A104272 -id:A104272 - cp's OEIS frontend

A191228 Greatest Ramanujan prime index less than x, eta(x).

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10

Offset: 1

John W. Nicholson, May 28 2011

nonn

a(n) is the greatest value k of A104272(k) less than x. The integer inverse function of A104272.

Starting at index m = a(A174602(n)) in A190874(m), the first instance of a count of n - 1 consecutive 1's is seen.

			a(17) = eta(17) = 3. Or, R_3 = 17.

J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.

Cf. A104272, A191225, A191226, A191227.

nn = 100; 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;
Table[Boole[MemberQ[A104272, k]], {k, 1, 100}] // Accumulate (* Jean-François Alcover, Nov 07 2018, using T. D. Noe's code for A104272 *)

A192821 3-Ramanujan primes; the interval (x/2,x] has at least n 2-Ramanujan primes for x >= a(n) but not for x = a(n)-1.

Original entry on oeis.org

41, 149, 227, 229, 233, 569, 571, 587, 593, 641, 643, 821, 937, 941, 1367, 1373, 1423, 1439, 1481, 1549, 1553, 2207, 2237, 2239, 2267, 2269, 2273, 2281, 2333, 2339, 2347, 2377, 2617, 2657, 3251, 3257, 3259, 3299, 3343, 3347, 3449, 3581, 3583, 3607, 3613

Offset: 1

T. D. Noe, Jul 11 2011

nonn

It is conjectured that primepi(a(n)) < 10*n for large n. - T. D. Noe, Aug 26 2011
The sequence is only conjectural without a proof of an upper bound on a(n) (like the bound A104272(n) < prime(3*n) proved by Laishram and used in computing Ramanujan primes). - Jonathan Sondow, Aug 26 2011
Subsequence of the 2-Ramanujan primes A192820, by the minimality of a(n). - Jonathan Sondow, Aug 21 2012

Cf. A104272 (Ramanujan primes), A192820 (2-Ramanujan primes), A192822, A192823, A192824.

Definition clarified by Jonathan Sondow, Aug 21 2012

A192822 4-Ramanujan primes; the interval (x/2,x] has at least n 3-Ramanujan primes for x >= a(n) but not for x = a(n)-1.

Original entry on oeis.org

569, 571, 587, 1367, 1373, 1423, 1439, 2207, 2237, 2239, 2267, 2269, 2273, 3251, 3257, 3259, 3299, 3343, 3347, 3449, 3581, 3583, 3607, 5639, 5641, 5647, 5651, 5653, 5689, 5693, 5737, 5779, 5783, 5821, 8059, 8069, 8209, 8527, 8537, 8597, 8599, 8731, 8747

Offset: 1

T. D. Noe, Jul 11 2011

nonn

It is conjectured that primepi(a(n)) < 22*n for large n. - T. D. Noe, Aug 26 2011
The sequence is only conjectural without a proof of an upper bound on a(n) (like the bound A104272(n) < prime(3*n) proved by Laishram and used in computing Ramanujan primes). - Jonathan Sondow, Aug 27 2011
Subsequence of the 3-Ramanujan primes A192821, by the minimality of a(n). - Jonathan Sondow, Aug 21 2012

Cf. A104272 (Ramanujan primes), A192820, A192821 (3-Ramanujan primes), A192823, A192824.

Definition clarified by Jonathan Sondow, Aug 21 2012

A192823 5-Ramanujan primes; the interval (x/2,x] has at least n 4-Ramanujan primes for x >= a(n) but not for x = a(n)-1.

Original entry on oeis.org

1367, 1373, 1423, 1439, 2207, 2237, 3251, 3257, 3259, 3299, 5639, 8059, 12739, 12781, 12799, 12809, 12821, 12823, 12829, 12907, 12911, 12917, 12919, 12953, 13147, 13163, 13171, 13669, 13687, 13691, 13693, 14009, 14029, 14057, 14081, 14143, 31957, 32183

Offset: 1

T. D. Noe, Jul 11 2011

It is conjectured that primepi(a(n)) < 50*n for large n. - T. D. Noe, Aug 26 2011
The sequence is only conjectural without a proof of an upper bound on a(n) (like the bound A104272(n) < prime(3*n) proved by Laishram and used in computing Ramanujan primes). - Jonathan Sondow, Aug 27 2011
Subsequence of the 4-Ramanujan primes A192822, by the minimality of a(n). - Jonathan Sondow, Aug 21 2012

Cf. A104272 (Ramanujan primes), A192820, A192821, A192822 (4-Ramanujan primes), A192824.

Definition clarified by Jonathan Sondow, Aug 21 2012

A192824 Least n-Ramanujan prime.

Original entry on oeis.org

2, 2, 11, 41, 569, 1367

Offset: 0

T. D. Noe, Jul 11 2011

We define the 0-Ramanujan primes to be the usual primes (A000040), and we define the 1-Ramanujan primes to be the Ramanujan primes (A104272).

The sequence is only conjectural without a proof of an upper bound on a(n) (like the bound A104272(n) < prime(3*n) proved by Laishram and used in computing Ramanujan primes). - Jonathan Sondow, Aug 27 2011

Cf. A104272 (Ramanujan primes), A192820 (2-Ramanujan primes), A192821 (3-Ramanujan primes), A192822 (4-Ramanujan primes), A192823 (5-Ramanujan primes), A225907 (least n-Ramanujan prime less than half the next n-Ramanujan prime, or 0 if none).

A204814 Number of decompositions of 2n into an unordered sum of two Ramanujan primes.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 2, 1, 0, 3, 0, 0, 1, 1, 0, 2, 0, 0, 1, 2, 0, 2, 2, 0, 4, 0, 0, 1, 2, 0, 2, 0, 1, 1, 3, 0, 2, 2, 0, 2, 0, 0, 1, 2, 0, 2, 1, 1, 2, 4, 0, 1, 2

Offset: 1

Donovan Johnson, Jan 27 2012

nonn

Suggested by John W. Nicholson.
There are 95 zeros in the first 10000 terms. Are there more? Related to Goldbach's conjecture. - T. D. Noe, Jan 27 2012
There are no other zeros in the first 10^8 terms. a(n) > 0 for n from 1313 to 10^8. - Donovan Johnson, Jan 27 2012

			a(29) = 3. 2*29 = 58 = 11+47 = 17+41 = 29+29 (11, 17, 29, 41 and 47 are all Ramanujan primes). 58 is the unordered sum of two Ramanujan primes in three ways.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A045917, A104272, A173634.

A228520 a(n) is the smallest number such that if x >= a(n), then pi^(x) - pi^(x/2) >= n, where pi^*(x) is the number of terms of A050376 <= x.

Original entry on oeis.org

2, 3, 11, 16, 23, 41, 47, 59, 67, 71, 79, 101, 107, 109, 127, 149, 167, 169, 179, 181, 227, 229, 233, 239, 256, 263, 269, 281, 283, 307, 347, 349, 359, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491, 521, 569, 587, 593, 599, 601, 607, 617, 641, 643, 647

Offset: 1

Vladimir Shevelev, Aug 24 2013

nonn

The sequence is a Fermi-Dirac analog of Ramanujan numbers (A104272), since terms of A050376 play a role of primes in Fermi-Dirac arithmetic (see comments in A050376).

Peter J. C. Moses, Table of n, a(n) for n = 1..10000

Cf. A104272.

a(n)<= R_n, where R_n is the n-th Ramanujan number (A104272); a(n)~A000040(2*n) as n goes to infinity.

More terms from Peter J. C. Moses

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

A233739 R(n) - prime(2n), where R(n) is the n-th Ramanujan prime and prime(n) is the n-th prime.

Original entry on oeis.org

-1, 4, 4, 10, 12, 10, 16, 14, 10, 26, 22, 18, 26, 42, 38, 36, 40, 30, 64, 56, 52, 46, 42, 40, 40, 42, 56, 48, 76, 68, 74, 62, 84, 72, 70, 72, 60, 56, 64, 78, 70, 70, 126, 114, 124, 114, 108, 98, 86, 100, 86, 78, 76, 66

Offset: 1

Jonathan Sondow, Dec 15 2013

sign

The sequence tends to decrease at runs of Ramanujan primes and at twin Ramanujan primes.
Is 4 the minimum value of a(n) for all n > 1? Is the sequence unbounded? What are its liminf and limsup? Is a(n)/n bounded?
Christian Axler has proved that the answers to the 1st, 2nd, and 4th questions are yes, and that liminf a(n) = limsup a(n) = infinity. - Jonathan Sondow, Feb 12 2014
a(n) > n, for 1 < n < 86853959 = limit. For limit, a(n) = 135595760, a(n) - n = 48741801. - John W. Nicholson, Dec 19 2013

			R(2) - prime(4) = 11 - 7 and R(3) - prime(6) = 17 - 13, so a(2) = a(3) = 4.

John W. Nicholson, Table of n, a(n) for n = 1..10000
Christian Axler, On generalized Ramanujan primes, arXiv:1401.7179 [math.NT], 2014.
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (2009), 630-635; arXiv:0907.5232 [math.NT], 2009-2010.
Jonathan Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.

Cf. A000040, A104272, A233740. Records are A233741.

nn = 60; 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]}];
R = R + 1;
Table[R[[n]] - Prime[2 n], {n, 1, nn}] (* Jean-François Alcover, Nov 07 2018, using T. D. Noe's code for R *)

a(n) = A104272(n) - A000040(2n).
a(n) = 2*A233740(n) for n > 1.
a(n) >= 2 for n > 1 (see "Ramanujan primes and Bertrand's postulate").
a(n)/p(2n) = R(n)/p(2n) - 1 -> 0 as n -> infinity (see same link).

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

cp's OEIS Frontend

A191228 Greatest Ramanujan prime index less than x, eta(x).

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A192821 3-Ramanujan primes; the interval (x/2,x] has at least n 2-Ramanujan primes for x >= a(n) but not for x = a(n)-1.

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A192822 4-Ramanujan primes; the interval (x/2,x] has at least n 3-Ramanujan primes for x >= a(n) but not for x = a(n)-1.

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A192823 5-Ramanujan primes; the interval (x/2,x] has at least n 4-Ramanujan primes for x >= a(n) but not for x = a(n)-1.

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A192824 Least n-Ramanujan prime.

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A204814 Number of decompositions of 2n into an unordered sum of two Ramanujan primes.

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A228520 a(n) is the smallest number such that if x >= a(n), then pi^*(x) - pi^*(x/2) >= n, where pi^*(x) is the number of terms of A050376 <= x.

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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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A233739 R(n) - prime(2n), where R(n) is the n-th Ramanujan prime and prime(n) is the n-th prime.

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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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A228520 a(n) is the smallest number such that if x >= a(n), then pi^(x) - pi^(x/2) >= n, where pi^*(x) is the number of terms of A050376 <= x.