A217566 -id:A217566 - cp's OEIS frontend

A195871 a(n) is the smallest prime(m) such that the interval (prime(m)n, prime(m+1)n) contains exactly one prime.

2, 2, 2, 17, 59, 29, 239, 227, 107, 149, 347, 191, 569, 461, 269, 659, 311, 1277, 2711, 821, 1427, 2711, 3581, 1019, 1451, 1319, 9281, 4931, 6269, 5849, 11549, 35729, 8537, 5441, 5741, 10007, 29759

Offset: 2

Vladimir Shevelev, Jan 02 2013

nonn

Conjecture: In the supposition that there are infinitely many twin primes, for n>=5 all terms are in A001359 (lesser of twin primes).
Note that a unique prime which is contained in an interval of the form (prime(m)*n, prime(m+1)*n) is called n-isolated (see author's link, where a heuristic proof is given that the number of n-isolated primes<=x approaches e^{-2(n-1)}x/log(x) as x goes to infinity (cf. Conjecture 25, Remark 26 and formula (47)). One can easily prove that a(n) is not bounded.
This conjecture seems hard, since it's not obvious how to find an upper bound for a(n) (see Conjecture 42 in the Shevelev link). - Charles R Greathouse IV, Jan 02 2013

			Let n=5, and consider intervals of the form (5*prime(m), 5*prime(m+1)).
For 2, 3, 5, ..., the intervals (10,15), (15,25), (25,35), (35,55), (55,65), (65,85), (85,95)... contain 2, 3, 2, 5, 2, 4, 1,... primes. Hence the smallest such prime is 17.

Charles R Greathouse IV, Table of n, a(n) for n = 2..100
V. Shevelev, Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes, Journal of Integer Sequences, Vol. 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, A217561, A217566, A217577.

a(n)=my(p=2,t);forprime(q=3,,t=0;for(i=p*n+1,q*n-1,if(isprime(i)&&t++>1,break));if(t==1,return(p));p=q) \\ Charles R Greathouse IV, Jan 02 2013

A217577 The only prime p such that ka < p < kb where a, b are consecutive primes, case k=5.

Original entry on oeis.org

89, 211, 359, 509, 541, 691, 751, 991, 1201, 1399, 1409, 1559, 1741, 2099, 2161, 2179, 2333, 2503, 2609, 2851, 3089, 3209, 3271, 4111, 4139, 4289, 4297, 4409, 5309, 5591, 6151, 6389, 6397, 6491, 6599, 7283, 7411, 7993, 8039, 8101, 8467, 8609, 8941, 9391, 9661

Offset: 1

Zak Seidov, Oct 07 2012

nonn

Corresponding values of b-a: 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 6, 4, 4, 2, 2, 2, 2, 6, 2, 2, 2, 4, 2, 2, 6, 2, 2, 4, 4, 2, 6. In most cases b-a = 2. Smallest n for which b-a = 2(2)26: 1, 10, 16, 62, 119, 414, 939, 2565, 1349, 1042, 10470, 22211, 23553. Also, at n = 43461, b-a = 32.

			89 is the only prime in the interval [5*17, 5*19] = [85,95],
211 is the only prime in the interval [5*41, 5*43] = [205,215],
359 is the only prime in the interval [5*71, 5*73] = [355,365].

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A166251 (k=2), A217561 (k=3), A217566 (k=4).

a = 2; b = 3; s = {}; k = 5; Do[If[(p = NextPrime[k*a]) < k*b && NextPrime[p] > k*b, AppendTo[s, p]]; a = b; b = NextPrime[b], {100}]; s

A217603 Consider sets of 3 consecutive primes a, b, c such that c - a = 100, then sequence gives the values of b.

Original entry on oeis.org

58831, 286927, 360653, 404941, 590489, 623107, 651587, 673747, 710119, 740801, 779413, 794831, 795427, 1040311, 1107269, 1185241, 1206869, 1320437, 1392007, 1568771, 1581829, 1599803, 1601953, 1613201, 1721081, 1744927, 1942273, 1951321, 1994299, 2024063

Offset: 1

Zak Seidov, Oct 08 2012

nonn

			a(1) = 58831 because {58789, 58831, 58889} is the first set of 3 consecutive primes a, b, c with c-a=100.
a(2) = 286927 because {286873, 286927, 286973} is the second set of 3 consecutive primes a, b, c with c-a=100.
a(1000) = 23090087 because {23090059, 23090087, 23090159} is the 1000th set of 3 consecutive primes a, b, c with c-a=100.

Zak Seidov, Table of n, a, b, c for n=1..1000
Zak Seidov, Table of n, a, b, c for n=1..1000

Cf. A166251, A217561, A217566, A217577.

s = {}; a = 2; b = 3; c = 5; Do[If[c - a == 100, AppendTo[s, b]; Print[{a, b, c}]]; a = b; b = c; c = NextPrime[c], {10^5}]; s
Select[Partition[Prime[Range[151000]],3,1],#[[3]]-#[[1]]==100&][[;;,2]] (* Harvey P. Dale, Jul 22 2024 *)

p=2;q=3;forprime(r=5,1e6,if(r-p==100,print1(q", "));p=q;q=r) \\ Charles R Greathouse IV, Nov 14 2012

A218275 a(n) is the smallest n-isolated prime, or a(n)=0 if there are no n-isolated primes.

Original entry on oeis.org

5, 7, 11, 89, 359, 211, 1913, 2053, 1087, 1657, 4177, 2503, 7993, 6917, 4327, 11213, 5623, 24281, 54251, 17257, 31397, 62383, 85991, 25523, 37747, 35617, 259907, 143053, 188107, 181361, 369581, 1179109, 290317, 190471, 206699, 370261, 1130863, 162143

Offset: 2

Vladimir Shevelev and Zak Seidov, Oct 25 2012

nonn

For a given n>=2, a prime p such that there is no other prime in the interval [n*prevprime(p/n), n*nextprime(p/n)] is called n-isolated.

Conjectures. 1) a(n) > 0; 2) a(n)/n is between 2 and 3 or between the smaller and larger member of a twin prime pair.

			a(5) = 89 because there are no primes except 89 in the interval [5*prevprime(89/5), 5*nextprime(89/5)] = [5*17, 5*19] = [85, 95].  And 89 is the smallest such prime - for example, if q = 37 < 89, then the interval [5*nextprime(q/5), 5*nextprime(q/5)] = [5*7,5*11] = [35,55] contains 4 primes other than 41, namely 37, 43, 47, and 53. - _Vladimir Shevelev_, Nov 04 2012.

Zak Seidov, Table of n, a(n) for n = 2..200
V. Shevelev, Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes, Journal of Integer Sequences, Vol. 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, A217561, A217566, A217577.

s = {}; Do[a = 2; b = 3; While[(p = NextPrime[k*a]) != NextPrime[k*b, -1], a = b; b = NextPrime[b]]; AppendTo[s, p], {k, 2, 40}]; s (* Zak Seidov, Nov 04 2012 *)

nextprime(a(n)/n) < nextprime(a(n))/n. For n>=5 and every prime q from the interval (3*n, a(n)), the interval (n*prevprime(q/n), n*nextprime(q/n)) contains a prime greater than q. - Vladimir Shevelev, Nov 04 2012

A218279 Let (p(n), p(n)+2) be the n-th twin prime pair. a(n) is the smallest k, such that there is only one prime in the interval (kp(n), k(p(n)+2)), or a(n)=0, if there is no such k.

Original entry on oeis.org

2, 4, 2, 2, 3, 2, 6, 5, 3, 5, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 5, 2, 2, 4, 3, 3, 2, 2, 2, 3, 6, 3, 2, 4, 3, 2, 2, 2, 3, 3, 2, 2, 2, 3, 5, 2, 2, 2, 3, 2, 3, 3, 6, 3, 4, 9, 5, 2, 5, 4, 2, 3, 2, 3, 3, 2, 4, 3, 2, 2, 5, 3, 4, 4, 4, 4, 3, 2, 6, 2, 7, 4, 2, 6, 4, 2

Offset: 1

Vladimir Shevelev, Oct 25 2012

nonn

Conjecture: a(n)>0 for all n.

			The first pair of twin primes is (3,5). For k=1 and 2, we have the intervals (3,5) and (6,10), such that not the first but the second interval contains exactly one prime(7). Thus a(1)=2. For n=2 and k=1 to 4, we have the intervals (5,7),(10,14),(15,21), and (20,28) and only the last interval contains exactly one prime(23). Thus, a(2)=4.

Zak Seidov, Table of n, a(n) for n = 1..10000
V. Shevelev, Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes, Journal of Integer Sequences, Vol. 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. A218275, A166251, A217561, A217566, A217577, A001359, A014574, A006512, A077800.

a(6) corrected and terms beyond a(11) contributed by Zak Seidov, Oct 25 2012

A217655 Consider sets of 3 consecutive primes a

Original entry on oeis.org

8917523, 17051707, 24662467, 25173593, 27001199, 37757639, 38089453, 42629173, 44310817, 67142137, 67816601, 76317653, 80517691, 82772143, 82843753, 83929789, 84980743, 90007363, 94905269, 99099773, 99524219, 117813307, 119401577, 129163457, 147426791, 162159887

Offset: 1

Zak Seidov, Oct 09 2012

nonn

			a(1) = 8917523 because {8917463, 8917523, 8917663} is the first set of 3 consecutive primes a, b, c with c-a = 200.

Zak Seidov, Table of n, a(n) for n = 1..1000
Zak Seidov, Table of n, a, b, c for n=1..1000

Cf. A166251, A217561, A217566, A217577, A217603.

a :=2 :
b := nextprime(a) :
c := nextprime(b) :
for i from 1 do
    if c-a= 200 then
        print(b);
    end if;
    a := b ;
    b := c ;
    c := nextprime(b) ;
end do:  # R. J. Mathar, Nov 07 2012

Select[Partition[Prime[Range[91*10^5]],3,1],#[[3]]-#[[1]]==200&][[;;,2]] (* Harvey P. Dale, Jul 17 2025 *)

A218333 The index of the smallest n-isolated prime p such that p/n is not between 2 and 3 and not between the smaller and greater primes of a twin prime pair, or 0 if no such p exists.

Original entry on oeis.org

5, 5, 8, 10, 2, 12, 7, 4, 37, 23, 5, 51, 3, 6, 34, 23, 5, 57, 9, 22, 49, 66, 64, 54, 5, 56, 43, 28, 46, 116, 56, 232, 92, 170, 65, 206, 181, 379, 170, 511, 190, 416, 187, 448, 89, 143, 200, 159, 434, 670, 145, 1081, 213, 1011, 680, 77

Offset: 2

Vladimir Shevelev, Oct 26 2012

nonn

For n>=2, a prime p is called n-isolated (cf. A166251 and the Shevelev link, Section 10) if there is no other prime in the interval (n*prevprime(p/n), n*nextprime(p/n)).
In particular, if a(n)=1, then the smallest n-isolated prime divided by n is not between 2 and 3 and not between the smaller and greater primes of a twin prime pair.
Suppose that for every n there exist infinitely many n-isolated primes.  Then if there exists n_0 such that a(n_0)=0, there are infinitely many twin primes. On the other hand, one can prove that the smallest n-isolated prime divided by n tends to infinity as n goes to infinity. Therefore, if there is not an N such that, for all n >= N, a(n)=1, then we also conclude that there are infinitely many twin primes.
Conjecture: all a(n) >= 2.

			Let n=2. The 2-isolated primes are in A166251.
The first 2-isolated prime is 5, and 5/2 is between 2 and 3.
The second 2-isolated prime is 7, and 7/2 is between 3 and 5.
The third 2-isolated prime is 23, and 23/2 is between 11 and 13.
The fourth 2-isolated prime is 37, and 37/2 is between 17 and 19.
The fifth 2-isolated prime is 79, and 79/2 is between 37 and 41.  Since (37,41) is not (2,3) and is not a twin prime pair, a(2)=5

V. Shevelev, Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes, Journal of Integer Sequences, Vol. 15 (2012), Article 12.5.4

Cf. A166251, A217561, A217566.

isoki(p, n) = (p==nextprime(n*precprime(p\n))) && (p==precprime(n*nextprime(p/n))); \\ A166251
nextp(p, n) = while(! isoki(p, n), p = nextprime(p+1)); p;
isokp(p, n) = {my(diff = nextprime(p/n) - precprime(p/n)); if ((diff == 1) || (diff == 2), return (0)); return (1);}
a(n) = {my(p = nextp(2, n), nb = 1); while (! isokp(p, n), p = nextp(nextprime(p+1), n); nb++;); nb;} \\ Michel Marcus, Dec 16 2018

a(6)-a(38) were calculated by Zak Seidov, Oct 28 2012

More terms from Michel Marcus, Dec 16 2018

cp's OEIS Frontend

A195871 a(n) is the smallest prime(m) such that the interval (prime(m)*n, prime(m+1)*n) contains exactly one prime.

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A217577 The only prime p such that k*a < p < k*b where a, b are consecutive primes, case k=5.

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A217603 Consider sets of 3 consecutive primes a, b, c such that c - a = 100, then sequence gives the values of b.

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A218275 a(n) is the smallest n-isolated prime, or a(n)=0 if there are no n-isolated primes.

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A218279 Let (p(n), p(n)+2) be the n-th twin prime pair. a(n) is the smallest k, such that there is only one prime in the interval (k*p(n), k*(p(n)+2)), or a(n)=0, if there is no such k.

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A217655 Consider sets of 3 consecutive primes a

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A218333 The index of the smallest n-isolated prime p such that p/n is not between 2 and 3 and not between the smaller and greater primes of a twin prime pair, or 0 if no such p exists.

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A195871 a(n) is the smallest prime(m) such that the interval (prime(m)n, prime(m+1)n) contains exactly one prime.

A217577 The only prime p such that ka < p < kb where a, b are consecutive primes, case k=5.

A218279 Let (p(n), p(n)+2) be the n-th twin prime pair. a(n) is the smallest k, such that there is only one prime in the interval (kp(n), k(p(n)+2)), or a(n)=0, if there is no such k.