A104272 -id:A104272 - cp's OEIS frontend

A164554 Ramanujan primes A104272(n) for which A104272(n) = A080359(n).

2, 71, 101, 181, 239, 241, 269, 349, 373, 409, 419, 433, 439, 491, 593, 599, 601, 607, 647, 653, 659, 823, 827, 857, 947, 1021, 1031, 1061, 1063, 1091, 1103, 1301, 1427, 1429, 1447, 1451, 1489, 1553, 1559, 1567, 1601, 1607, 1609, 1789, 1867, 1871, 1913, 1999, 2003

Offset: 1

Vladimir Shevelev, Aug 15 2009

nonn

For every n>=1, A104272(n) >= A080359(n), and the sequence shows where the inequality becomes an equality.

Let prime(m) < a(n)/2 < prime(m+1); then there exist primes p

For example, a(2) = 71, 31 < a(2)/2 < 37 and intervals (62,71), (71,74) contain the primes p = 67 and q = 73 respectively.

Let us call a prime p compatible with another prime q, if the intervals (p/2,q/2) and (p,q], if q>p, (or intervals (q/2,p/2) and (q,p], if qVladimir Shevelev, Apr 25 2012]

			a(2)=71, such that 31<71/2<37, and we see that p=67 is in interval (62, 71) and q=73 is in interval (71, 74).

V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT] [From _Vladimir Shevelev_, Aug 20 2009]
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. A104272, A080359, A164368, A164333, A164288, A164294, A212493, A212541.

nn = 200; t = Table[0, {nn+1}]; s = 0;
Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s <= nn && t[[s+1]] == 0, t[[s+1]] = k], {k, Prime[3nn]}
];
A080359 = Rest[t];
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[3nn]}
];
A104272 = R+1;
Intersection[A104272, A080359] (* Jean-François Alcover, Oct 28 2018, after T. D. Noe in A104272 *)

All solutions of the equation A104272(x)=A080359(x) are x=pi(a(n))-pi(a(n)/2). - Vladimir Shevelev, Apr 25 2012

Terms beyond 659 from R. J. Mathar, Dec 17 2009

A179196 Number of primes up to the n-th Ramanujan prime: A000720(A104272(n)).

Original entry on oeis.org

1, 5, 7, 10, 13, 15, 17, 19, 20, 25, 26, 28, 31, 35, 36, 39, 41, 42, 49, 50, 51, 52, 53, 56, 57, 60, 63, 64, 69, 70, 73, 74, 79, 80, 81, 83, 84, 85, 89, 93, 94, 96, 104, 105, 107, 108, 109, 110, 111, 116, 117, 118, 119, 120, 123, 128, 129, 131, 133, 136, 140, 142, 143

Offset: 1

John W. Nicholson, Jul 02 2010

nonn

a(n) = k = pi(p_k) = pi(R_n), where pi is the prime number counting function and R_n is the n-th Ramanujan prime. I.e., p_k, the k-th prime, is the n-th Ramanujan prime.

Prime index of A168421(n), that is A000720(A168421(n)), is equal to a(n) - n + 1. - John W. Nicholson, Sep 16 2015

			The 10th Ramanujan prime is 97, and pi(97) = 25, so a(10) = 25.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Christian Axler, On the number of primes up to the nth Ramanujan prime, arXiv:1711.04588 [math.NT], 2017.
Christian Axler, On Ramanujan primes, Functiones et Approximatio Commentarii Mathematici (2019).
S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
H. W. Shapiro, Iterates of arithmetic functions and a property of the sequence of primes, Pacific J. Math. Volume 3, Number 3 (1953), 647-655.
J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 7(2009), 630-635.
J. Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009, 2010.
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.
Wikipedia, Ramanujan prime

Cf. A168421, A168425.

f[n_] := With[{s = Table[{k, PrimePi[k] - PrimePi[k/2]}, {k, Prime[3 n]}]}, Table[1 + First@ Last@ Select[s, Last@ # == i - 1 &], {i, n}]]; PrimePi@ f@ 63 (* Michael De Vlieger, Nov 14 2017, after Jonathan Sondow at A104272 *)

use ntheory ":all"; say prime_count(nth_ramanujan_prime($)) for 1..100; # _Dana Jacobsen, Dec 25 2015

a(n) = A000720(A104272(n)).
a(n) = rho(n) in the paper by Sondow, Nicholson, and Noe.
prime(a(n)) = R_n = A104272(n).
a(n) = A000720(A168421(n)) + n - 1. - John W. Nicholson, Sep 16 2015

A190874 First differences of A179196, pi(R_(n+1)) - pi(R_n) where R_n is A104272(n).

Original entry on oeis.org

4, 2, 3, 3, 2, 2, 2, 1, 5, 1, 2, 3, 4, 1, 3, 2, 1, 7, 1, 1, 1, 1, 3, 1, 3, 3, 1, 5, 1, 3, 1, 5, 1, 1, 2, 1, 1, 4, 4, 1, 2, 8, 1, 2, 1, 1, 1, 1, 5, 1, 1, 1, 1, 3, 5, 1, 2, 2, 3, 4, 2, 1, 1, 3, 1, 4, 7, 1, 1, 2, 3, 3, 2, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 5, 2, 3

Offset: 1

John W. Nicholson, May 22 2011

nonn

The count of primes of the interval (R_n,R_(n+1)] where R_n is A104272(n).
The sequence A182873 is the first difference of Ramanujan primes R_(n+1)- R_n. While each non-Ramanujan prime is bound by Ramanujan primes, the maximal non-Ramanujan prime gap is less than the maximal Ramanujan prime gap, A182873, and the ratio of a(n)/A182873(n) is the average gap size at R_n.
Record terms of n, a(n) are in A202186, A202187. Each record term value of a(n) - 1 is the index m of A168425(m). A202188 is the index of A168425 when A174641(n) = A168425(m), it has repeated values of A202187.
Starting at index n = A191228(A174602(m)) in this sequence, the first instance of a count of m - 1 consecutive 1's is seen.
Limit inferior of a(n) is positive, because there are infinitely many Ramanujan primes and each term of the sequence is >= 1.
Limit superior of a(n)/log(pi(R_n)) is positive infinity. Equivalently, there are infinitely many n > 0 such that pi(R_(n+1)) > pi(R_n) + t log(pi(R_n)), for every t > 0.
For all n > 3, a(n) < n.
a(n) = rho(n+1) - rho(n) using rho(x) as defined in Sondow, Nicholson, Noe.

			R(4) = 29, the fourth Ramanujan prime, the next Ramanujan prime is a(4) = 3 primes away or R(5) = 41.

T. D. Noe, 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, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.

Cf. A179196, A104272, A000720, A168421, A168425, A182873, A202186, A202187, A202188, A174641, A191228, A174602.

nn = 100;
R = Table[0, {nn}]; s = 0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[sJean-François Alcover, Nov 11 2018, after T. D. Noe in A104272 *)

a(n) = pi(R_(n+1)) - pi(R_n) or
a(n) = A000720(A104272(n+1)) - A000720(A104272(n)).
a(n) = A179196(n+1) - A179196(n).

A194184 a(n) = A104272(n)-A193507(n).

Original entry on oeis.org

0, 8, 4, 10, 10, 4, 6, 6, 0, 24, 0, 4, 18, 18, 0, 10, 6, 0, 36, 0, 0, 0, 0, 12, 0, 10, 24, 0, 34, 0, 14, 0, 22, 0, 0, 10, 0, 0, 18, 24, 0, 4, 60, 0, 10, 0, 0, 0, 0, 28, 0, 0, 0, 0, 16, 36, 0, 6, 8, 12, 36, 10, 0, 0, 24, 0, 22, 54, 0, 0, 14, 12, 18, 6, 0, 0, 16

Offset: 1

Vladimir Shevelev, Aug 18 2011

nonn

Conjecture: The sequence is unbounded.
Records are 0, 8, 10, 24, 36, 60, 64, ... with indices 1, 2, 4, 10, 19, 43, 95, ...
Conjecture: The lower asymptotic density of nonzero terms is >0.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A104272, A193507, A194186.

A182873 First differences of the Ramanujan primes, A104272.

Original entry on oeis.org

9, 6, 12, 12, 6, 12, 8, 4, 26, 4, 6, 20, 22, 2, 16, 12, 2, 46, 2, 4, 6, 2, 22, 6, 12, 26, 4, 36, 2, 18, 6, 28, 8, 10, 12, 2, 6, 22, 26, 4, 12, 66, 2, 16, 6, 6, 2, 6, 34, 2, 4, 6, 6, 18, 42, 8, 12, 12, 18, 40, 12, 2, 4, 26, 4, 24, 56, 4, 6, 20, 16, 26, 10, 2

Offset: 1

T. D. Noe, Dec 09 2010

nonn

That is, the gaps between adjacent Ramanujan primes.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A182874, A182875, A182876, A182877.
Cf. A190874.
Cf. A001223 (differences between primes).

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

A165959 Size of the range of the Ramanujan Prime Corollary, 2*A168421(n) - A104272(n).

Original entry on oeis.org

2, 3, 5, 5, 5, 11, 3, 7, 3, 9, 5, 11, 7, 9, 7, 11, 15, 13, 27, 25, 21, 15, 13, 11, 5, 17, 7, 3, 11, 9, 15, 9, 21, 13, 3, 15, 13, 7, 5, 15, 11, 11, 17, 15, 27, 21, 15, 13, 7, 21, 19, 15, 9, 3, 17, 15, 7, 7, 7, 9, 9, 17, 15, 11, 9, 5, 5, 21, 17, 11, 7, 15, 9

Offset: 1

John W. Nicholson, Sep 12 2011

nonn

All but the first term is odd because A104272 has only one even term, 2. Because of all primes > 2 are odd, 1 can be subtracted from each term.
If this sequence has an infinite number of terms in which a(n) = 3, then the twin prime conjecture can be proved.
R_n is the sequence A104272(n) and k = pi(R_n)= A000720(R_n) with i>k.
By comparing the fractions we can see that (p_(i+1)-p_i)/(2*sqrt(p_i)) and a(n)/(2*sqrt(p_k)) are < 1 for all n > 0, in fact a(n)/(1.8*sqrt(p_k)) < 1 for all n > 0. When taking into account numbers in A182873(n) and A190874(n) to sqrt(R_n) we see that A182873(n)/(A190874(n)*sqrt(R_n)) < 1 for all n > 1.

			A168421(19) = 127, A104272(19) = 227; so a(19) = 2*A168421(19) - A104272(19) = 254 - 227 = 27. Note: for n = 20, 21, 22, 23, A168421(n) = 127. Because A168421 remains the same for these n and A104272 increases, the size of the range for a(n) for these n decreases. Note: a(18) = 2*97 - 181 = 194 - 181 = 13. This is nearly half a(19). The actual gap betweens A104272(19) and the next prime, 229, is 2.

T. D. Noe, Table of n, a(n) for n = 1..10000
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-201; Amer. Math. Monthly 116 (2009) 630-635.
Jonathan Sondow, John W. Nicholson, and Tony D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.
Wikipedia, Ramanujan Prime
Marek Wolf, A Note on the Andrica Conjecture, arXiv:1010.3945 [math.NT], 2010.

Cf. A168421, A104272, A182873, A190874.

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; t = Table[0, {nn}];
Do[m = PrimePi[2 n] - PrimePi[n]; If[0 < m <= nn, t[[m]] = n], {n, 15 nn}];
A168421 = NextPrime[Join[{1}, t]] // Most;
A165979 = 2 A168421 - A104272 (* Jean-François Alcover, Nov 07 2018, after T. D. Noe in A104272 *)

a(n) = 2*A168421(n) - A104272(n).

A173634 Even numbers that are not the sum of 2 Ramanujan primes (A104272).

Original entry on oeis.org

2, 6, 8, 10, 12, 14, 16, 18, 20, 24, 26, 30, 32, 36, 38, 42, 44, 48, 50, 54, 56, 60, 62, 66, 68, 72, 74, 80, 86, 90, 92, 98, 102, 104, 110, 116, 120, 122, 128, 132, 140, 146, 150, 152, 158, 170, 176, 182, 188, 200, 206, 212, 230, 232, 236, 242, 260, 266, 272, 284, 290, 314, 320, 344, 350, 372, 386, 398, 424, 428, 452, 484, 512, 542, 556, 564, 572, 626, 632, 644, 686, 692, 764, 962, 986, 1022, 1028, 1070, 1532, 1712, 1742, 1766, 2078, 2582, 2624

Offset: 1

Donovan Johnson, Nov 23 2010

nonn

No other terms < 2*10^8. Conjectured to be complete.

a(n) = 2*(n of A204814) when A204814(n) = 0. Related to Goldbach's conjecture in that (Conjecture:) even numbers 2626 and greater are the sum of two Ramanujan primes. - John W. Nicholson, Jan 26 2017

			68 is a term because no 2 Ramanujan primes sum to 68. 70 is not a term because 11 + 59 = 70. 11 and 59 are both Ramanujan primes.

Eric Weisstein's World of Mathematics, Ramanujan Prime
Wikipedia, Ramanujan prime

Cf. A104272, A204814.

A214756 a(n) = largest Ramanujan prime R_k in A104272 that is <= A002386(n).

Original entry on oeis.org

2, 2, 2, 17, 71, 107, 503, 881, 1103, 1301, 9521, 15671, 19543, 31387, 155849, 360289, 370061, 492067, 1349147, 1356869, 2010553, 4652239, 17051297, 20831119, 47326519, 122164649, 189695483, 191912659

Offset: 1

John W. Nicholson, Jul 27 2012

nonn

While many values in A214757(n) are equal to A000101(n), here it seems the only value such that A002386(n) is equal to a(n) is a(1) = R_k = A002386(1) = 2.

See "Let rho(m) = A179196(m)" comment at A001223.

			A104272(94) = 1301 < 1327 = A002386(10), so a(10) = 1301.

Dana Jacobsen, Table of n, a(n) for n = 1..59

Cf. A002386, A104272, A001223.

Cf. A214925, A214926, A214757, A214924.

use ntheory ":all";  sub a_from_2386 { my $n = shift; $n = prev_prime($n) while !is_ramanujan_prime($n); $n } # Dana Jacobsen, Jul 13 2016

perl -Mntheory=:all -nE 'my $n=$1 if /(\d+)$/; $r=ramanujan_primes($n>1e6 ? $n-1e6 : 2, $n); say ++$x," ",$r->[-1];' b002386.txt # Dana Jacobsen, Jul 13 2016

Edited by N. J. A. Sloane, Aug 06 2012

a(16)-a(28) from Donovan Johnson, Nov 04 2012

A214757 a(n) = smallest Ramanujan prime R_k in A104272 that is >= A000101(n).

Original entry on oeis.org

11, 11, 11, 29, 97, 127, 569, 937, 1151, 1367, 9613, 15727, 19681, 31481, 156007, 360769, 370387, 492251, 1349669, 1357333, 2010881, 4652507, 17051981, 20831639, 47326913, 122165059, 189695893, 191913047

Offset: 1

John W. Nicholson, Jul 27 2012

nonn

While many values in a(n) are equal to A000101(n), for A214756 it seems the only value such that A002386(n) is equal to A214756(n) is A214756(1) = R_k = A002386(1) = 2.

See "Let rho(m) = A179196(m)" comment at A001223.

			A104272(95) = R_k = 1367 > 1361 = A000101(10), so a(10) = 1367.

Cf. A000101, A104272, A214756, A001223.

Cf. A002386, A214924, A214925, A214926, A214757.

Edited by N. J. A. Sloane, Aug 06 2012

a(16)-a(28) from Donovan Johnson, Nov 04 2012

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cp's OEIS Frontend

A164554 Ramanujan primes A104272(n) for which A104272(n) = A080359(n).

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A179196 Number of primes up to the n-th Ramanujan prime: A000720(A104272(n)).

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A190874 First differences of A179196, pi(R_(n+1)) - pi(R_n) where R_n is A104272(n).

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A194184 a(n) = A104272(n)-A193507(n).

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A182873 First differences of the Ramanujan primes, A104272.

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

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A165959 Size of the range of the Ramanujan Prime Corollary, 2*A168421(n) - A104272(n).

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A173634 Even numbers that are not the sum of 2 Ramanujan primes (A104272).

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A214756 a(n) = largest Ramanujan prime R_k in A104272 that is <= A002386(n).

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