Alain Rocchelli - cp's OEIS frontend

A385093 Primes preceded and followed by gaps whose quotient (value greater or equal to 1) is greater than 2 and less than 3.

331, 953, 1381, 1861, 2161, 2357, 2371, 2423, 2879, 3229, 3271, 3407, 3491, 3607, 3643, 3691, 3889, 4057, 4073, 4139, 4201, 4507, 4567, 4751, 4831, 4903, 4987, 5059, 5153, 5297, 5309, 5683, 5897, 6029, 6053, 6067, 6173, 6229, 6529, 6599, 6653, 6857, 7079, 7151, 7159, 7193, 7283, 7417, 7717, 7867, 7963

Offset: 1

Alain Rocchelli, Jun 17 2025

nonn

Primes prime(k) such that 2 < Max(prime(k)-prime(k-1), prime(k+1)-prime(k)) / Min(prime(k)-prime(k-1), prime(k+1)-prime(k)) < 3.

			331 is a term because Max(331-317,337-331)/Min(331-317,337-331) = 14/6 = 2.3333.
953 is a term because Max(953-947,967-953)/Min(953-947,967-953) = 14/6 = 2.3333.
1381 is a term because Max(1381-1373,1399-1381)/Min(1381-1373,1399-1381) = 18/8 = 2.25.

Cf. A384603, A384618.

Prime/@Select[Range[3,1000],2James C. McMahon, Jun 29 2025 *)

forprime(P=3, 8000, my(M=P-precprime(P-1), Q=nextprime(P+1)-P, AR=max(M,Q)/min(M,Q)); if(AR>2 && AR<3, print1(P,", ")));

Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 1/6.

A384618 Primes preceded and followed by gaps whose quotient (value greater or equal to 1) is greater than 2.

Original entry on oeis.org

29, 31, 59, 61, 73, 113, 127, 137, 139, 149, 151, 179, 181, 191, 199, 223, 239, 241, 269, 271, 283, 307, 317, 331, 347, 419, 421, 431, 433, 467, 521, 523, 541, 569, 571, 599, 601, 619, 641, 659, 661, 673, 773, 809, 811, 821, 829, 853, 863, 877, 887, 907, 953, 967

Offset: 1

Alain Rocchelli, Jun 04 2025

nonn

Primes prime(k) such that Max(prime(k)-prime(k-1),prime(k+1)-prime(k)) / Min(prime(k)-prime(k-1),prime(k+1)-prime(k)) > 2.

			19 is not a term because Max(19-17,23-19)/Min(19-17,23-19) = 4/2 = 2.
29 is a term because Max(29-23,31-29)/Min(29-23,31-29) = 6/2 = 3.
31 is a term because Max(31-29,37-31)/Min(31-29,37-31) = 6/2 = 3.
37 is not a term because Max(37-31,41-37)/Min(37-31,41-37) = 6/4 = 1.5.

Cf. A384603.

forprime(P=3, 1000, my(M=P-precprime(P-1), Q=nextprime(P+1)-P, AR=max(M, Q)/min(M, Q), AR0=2); if(AR>AR0, print1(P, ", ")));

from itertools import islice
from sympy import nextprime
def A384618_gen(): # generator of terms
    p,q,r = 2,3,5
    while True:
        s, t = q-p, r-q
        if s>(t<<1) or t>(s<<1): yield q
        p, q, r = q, r, nextprime(r)
A384618_list = list(islice(A384618_gen(),54)) # Chai Wah Wu, Jun 10 2025

Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 2/3.

A384603 Primes preceded and followed by gaps whose quotient (value greater or equal to 1) is less than 2.

Original entry on oeis.org

5, 23, 37, 47, 53, 67, 79, 83, 89, 131, 157, 163, 167, 173, 211, 233, 251, 257, 263, 277, 293, 337, 353, 359, 367, 373, 379, 383, 389, 409, 439, 443, 449, 479, 503, 547, 557, 563, 577, 587, 593, 607, 613, 631, 647, 653, 677, 683, 691, 701, 709, 719, 727, 733, 739, 751, 757, 787, 797

Offset: 1

Alain Rocchelli, Jun 04 2025

nonn

Primes prime(k) such that Max(prime(k)-prime(k-1),prime(k+1)-prime(k)) / Min(prime(k)-prime(k-1),prime(k+1)-prime(k)) < 2.

			5 is a term because Max(5-3,7-5)/Min(5-3,7-5) = 2/2 = 1.
23 is a term because Max(23-19,29-23)/Min(23-19,29-23) = 6/4 = 1.5.
37 is a term because Max(37-31,41-37)/Min(37-31,41-37) = 6/4 = 1.5.

Cf. A383215.

forprime(P=3, 1000, my(M=P-precprime(P-1), Q=nextprime(P+1)-P, AR=max(M,Q)/min(M,Q), AR0=2); if(AR

from itertools import islice
from sympy import nextprime
def A384603_gen(): # generator of terms
    p,q,r = 2,3,5
    while True:
        s, t = q-p, r-q
        if s<(t<<1) and t<(s<<1): yield q
        p, q, r = q, r, nextprime(r)
A384603_list = list(islice(A384603_gen(),59)) # Chai Wah Wu, Jun 10 2025

Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 1/3.

A381850 Primes p preceded and followed by primes whose difference is less than 2*log(p).

Original entry on oeis.org

41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 151, 163, 167, 179, 193, 197, 227, 229, 233, 239, 269, 271, 277, 281, 311, 313, 349, 353, 379, 383, 419, 421, 431, 433, 439, 443, 457, 461, 463, 487, 491, 499, 503, 563, 569, 571, 593, 599, 601, 607, 613, 617, 641, 643, 647, 653

Offset: 1

Alain Rocchelli, May 06 2025

nonn

Primes prime(k) such that prime(k+1) - prime(k-1) < 2*log(prime(k)).

Since the geometric mean is never greater than the arithmetic mean: this sequence is a subsequence of A383652.

			19 is not a term because 23-17=6 and 2*log(19)=5.8889.
41 is a term because 43-37=6 and 2*log(41)=7.4271.
131 is not a term because 137-127=10 and 2*log(131)=9.7504.
137 is a term because 139-131=8 and 2*log(137)=9.8400.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A031131, A383652.

A288907 is a subsequence.

P:= select(isprime,[2,seq(i,i=3..1000,2)]):
P[select(i -> is(P[i+1]-P[i-1] < 2*log(P[i])), [$2..nops(P)-1])]; # Robert Israel, Jun 06 2025

Select[Prime[Range[120]],NextPrime[#] - NextPrime[#,-1] < 2Log[#] &] (* Stefano Spezia, May 06 2025 *)

forprime(P=3, 800, my(M=precprime(P-1), Q=nextprime(P+1)); if(Q-M<2*log(P), print1(P,", ")));

Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 1-(3/e^2).

A383652 Primes p preceded and followed by gaps whose product is less than (log(p))^2.

Original entry on oeis.org

17, 19, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 139, 149, 151, 163, 167, 179, 181, 191, 193, 197, 199, 227, 229, 233, 239, 241, 269, 271, 277, 281, 283, 311, 313, 347, 349, 353, 379, 383, 397, 401, 419, 421, 431, 433, 439, 443, 457, 461, 463, 487, 491, 499, 503, 521, 523, 563, 569, 571, 593, 599

Offset: 1

Alain Rocchelli, May 04 2025

nonn

Since the geometric mean is never greater than the arithmetic mean: A381850 is a subsequence.

			17 is a term because (17-13)*(19-17)=8 is less than (log(17))^2=8.0271.
19 is a term because (19-17)*(23-19)=8 is less than (log(19))^2=8.6697.
29 is not a term because(29-23)*(31-29)=12 is greater than (log(29))^2=11.3387.

A288907 and A381850 are subsequences.

Cf. A083550.

Select[Range[2, 110] // Prime, (# - NextPrime[#, -1])(NextPrime[#] - #) < Log[#]^2 &] (* Stefano Spezia, May 04 2025 *)

forprime(P=3, 600, my(M=P-precprime(P-1), Q=nextprime(P+1)-P, AR=M*Q, AR0=(log(P))^2); if(AR

Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 0.720268...

A381044 Primes prime(k) followed by a gap, prime(k+1)-prime(k), smaller than the local geometric average gap between consecutive primes: log(prime(k))/e^(gamma).

Original entry on oeis.org

41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1279, 1289, 1297, 1301, 1303, 1319, 1423, 1427, 1429, 1447, 1451, 1481, 1483, 1487, 1489

Offset: 1

Alain Rocchelli, Apr 14 2025

nonn

Primes prime(k) such that log(prime(k+1)-prime(k)) < log(log(prime(k)))-gamma, where log is the natural logarithm and gamma is Euler’s constant (A001620).
Except for terms less than 41, A001359 (Lesser of twin primes) is a subsequence. From 41, the first term not included is 1279.
It has been conjectured that primes are distributed around their average spacing in a Poisson distribution (cf. D. A. Goldston and A. H. Ledoan). This is the basis of the conjecture that, for k tending to infinity, the asymptotic limit of the average of log(prime(k+1)-prime(k)) is log(log(prime(k))) - gamma (where gamma is Euler's constant). Also, the geometric mean of the gap between consecutive primes [p(k+1)-p(k)] is equivalent to log(prime(k)) / e^gamma.

			29 is not a term because log(31-29) > log(log(29))-0.5772156649, i.e.: 0.693147 > 0.636894.
41 is a term because log(43-41) < log(log(41))-0.5772156649, i.e.: 0.693147 < 0.734779.

D. A. Goldston and A. H. Ledoan, On the differences between consecutive prime numbers, I, arXiv:1111.3380 [math.NT], 2011-2012.
Carlos Rivera, Conjecture 82. Average of log Dn / log(logPn) equal R = 0,877 08..., The Prime Puzzles & Problems Connection.

Cf. A001359, A046869, A001223, A001620.

Select[Prime[Range[237]],Log[NextPrime[#]-#]James C. McMahon, May 02 2025 *)

forprime(P=3, 1500, my(Q=nextprime(P+1), LNDP=log(Q-P)); if(LNDP

Limit_{n->oo} n / PrimePi(a(n)) = 1-e^(-1/e^gamma).

A383215 Primes p preceded and followed by gaps whose difference (absolute value) is greater than log(p).

Original entry on oeis.org

7, 29, 31, 113, 127, 139, 149, 181, 191, 199, 223, 241, 283, 307, 317, 331, 347, 419, 421, 431, 467, 521, 523, 541, 619, 641, 661, 673, 773, 809, 811, 821, 829, 853, 863, 877, 887, 907, 953, 967, 1009, 1021, 1031, 1049, 1051, 1061, 1069, 1087, 1129, 1151, 1153, 1213, 1259, 1277

Offset: 1

Alain Rocchelli, Apr 19 2025

nonn

Primes prime(k) such that abs(prime(k-1)-2*prime(k)+prime(k+1)) > log(prime(k)), where log is the natural logarithm.

a(n) ~ prime(round(n*e)) as n tends to infinity, where e is Euler's number.

			7 is a term because abs(5-2*7+11)=2 and log(7)=1.9459.
29 is a term because abs(23-2*29+31)=4 and log(29)=3.3673.

Cf. A036263, A383216, A068985.

Select[Prime[Range[2,206]],Abs[NextPrime[#,-1]-2#+NextPrime[#]]>Log[#]&] (* James C. McMahon, Apr 29 2025 *)

forprime(P=3, 1300, my(M=P-precprime(P-1), Q=nextprime(P+1)-P, AR1=min(M,Q), AR2=max(M,Q), AR0=log(P)); if(AR2-AR1>AR0, print1(P,", ")));

Limit_{n->oo} n / PrimePi(a(n)) = 1/e (A068985).

A383216 Primes p which are preceded and followed by gaps whose difference is greater than 2*log(p).

Original entry on oeis.org

113, 127, 523, 887, 907, 1087, 1129, 1151, 1277, 1327, 1361, 1669, 1693, 1931, 1951, 1973, 2203, 2311, 2333, 2477, 2557, 2971, 2999, 3163, 3251, 3299, 3469, 4049, 4297, 4327, 4523, 4547, 4783, 4861, 5119, 5147, 5237, 5351, 5381, 5531, 5557, 5591, 5749, 5779, 5981

Offset: 1

Alain Rocchelli, Apr 19 2025

nonn

Primes prime(k) such that abs(prime(k-1)-2*prime(k)+prime(k+1)) > 2*log(prime(k)), where log is the natural logarithm.

			113 is a term because abs(109-2*113+127)=12 and 2*log(113)=9.4548.
127 is a term because abs(113-2*127+131)=10 and 2*log(127)=9.6884.

Cf. A036263, A383215, A092553.

Select[Prime[Range[2,782]],Abs[NextPrime[#,-1]-2#+NextPrime[#]]>2Log[#]&] (* James C. McMahon, Apr 27 2025 *)

forprime(P=3, 6000, my(M=P-precprime(P-1), Q=nextprime(P+1)-P, AR1=min(M,Q), AR2=max(M,Q), AR0=2*log(P)); if(AR2-AR1>AR0, print1(P,", ")));

Limit_{n->oo} n / PrimePi(a(n)) = 1/e^2 (A092553).

A382052 Primes prime(k) such that klog(k)/prime(k) > (k-1)log(k-1)/prime(k-1).

Original entry on oeis.org

3, 5, 7, 13, 19, 31, 41, 43, 47, 61, 71, 73, 83, 101, 103, 107, 109, 113, 131, 139, 151, 167, 181, 193, 197, 199, 227, 229, 233, 241, 271, 281, 283, 311, 313, 317, 337, 349, 353, 359, 373, 379, 383, 389, 401, 421, 433, 439, 443, 449, 461, 463, 467, 491, 503, 509, 523, 547, 563, 569, 571, 577, 593, 599

Offset: 1

Alain Rocchelli, Mar 13 2025

nonn

All terms of this sequence are contained in A060770.

a(n) ~ prime(round(n*e/(e-1))) as n tends to infinity, cf. A185393.

			3 is a term because 2*log(2)/3 > 1*log(1)/2 and 3 is the 2nd prime following 2.
5 is a term because 3*log(3)/5 > 2*log(2)/3 and 5 is the 3rd prime following 3.

Cf. A382051, A060770, A068996, A185393.

Select[Prime[Range[2, 109]],PrimePi[#]*Log[PrimePi[#]]/#>(PrimePi[#]-1)*Log[PrimePi[#]-1]/NextPrime[#,-1]&] (* James C. McMahon, Apr 14 2025 *)

my(N=1); forprime(P=3, 600, my(Q=precprime(P-1), AR0=N*log(N)/Q, AR=(N+1)*log(N+1)/P); N++; if(AR>AR0, print1(P,", ")));

Limit_{n->oo} n / PrimePi(a(n)) = 1-1/e (A068996).

A382051 Primes prime(k) such that klog(k)/prime(k) < (k-1)log(k-1)/prime(k-1).

Original entry on oeis.org

11, 17, 23, 29, 37, 53, 59, 67, 79, 89, 97, 127, 137, 149, 157, 163, 173, 179, 191, 211, 223, 239, 251, 257, 263, 269, 277, 293, 307, 331, 347, 367, 397, 409, 419, 431, 457, 479, 487, 499, 521, 541, 557, 587, 631, 641, 673, 691, 701, 709, 719, 727, 751, 769, 787, 797

Offset: 1

Alain Rocchelli, Mar 13 2025

nonn

a(n) ~ prime(round(n*e)) as n tends to infinity, where e is Euler's number.

			11 is a term because 5*log(5)/11 < 4*log(4)/7 and 11 is the 5th prime following 7.
17 is a term because 7*log(7)/17 < 6*log(6)/13 and 17 is the 7th prime following 13.

A subsequence is A060769.

Cf. A382052, A060769, A001113, A068985.

Select[Prime[Range[2,139]],PrimePi[#]*Log[PrimePi[#]]/#<(PrimePi[#]-1)*Log[PrimePi[#]-1]/NextPrime[#,-1]&] (* James C. McMahon, Apr 08 2025 *)

my(N=1); forprime(P=3, 800, my(Q=precprime(P-1), AR0=N*log(N)/Q, AR=(N+1)*log(N+1)/P); N++; if(AR

Limit_{n->oo} n / PrimePi(a(n)) = 1/e.

cp's OEIS Frontend

User: Alain Rocchelli

A385093 Primes preceded and followed by gaps whose quotient (value greater or equal to 1) is greater than 2 and less than 3.

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A384618 Primes preceded and followed by gaps whose quotient (value greater or equal to 1) is greater than 2.

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A384603 Primes preceded and followed by gaps whose quotient (value greater or equal to 1) is less than 2.

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A381850 Primes p preceded and followed by primes whose difference is less than 2*log(p).

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A383652 Primes p preceded and followed by gaps whose product is less than (log(p))^2.

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A381044 Primes prime(k) followed by a gap, prime(k+1)-prime(k), smaller than the local geometric average gap between consecutive primes: log(prime(k))/e^(gamma).

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A383215 Primes p preceded and followed by gaps whose difference (absolute value) is greater than log(p).

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A382052 Primes prime(k) such that klog(k)/prime(k) > (k-1)log(k-1)/prime(k-1).

A382051 Primes prime(k) such that klog(k)/prime(k) < (k-1)log(k-1)/prime(k-1).