John W. Nicholson - cp's OEIS frontend

A350582 Decimal expansion of sum of reciprocals of maximal prime gap primes: Sum_{n>=1} 1/A000101(n).

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

Offset: 0

John W. Nicholson, Jan 07 2022

			0.68172979491338871...

Cf. A000101, A350581.

v = readvec("b000101.txt");s=0; for(i=1,#v,s= 1/v[i]+s);s*1.
\\ PARI's "readvec" doesn't work with the 2-column original OEIS b-file "b000101.txt". One needs to strip the index column first from b-file.

a(15)-a(17) from John W. Nicholson, Feb 25 2025

A350581 Decimal expansion of sum of reciprocals of maximal prime gap primes: Sum_{n>=1} 1/A002386(n).

Original entry on oeis.org

1, 0, 4, 4, 7, 0, 0, 5, 8, 5, 0, 8, 1, 1, 9

Offset: 1

John W. Nicholson, Jan 07 2022

			1.04470058508119...

Cf. A002386, A350582.

B2386A = readvec("b002386.txt");s=0;for(i=1,80,s= 1/B2386A[i]+s);s*1.
\\ PARI's "readvec" doesn't work with the 2-column original OEIS b-file "b002386.txt". One needs to strip the index column first from b-file.

A326915 Number of terms of A002386 (primes preceding record prime gaps) in the interval (2^n, 2^(n+1)].

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 1, 3, 0, 3, 0, 1, 0, 2, 1, 1, 2, 2, 0, 2, 2, 1, 1, 3, 1, 1, 2, 5, 2, 3, 1, 1, 1, 1, 1, 1, 2, 0, 0, 3, 0, 0, 0, 0, 2, 1, 2, 4, 1, 1, 0, 2, 3

Offset: 0

John W. Nicholson, Oct 20 2019

nonn

The record prime gaps are in A005250; the corresponding primes are in A002386.

			For n=3, there are no primes p_m in A002386 in the range 2^3 = 8 < p_m <= 16 = 2^4, so a(3)=0.
For n=6, there are 2 primes p_m in A002386 in the range 2^6 = 64 < p_m <= 128 = 2^7, namely p_m = 89, 113, so a(6)=2.

Cf. A000101 (upper ends), A005250 (record gaps).

A280380 First occurrence of A280379(k) = n.

Original entry on oeis.org

1, 2, 30, 46, 374, 2146, 5945, 14855, 24702

Offset: 0

John W. Nicholson, Jan 09 2017

nonn

n < log(a(n)) for a(n) < 1.1*10^6.

			For prime(30)=113, A056171(113) = 14, A104272(12)=107 and A104272(13) = 127, so 14 - 12 = 2 (First occurrence).

Cf. A056171, A104272.

\\RR[x] is a list of Ramanujan primes, A104272.
{plimit=1.1*10^6;i=n=s=0;
forprime(p=2,plimit,
s++;
if(p==RR[n+1],n++);
if(i==s-primepi(floor(p/2))-n,print(i," ",s);i++)
)
}

A280379 a(n) = A056171(k) - m, where k=prime(n) and m is the Ramanujan prime index to the greatest Ramanujan prime R(m) <= k.

Original entry on oeis.org

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

Offset: 1

John W. Nicholson, Jan 01 2017

nonn

a(n)=0 corresponds to the Ramanujan prime R_m = A104272(m) = A056171(k).

			For prime(30)=113, A056171(113) = 14, 107 is R_12 and 127 is R_13, so 14 -12 = 2 (first occurrence).

Cf. A056171, A104272.

\\RR[x] is a list of Ramanujan primes, A104272.
{plimit=1.1*10^4;n=s=0;
forprime(p=2,plimit,
    s++;
    if(p==RR[n+1],n++);
    print1(s-primepi(floor(p/2))-n,", ");
    )
}

a(n) = A056171(k) - m, where k=prime(n) and m is the Ramanujan prime index to the greatest Ramanujan prime R_m = A104272(m) <= k.

A277719 Index for the bound for the first k-Ramanujan prime.

Original entry on oeis.org

3, 5, 7, 10, 12, 16, 31, 35, 47, 48, 63, 67, 100, 218, 264, 298, 328, 368, 430, 463, 591, 651, 739, 758, 782, 843, 891, 929, 1060, 1184, 1230, 1316, 1410, 1832, 2226, 3386, 3645, 3794, 3796, 4523, 4613, 4755, 5009, 5950

Offset: 1

John W. Nicholson, Oct 27 2016

nonn

The index a(n) is h(n), the prime A277718(n) is p_h(n). If 1 <= n <= 43 and k in [p_{h(n+1)}/p_{h(n+1)-1}, p_{h(n)}/p_{h(n)-1}), then the first k-Ramanujan prime R^{(k)}1 = p{h(n)}. Extra terms require improvements of prime numbers in short intervals.

			With n = 3, we see p_h(3) = 17, p_h(4) = 29, so that 29/23 <= k < 17/13. If k = 1.3 then R^(1.3)_1 = 17 = p_h(3).

Christian Axler and Thomas Leßmann, An explicit upper bound for the first k-Ramanujan prime, arXiv:1504.05485 [math.NT], 2015.
Christian Axler and Thomas Leßmann, On the first k-Ramanujan prime, Amer. Math. Monthly, 124 (2017), 642-646.

Cf. A277718, A164952, A104272, A290394 (first (1 + 1/n)-Ramanujan prime).

A277718 Bounding prime for the first k-Ramanujan prime.

Original entry on oeis.org

5, 11, 17, 29, 37, 53, 127, 149, 211, 223, 307, 331, 541, 1361, 1693, 1973, 2203, 2503, 2999, 3299, 4327, 4861, 5623, 5779, 5981, 6521, 6947, 7283, 8501, 9587, 10007, 10831, 11777, 15727, 19661, 31469, 34123, 35671, 35729, 43391, 44351, 45943, 48731, 58889

Offset: 1

John W. Nicholson, Oct 27 2016

nonn

The index A277719(n) is h(n), the prime a(n) is p_h(n). If 1 <= n <= 43 and k in [p_{h(n+1)}/p_{h(n+1)-1}, p_{h(n)}/p_{h(n)-1}), then the first k-Ramanujan prime R^{(k)}1 = p{h(n)}. Extra terms require improvements of prime numbers in short intervals.

			With n = 3, we see p_h(3) = 17, p_h(4) = 29, so that 29/23 <= k < 17/13. If k = 1.3 then R^(1.3)_1 = 17 = p_h(3).

Christian Axler and Thomas Leßmann, An explicit upper bound for the first k-Ramanujan prime, arXiv:1504.05485 [math.NT], 2015.
Christian Axler and Thomas Leßmann, On the first k-Ramanujan prime, Amer. Math. Monthly, 124 (2017), 642-646.

Cf. A277719, A164952, A104272, A290394 (first (1 + 1/n)-Ramanujan prime).

A267549 Primes prime(k) such that floor( (prime(k)/k)^2 ) <= prime(k+1) - prime(k).

Original entry on oeis.org

3, 5, 7, 13, 23, 113

Offset: 1

John W. Nicholson, Jan 16 2016

Prime index A000720 is: 2, 3, 4, 6, 9, and 30.
floor( (prime(k)/k)^2 ) is: 2, 2, 3, 4, 6, and 14.
Similarly, ceiling( (prime(k)/k)^2 ) > prime(k+1) - prime(k) holds for all prime(k) < 10^8 with the exception of prime(k) = 7. For prime(k) = 7, 4 = ceiling((prime(k) / k)^2) = prime(k+1) - prime(k).
Stronger than Firoozbakht's conjecture which states that the sequence prime(k)^(1/k) is strictly decreasing.
Conjecture: list is complete. If so, subsequence of A124147 and A174635.
Andrew Granville conjectures that lim sup (prime(n+1)-prime(n))/log(prime(n))^2 >= 2/e^gamma = 1.1229189.... If so (or at least if the lim sup is greater than 1) then this sequence is infinite. - Charles R Greathouse IV, Feb 18 2016

			For a(3) = 7, floor((7 / 4)^2) = 3 < 4 = 11 - 7. Note that all other a(n) use = instead of <.

Alexei Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, arXiv preprint arXiv:1506.03042 [math.NT], 2015.
John Nicholson, A Conjecture Sharper than Cramér's and Firoozbakht's, Mathematics Stack Exchange.

Cf. A001223, A001248, A124147.

Select[Prime@ Range[10^5], Floor[(#/PrimePi@ #)^2] <= NextPrime@ # - # &] (* Michael De Vlieger, Jan 21 2016 *)

L=10^11;p=2;forprime(q=3,L,a=floor((p/primepi(p))^2.);if(a<=q-p, print1(p, ", "));p=q)

Floor((A000040(k) / k)^2) <= A000040(k+1)-A000040(k), where k = A000720.

Floor(A001248(k) / A000290(k)) <= A001223(k), where k = A000720.

A238734 Log of twice the twin prime constant, C_2, log(2*A005597).

Original entry on oeis.org

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

Offset: 0

John W. Nicholson, Mar 03 2014

The value occurs as term in equation (15) in the Wolf paper. - Ralf Stephan, Mar 28 2014

			0.2778768820732319619323108667032534203602062941473682988245270533677164980...

Marek Wolf, Nearest-neighbor-spacing distribution of prime numbers and quantum chaos, Phys. Rev. E 89, 022922 (2014); arXiv preprint, arXiv:1212.3841 [math.NT], 2012-2014.

Cf. A005597, A114907.

digits = 113;
s[n_] := (1/n)*N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], digits + 50];
C2 = (175/256)*Product[(Zeta[n]*(1 - 2^(-n))*(1 - 3^(-n))*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[n]), {n, 2, digits + 50}];
RealDigits[Log[2 C2]][[1]][[1 ;; digits]] (* Jean-François Alcover, Feb 16 2019 *)

default(realprecision,1000);
result={175/256*prod(k=2, 500, (zeta(k)*(1-1/2^k)*(1-1/3^k)*(1-1/5^k)*(1-1/7^k))^(-sumdiv(k, d, moebius(d)*2^(k/d))/k))};log(2*result)

log(2 * prodeulerrat(1-1/(p-1)^2, 1, 3)) \\ Amiram Eldar, Mar 16 2021

Equals log(2*A005597).

A234298 Ramanujan prime R_k such that pi(R_(k+1)) - pi(R_k) are record values: record Ramanujan prime A190874(k).

Original entry on oeis.org

2, 71, 181, 503, 2531, 5273, 9887, 11587, 55339, 174917, 225961, 534883, 2492311, 5409337, 130449773, 141833603, 212583797, 658046911, 1183597123, 2897211971, 5602581277, 46992178547, 70637059291, 158465541049, 182591976709, 339683208863

Offset: 1

John W. Nicholson, Dec 22 2013

nonn

These are the primes preceding the unique values of A174641. That sequence is the start of a run of non-Ramanujan primes, so the previous prime is the Ramanujan prime. - Dana Jacobsen, Jul 14 2016

Record values are in A202186, index of A190874 at record terms in A202187, A202188 is the index of A168425 when A174641(n) = A168425(m); A202188(n) = m. A202187 is also the index of a(n).

perl -Mntheory=:all -nE 'my $n = $1 if /(\d+)$/; say ++$x," ",prev_prime($n) unless $seen{$n}++;' b174641.txt  # Dana Jacobsen, Jul 14 2016

use ntheory ":all"; my($max,$r)=(0,ramanujan_primes(1e7)); for (0..$#$r-1) { my $d=prime_count($r->[$+1])-prime_count($r->[$]); if ($d > $max) { say $r->[$]; $max=$d; } } # _Dana Jacobsen, Jul 14 2016

a(20) to a(26) from Dana Jacobsen, Jul 14 2016

cp's OEIS Frontend

User: John W. Nicholson

A350582 Decimal expansion of sum of reciprocals of maximal prime gap primes: Sum_{n>=1} 1/A000101(n).

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A350581 Decimal expansion of sum of reciprocals of maximal prime gap primes: Sum_{n>=1} 1/A002386(n).

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A326915 Number of terms of A002386 (primes preceding record prime gaps) in the interval (2^n, 2^(n+1)].

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A280380 First occurrence of A280379(k) = n.

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A280379 a(n) = A056171(k) - m, where k=prime(n) and m is the Ramanujan prime index to the greatest Ramanujan prime R(m) <= k.

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A277719 Index for the bound for the first k-Ramanujan prime.

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A277718 Bounding prime for the first k-Ramanujan prime.

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A267549 Primes prime(k) such that floor( (prime(k)/k)^2 ) <= prime(k+1) - prime(k).

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A238734 Log of twice the twin prime constant, C_2, log(2*A005597).

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