresponds to the case of "k=oo". These numbers possess the following properties: R_n^(k)~p_((k/(k-1))n) as n tends to the infinity; if A_k(x) is the counting function of the Ramanujan k-primes not exceeding x, then A_k(x)~(1-1/k)\pi(x) as x tends to the infinity; let p be a Ramanujan k-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 (p, k*p_(m+1)) contains a prime. Conjecture. For every k>=2 there exist non-Ramanujan k-primes, which possess the latter property. For example, for k=2, the smallest such prime is 109. Problem. For every k>2 to estimate the smallest non-Ramanujan k-prime which possesses the latter property. [From Vladimir Shevelev, Sep 01 2009]

All Ramanujan 3-primes are in the sequence.

			If p=61, the p/3 is in the interval (19, 23); we see that the interval (61,69) contains a prime (67).

Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv:1212.2785

Cf. A104272, A164368, A164288.

nn=1000; t=Table[0, {nn}]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/3], s--]; If[s

Extended and edited by T. D. Noe, Nov 22 2010

Comments edited by Jonathan Sondow, Aug 27 2011

A195271 1.5-gap primes: Prime p is a term iff there is no prime between 1.5p and 1.5q, 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

Vladimir Shevelev, Sep 14 2011

nonn

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.

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

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

A194659 a(n) = A104272(n) - A194658(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 12, 0, 0, 0, 0, 36, 32, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 18, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 34, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 44, 40

Offset: 1

Vladimir Shevelev, Sep 01 2011

nonn

Conjecture 1. The sequence is unbounded.
Records are 0, 18, 36, 48, 64, 84, 114, 138, 184, 202, 214, 268, 282, 366, 374, 378, 412, 444, 528, ... with indices 1, 13, 19, 43, 144, 145, 167, 560, 635, 981, 982, 2605, 3967, 4582, 7422, 7423, 7424, 7425, 10320, ... .
The places of nonzero terms correspond to places of those terms of A194658 which are in A164288. Moreover, for n>=1, places of nonzero terms of A194659 and A194186(n+1) coincide. This means that these sequences have the same lengths of the series of zeros.
Conjecture 2. The asymptotic density of nonzero terms is 2/(e^2+1).

Antti Karttunen, Table of n, a(n) for n = 1..16385

Cf. A104272, A194658, A194186, A164288.

up_to = 65537;
A104272list(n) = { my(L=vector(n), s=0, k=1); for(k=1, prime(3*n)-1, if(isprime(k), s++); if(k%2==0 && isprime(k/2), s--); if(sA104272 by Satish Bysany, Mar 02 2017
v104272 = A104272list(65537);
A104272(n) = v104272[n];
A080359(n) = {my(x = 1); while((primepi(x) - primepi(x\2)) != n, x++; ); x; }; \\ From A080359
A194658(n) = precprime(A080359(1+n)-1);
A194659(n) = (A104272(n) - A194658(n)); \\ Antti Karttunen, Sep 21 2018

A164371 The smaller members of twin prime pairs which are non-Ramanujan primes.

Original entry on oeis.org

3, 5, 137, 191, 197, 521, 617, 1319, 1619, 1697, 1931, 1949, 2111, 2549, 2591, 2969, 3119, 3329, 3389, 3557, 3929, 4157, 4217, 4271, 4517, 5279, 5417, 7331, 7349, 7757, 8009, 8429, 8837, 9011, 9281, 9437, 9857, 9929, 10007, 10037, 10457, 10529, 10859

Offset: 1

Vladimir Shevelev, Aug 14 2009, Aug 17 2009

nonn

The terms greater than 5 of the sequence are in A164288.

Dana Jacobsen, Table of n, a(n) for n = 1..10000
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2

Cf. A104272, A001359, A164288, A164554

nn = 1000;
A001359 = Select[Prime[Range[2 nn]], PrimeQ[# + 2] &];
R = Table[0, {nn}]; s = 0;
Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s < nn, R[[s + 1]] = k], {k, Prime[3 nn]}];
A104272 = R + 1;
Complement[A001359, A104272] (* Jean-François Alcover, Oct 28 2018, after T. D. Noe in A104272 *)

use ntheory ":all"; for (@{twin_primes(1e4)}) { say if !is_ramanujan_prime($) } # _Dana Jacobsen, Dec 30 2015

A001359 \ A104272. - R. J. Mathar, Sep 17 2009

Extended by R. J. Mathar, Sep 17 2009

A182365 The largest prime in some interval of the form (2prime(k),2prime(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

Vladimir Shevelev, Apr 26 2012

nonn

These are called "left primes" in A166251.

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

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

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 *)

A164917 Smallest number of steps to reach prime(n) by applying the map x->A060308(x) starting from any member of A164368.

Original entry on oeis.org

0, 1, 2, 3, 0, 4, 0, 1, 5, 0, 1, 2, 0, 6, 0, 1, 0, 2, 0, 0, 3, 1, 7, 1, 0, 0, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 2, 8, 0, 2, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 0, 0, 3, 9, 1, 3, 0, 0, 1, 1, 0, 0, 1, 2, 1, 2, 0, 0, 0, 2, 0, 0, 0, 3, 1, 1, 0, 1, 1, 1, 0, 0, 2, 0, 3, 0, 1, 2, 3, 1, 1, 0, 0, 2

Offset: 1

Vladimir Shevelev, Aug 31 2009

nonn

Starting from some prime, iterated application of A060308 (or of the equivalent A059788) generates a chain of increasing prime numbers.
The nature of these chains is to reach higher in the list of primes, sometimes "over-satisfying" Bertrand's postulate by skipping some nearer primes, almost doubling of possible. On the other hand, A164368 contains the primes that would be skipped by a chain which contains the prime slightly above half of their value. The sequence shows how far up in chains starting from some member of A164368 we find prime(n), or equivalently, how many inverse applications of the map we need to hit a member of A164368 if starting at prime(n).
Note that by construction A164368(k) starts with the smallest prime that is not member of any chain started from any previous A164368. So each prime exists at some place in one of these chains, and the number of steps a(n) to reach it from the start of its chain is well defined.

			The first prime chains of the mapping with A060308 initialized with members of A164368 are
2->3->5->7->13->23->43->83->163->317->631->1259->2503->..
11->19->37->73->139->277->547->1093->2179->4357->8713->17419->..
17->31->61->113->223->443->883->1759->3517->7027->14051->28099->..
29->53->103->199->397->787->1571->3137->6271->12541->25073->..
41->79->157->313->619->1237->2473->4943->9883->19763->39521->..
47->89->173->337->673->1327->2647->5281->10559->21107->..
The a(1) to a(4) representing the first 4 primes are all on the first chain, and need 0 to 3 steps to be reached from 2 = A164368(1). a(5) asks for the number of steps for A000040(5)=11 which is on the second chain, and needs 0 steps.

V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.

Cf. A006992, A104272, A164368, A164288.

A060308 := proc(n) prevprime(2*n+1) ; end:
isA164368 := proc(p) local q ; q := nextprime(floor(p/2)) ; RETURN(numtheory[pi](2*q) -numtheory[pi](p) >= 1); end:
A164368 := proc(n) option remember; local a; if n = 1 then 2; else a := nextprime( procname(n-1)) ; while not isA164368(a) do a := nextprime(a) ; od: RETURN(a) ; fi; end:
A164917 := proc(n) local p,a,j,q,itr ; p := ithprime(n) ; a := 1000000000000000 ; for j from 1 do q := A164368(j) ; if q > p then break; fi; itr := 0 ; while q < p do q := A060308(q) ; itr := itr+1 ; od; if q = p then if itr < a then a := itr; fi; fi; od: a ; end:
seq(A164917(n),n=1..120) ; # R. J. Mathar, Sep 24 2009

A060308[n_] := NextPrime[2*n + 1, -1];
isA164368[p_] := Module[{q}, q = NextPrime[Floor[p/2]]; Return[PrimePi[2*q] - PrimePi[p] >= 1]];
A164368[n_] := A164368[n] = Module[{a}, If[n == 1, 2, a = NextPrime[ A164368[n-1]]; While[Not @ isA164368[a], a = NextPrime[a]]; Return[a]]];
A164917[n_] := Module[{p, a, j, q, itr}, p = Prime[n]; a = 10^15; For[j = 1 , True, j++, q = A164368[j]; If[q > p, Break[]]; itr = 0; While[q < p, q = A060308[q]; itr++]; If[q == p, If[itr < a, a = itr]]]; a];
Table[A164917[n], {n, 1, 120}] (* Jean-François Alcover, Dec 14 2017, after R. J. Mathar *)

Edited, examples added and extended by R. J. Mathar, Sep 24 2009

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

Vladimir Shevelev, Sep 01 2011

nonn

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.

V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes

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

A164918 The smallest starting prime which reaches prime(n) by repeated application of the map x->A060308(x).

Original entry on oeis.org

2, 2, 2, 2, 11, 2, 17, 11, 2, 29, 17, 11, 41, 2, 47, 29, 59, 17, 67, 71, 11, 41, 2, 47, 97, 101, 29, 107, 109, 17, 127, 67, 137, 11, 149, 151, 41, 2, 167, 47, 179, 181, 191, 97, 197, 29, 107, 17, 227, 229, 233, 239, 241, 127, 67, 263, 269, 137, 11, 281, 283, 149, 307, 311, 41

Offset: 1

Vladimir Shevelev, Aug 31 2009

nonn

a(n) is the starting value of the prime chain described in A164917 which contains (touches) prime(n).

By construction, each member of this sequence here is one of the values of A164368, the head elements of all chains of this map.

			The first four values are 2 because prime(1)=2, prime(2)=3, prime(3)=5 and prime(4)=7 are all in the prime chain starting at 2.

V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.

Cf. A006992, A164917, A104272, A164368, A164288.

A060308 := proc(n) prevprime(2*n+1) ; end:
isA164368 := proc(p) local q ; q := nextprime(floor(p/2)) ; return (numtheory[pi](2*q) -numtheory[pi](p) >= 1); end proc:
A164368 := proc(n) option remember; local a; if n = 1 then 2; else a := nextprime( procname(n-1)) ; while not isA164368(a) do a := nextprime(a) ; end do : RETURN(a) ; end if; end proc:
A164918 := proc(n) local p, a, j, q, itr ; p := ithprime(n) ; a := 1000000000000000 ; for j from 1 do q := A164368(j) ; if q > p then break; end if; itr := 0 ; while q < p do q := A060308(q) ; itr := itr+1 ; end do; if q = p then return A164368(j) ; end if; end do: end proc:
seq(A164918(n), n=1..120) ; # R. J. Mathar, Mar 12 2010

lp[n_] := NextPrime[2n, -1];
a[n_] := For[pn = Prime[n]; p = 2, p <= pn, p = NextPrime[p], nwl = NestWhileList[lp, p, # <= Prime[n]&]; If[MemberQ[nwl, pn], Return[p]]];
Array[a, 120] (* Jean-François Alcover, Dec 01 2017 *)

Edited and extended by R. J. Mathar, Mar 12 2010

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

Vladimir Shevelev, Sep 02 2009

nonn

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.

			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.

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

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]

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

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

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A164952 Primes p with the property: 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 (p, 3p_(m+1)) contains a prime.

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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.

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A194659 a(n) = A104272(n) - A194658(n).

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A164371 The smaller members of twin prime pairs which are non-Ramanujan primes.

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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.

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A164917 Smallest number of steps to reach prime(n) by applying the map x->A060308(x) starting from any member of A164368.

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

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A164918 The smallest starting prime which reaches prime(n) by repeated application of the map x->A060308(x).

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Maple

Mathematica

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

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