A164368 -id:A164368 - cp's OEIS frontend

A217564 Number of primes between prime(n)/2 and prime(n+1)/2.

0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 2, 0, 1, 0, 2, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 2, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1

Offset: 1

Hans Havermann, Oct 06 2012

nonn

Conjecture: this sequence is unbounded, as implied by Dickson's conjecture. - Charles R Greathouse IV, Oct 09 2012
Conjecture: 0 appears infinitely often. - Jon Perry, Oct 10 2012
First differences of A079952. - Peter Munn, Oct 19 2017

			a(30) = 2 because there are two primes between prime(30)/2 [=113/2] and prime(31)/2 [=127/2]; i.e., the numbers 59 and 61.

Hans Havermann, Table of n, a(n) for n = 1..10000

Cf. A079952, A102820.
Cf. A215237 (location of first n).
A164368 lists the prime(n) corresponding to the zero terms.

q = 2; Table[p = q; q = NextPrime[p]; Length[Position[PrimeQ[Range[p + 1, q - 1, 2]/2], True]], {105}]
Table[PrimePi[Prime[n + 1]/2] - PrimePi[Prime[n]/2], {n, 105}] (* Alonso del Arte, Oct 08 2012 *)

a(n) = pi(prime(n + 1)/2) - pi(prime(n)/2), where pi is the prime counting function and prime(n) is the n-th prime.
Equivalently, a(n) = A079952(n+1) - A079952(n). - Peter Munn, Oct 19 2017
The average order of a(n) is 1/2, that is, a(1) + a(2) + ... + a(n) ~ n/2. - Charles R Greathouse IV, Oct 09 2012

A229607 Square array read by antidiagonals downwards in which each row starts with the least prime not in a previous row, and each prime p in a row is followed by the greatest prime < 2*p.

Original entry on oeis.org

2, 3, 11, 5, 19, 17, 7, 37, 31, 29, 13, 73, 61, 53, 41, 23, 139, 113, 103, 79, 47, 43, 277, 223, 199, 157, 89, 59, 83, 547, 443, 397, 313, 173, 113, 67, 163, 1093, 883, 787, 619, 337, 223, 131, 71, 317, 2179, 1759, 1571, 1237, 673, 443, 257, 139, 97, 631

Offset: 1

Clark Kimberling, Sep 26 2013

Conjectures: (row 1) = A006992, (column 1) = A104272, and for each row r(k), the limit of r(k)/2^k exists. For rows 1 to 4, the respective limits are 0.303976..., 4.249137..., 6.857407..., 12.235210... .
From Pontus von Brömssen, Jan 18 2025: (Start)
Regarding the conjectures above:
  - Row 1 is A006992 by definition.
  - Column 1 is A164368, not A104272. It seems that the first column would be A104272 if no duplicates were allowed, i.e., if the prime p in a row were followed by the largest prime < 2*p not in a previous row; see A380277.
  - The existence of the limits should follow from a strong version of Bertrand's postulate. For row 1, see formula in A006992.
(End)

			Northwest corner:
   2,    3,    5,    7,   13,   23,   43,   83, ...
  11,   19,   37,   73,  139,  277,  547, 1093, ...
  17,   31,   61,  113,  223,  443,  883, 1759, ...
  29,   53,  103,  199,  397,  787, 1571, 3137, ...
  41,   79,  157,  313,  619, 1237, 2473, 4943, ...
  47,   89,  173,  337,  673, 1327, 2647, 5281, ...

Pontus von Brömssen, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals)
Wikipedia, Bertrand's postulate.

Cf. A006992, A104272, A164368, A229608, A229609, A229610, A380277.

seqL = 14; arr1[1] = {2}; Do[AppendTo[arr1[1], NextPrime[2*Last[arr1[1]], -1]], {seqL}]; Do[tmp = Union[Flatten[Map[arr1, Range[z]]]]; arr1[z] = {Prime[NestWhile[# + 1 &, 1, PrimePi[tmp[[#]]] - # == 0 &]]}; Do[AppendTo[arr1[z], NextPrime[2*Last[arr1[z]], -1]], {seqL}], {z, 2, 12}]; m = Map[arr1, Range[12]]; m // TableForm
t = Table[m[[n - k + 1]][[k]], {n, 12}, {k, n, 1, -1}] // Flatten (* Peter J. C. Moses, Sep 26 2013 *)

Incorrect comment deleted by Peter Munn, Aug 15 2017

A182426 Lengths of runs of consecutive isolated primes beginning with A166251(n).

Original entry on oeis.org

2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 2, 1, 2, 1, 2, 1, 4, 3, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 3, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 3, 2, 1, 2, 1, 1, 2, 1, 2, 1, 3, 2, 1

Offset: 1

Vladimir Shevelev, Apr 28 2012

nonn

Theorem. If the sequence is unbounded, then there exist arbitrarily long sequences of consecutive primes p_k, p_(k+1),...,p_m such that every interval (p_i/2, p_(i+1)/2), i=k,k+1,...,m-1, contains a prime.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4.
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. A166251, A182423, A182405, A164368, A194598.

Cf. A049084.

import Data.List (group)
a182426 n = a182426_list !! (n-1)
a182426_list = concatMap f $ group $ zipWith (-) (tail ips) ips where
   f xs | head xs == 1 = reverse $ enumFromTo 2 $ length xs + 1
        | otherwise    = take (length xs) $ repeat 1
   ips = map a049084 a166251_list
-- Reinhard Zumkeller, May 18 2012

Data corrected: a(49)=2.

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

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

Vladimir Shevelev, Sep 15 2011

nonn

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.

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 7prime(k)/2 and 7prime(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

Vladimir Shevelev, Sep 17 2011

nonn

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

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

Corrected by R. J. Mathar, Sep 20 2011

A164962 a(n) is the least prime from the union {2,3} and A164333, beginning with which the n-th prime p_n is obtained by some number of iterations of the S operator g(see A164960).

Original entry on oeis.org

2, 3, 2, 3, 2, 13, 3, 19, 2, 13, 31, 3, 19, 43, 2, 53, 13, 61, 31, 71, 73, 3, 19, 43, 2, 101, 103, 53, 109, 113, 13, 131, 31

Offset: 1

Vladimir Shevelev, Sep 02 2009

The sequence is connected with our sieve selecting the primes of the union {2,3} and A164333 from all primes. Note that a(n)=n iff p_n is in the considered union, which corresponds to 0's iterations of g.

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

Cf. A164333, A055496, A164960, A104272, A080359, A164918, A006992, A164917, A164368, A164288.

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A217564 Number of primes between prime(n)/2 and prime(n+1)/2.

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A229607 Square array read by antidiagonals downwards in which each row starts with the least prime not in a previous row, and each prime p in a row is followed by the greatest prime < 2*p.

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A182426 Lengths of runs of consecutive isolated primes beginning with A166251(n).

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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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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.

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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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A195329 Records of A195325.

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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.

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