A051634 -id:A051634 - cp's OEIS frontend

A109270 Numbers k such that k^2 > (1/2)*(prevprime(k^2) + nextprime(k^2)).

4, 6, 10, 11, 14, 16, 17, 20, 22, 24, 26, 28, 30, 31, 36, 38, 39, 40, 45, 48, 50, 52, 54, 56, 57, 59, 62, 65, 66, 67, 70, 73, 74, 76, 79, 81, 84, 85, 87, 90, 91, 94, 95, 96, 97, 99, 100, 104, 105, 106, 109, 110, 111, 114, 115, 116, 120, 122, 123, 124, 125, 126, 130, 134

Offset: 1

Zak Seidov, Jun 24 2005

One may call these k^2 the "strong squares" by analogy with A051634 (strong primes).

			4^2=16>(13+17)/2 so 4 is a term;
5^2 < (23+29)/2=26, so 5 is not a term;
6^2=36>(31+37)/2 so 6 is a term, etc.

Cf. A051634, A033597, A075190, A109269.

Cf. A024675.

a:=proc(n) if n^2 > (1/2)*(prevprime(n^2)+nextprime(n^2)) then n else fi end: seq(a(n),n=2..150); # Emeric Deutsch, Jun 26 2005

prQ[n_]:=Module[{n2=n^2},n2>(NextPrime[n2]+NextPrime[n2,-1])/2]; Select[ Range[2,150],prQ] (* Harvey P. Dale, Feb 19 2012 *)

More terms from Emeric Deutsch, Jun 26 2005

A159686 Sum of strong primes < 10^n.

Original entry on oeis.org

0, 508, 33551, 2751328, 216056493, 18084221125, 1548424793743, 135655041210402, 12054551765023934, 1084635554912125542, 98583402030663969332, 9035771475185456034956

Offset: 1

Cino Hilliard, Apr 19 2009

Given 3 consecutive primes p1, p2, and p3, p2 is a strong prime if p2 > (p1+p2)/2.
Or, primes that are greater than the arithmetic mean of their immediate surrounding primes.
The number of strong primes < n ~ sum of strong primes < sqrt(n). For number of strong primes < 10^11 = 2014200162 and the sum of strong primes < 10^5.5 = 1972716560, for an error of 0.0206

			The strong primes < 10^2 are 11, 17, 29, 37, 41, 59, 67, 71, 79, 97. These add up to 508 which is the second term in the sequence.

Cino Hilliard, Sum of Strong Primes. [broken link]

Cf. A051634, A159687.

lista(pmax) = {my(s = 0, pow = 10, p1 = 2, p2 = 3); forprime(p3 = 5, pmax, if(p2 > pow,print1(s, ", "); pow *= 10); if(2*p2 > p1+p3, s += p2); p1 = p2; p2 = p3);} \\ Amiram Eldar, Jul 02 2024

Edited by N. J. A. Sloane, Apr 20 2009

a(11)-a(12) from Amiram Eldar, Jul 02 2024

A159687 Number of strong primes < 10^n.

Original entry on oeis.org

0, 10, 73, 574, 4543, 37723, 320991, 2796946, 24758534, 222126290, 2014200162, 18425778658

Offset: 1

Cino Hilliard, Apr 19 2009

Given 3 consecutive primes p1, p2, and p3, p2 is a strong prime if p2 > (p1+p2)/2.
Or, primes that are greater than the arithmetic mean of their immediate surrounding primes.
The number of strong primes < n ~ sum of strong primes < sqrt(n). The number of strong primes < 10^11 = 2014200162 and the sum of strong primes < 10^5.5 = 1972716560, for an error of 0.0206.

			a(2) = 10 because there are 10 strong primes < 10^2: 11, 17, 29, 37, 41, 59, 67, 71, 79, and 97.

Cino Hilliard, Sum of Strong Primes. [broken link]

Cf. A051634, A159686.

See the link for Gcc programs that count and sum these primes.

lista(pmax) = {my(c = 0, pow = 10, p1 = 2, p2 = 3); forprime(p3 = 5, pmax, if(p2 > pow,print1(c, ", "); pow *= 10); if(2*p2 > p1+p3, c++); p1 = p2; p2 = p3);} \\ Amiram Eldar, Jul 02 2024

Edited by N. J. A. Sloane, Apr 20 2009
a(11) corrected by Bill McEachen, Oct 18 2023
a(12) from Amiram Eldar, Jul 02 2024

A175145 Primes associated with A175102.

Original entry on oeis.org

281, 311, 495511, 495557, 496187, 496229, 496259, 496303, 496333, 496343, 496399, 496439, 496459, 496499, 496549, 496583, 496631, 496763, 497153, 497177, 497239, 497261, 497279, 497291, 497297, 497303, 497411, 497417, 497449, 497461, 497479, 498073, 498119, 498181, 498227, 498259, 498331, 498361, 498391, 498409, 498803, 498881, 507607

Offset: 1

G. L. Honaker, Jr., Dec 02 2010

nonn

Prime Curios for 281.

Cf. A051634, A051635, A175102.

my(q=3, r=2, s=0); forprime(p=5,default(primelimit),(s+=sign(r+0-2*(r=q)+q=p))||print1(r, ", "))

More terms from Chris K. Caldwell

A362017 a(n) is the leading prime in the n-th prime sublist defined in A348168.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 23, 29, 37, 53, 59, 67, 79, 89, 97, 127, 137, 149, 157, 163, 173, 179, 191, 197, 211, 223, 239, 251, 293, 307, 331, 347, 353, 359, 367, 397, 409, 419, 431, 439, 449, 457, 479, 521, 541, 557, 587, 631, 673, 683, 691, 701, 719, 787, 809, 821

Offset: 1

Ya-Ping Lu, Apr 04 2023

nonn

If Conjecture 2 in A348168 is true, lim_{n->infinity} a(n)/prime(round((n-1)*e)+1) = 1, where e is Euler's number.
If a term p (>2) is from a single-prime sublist (A356271), then p is a weak prime (A051635) or a balanced prime (A006562). Otherwise, p is a strong prime (A051634).
The definition divides the primes into maximal sublists such that gaps between adjacent primes in a sublist are smaller than the gap that precedes the sublist and no larger than the first gap within the sublist. - Peter Munn, Jul 07 2025

			According to the definition in A348168, prime numbers are divided into sublists, L_1, L_2, L_3,..., in which L_n = [p(n,1), p(n,2), ..., p(n,m(n))], where p(n,k) is the k-th prime and m(n) the number of primes in the n-th sublist L_n. Thus, a(n) = p(n,1). The first sublist L_1 = [2]. If p(n,1) <= (prevprime(p(n,1)) + nextprime(p(n,1)))/2, then L_n has only 1 prime, p(n,1). Otherwise, m(n) is the largest integer such that g(n,1) >= g(n,i), where g(n,i) = p(n,i+1) - p(n,i) and 2 <= i <= m(n).
The first 32 primes, for example, are divided into 16 prime sublists:
  [2],
  [3],
  [5],
  [7],
  [11,13],
  [17,19],
  [23],
  [29,31],
  [37,41,43,47],
  [53],
  [59,61],
  [67,71,73],
  [79,83],
  [89],
  [97,101,103,107,109,113],
  [127,131].
The leading primes in these sublists are: 2, 3, 5, 7, 11, 17, 23, 29, 37, 53, 59, 67, 79, 89, 97, 127. Therefore, a(1) = 2, a(2) = 3, ..., and a(16) = 127.

Cf. A006562, A051634, A051635, A348168, A356271.

from sympy import nextprime; R = [2]; L = [2]
for n in range(2, 57):
    p0 = L[-1]; p1 = nextprime(p0); M = [p1]; g0 = p1-p0; p = nextprime(p1); g1 = p-p1
    while g1 < g0 and p-p1 <= g1: M.append(p); p1 = p; p = nextprime(p)
    L = M; R.append(L[0])
print(*R, sep =', ')

A364778 Products of two distinct strong primes.

Original entry on oeis.org

121, 187, 289, 319, 407, 451, 493, 629, 649, 697, 737, 781, 841, 869, 1003, 1067, 1073, 1111, 1139, 1177, 1189, 1207, 1343, 1369, 1397, 1507, 1517, 1639, 1649, 1681, 1711, 1717, 1793, 1819, 1943, 1969, 2059, 2101, 2159, 2167, 2183, 2291, 2329, 2419, 2453, 2479, 2497, 2533, 2627, 2629, 2747

Offset: 1

Massimo Kofler, Aug 07 2023

nonn

Strong primes: prime(n) > (prime(n-1) + prime(n+1))/2.

			121 = 11^2 and 11 > (7+13)/2.
187 = 11*17 and 11 > (7+13)/2, 17 > (13+19)/2.
493 = 17*29 and 17 > (13+19)/2, 29 > (23+31)/2.

Cf. A001358, A051634, A363167, A363782.

strongQ[p_] := p > 2 && 2*p > Total[NextPrime[p, {-1, 1}]]; Select[Range[1, 3000, 2], MemberQ[{{1, 1}, {2}}, (f = FactorInteger[#])[[;; , 2]]] && AllTrue[f[[;; , 1]], strongQ] &] (* Amiram Eldar, Aug 07 2023 *)

Definition clarified by N. J. A. Sloane, Oct 08 2023

A054816 Fourth term of strong prime sextets: p(m-2)-p(m-3) > p(m-1)-p(m-2) > p(m)-p(m-1) > p(m+1)-p(m) > p(m+2)-p(m+1).

Original entry on oeis.org

1867, 2531, 3457, 9613, 21487, 23011, 25237, 26107, 32183, 33403, 33931, 39097, 40813, 41227, 44263, 47287, 48817, 55897, 57787, 67033, 70117, 74287, 74707, 74713, 75149, 75161, 82981, 84313, 87869, 88411, 88657, 103801, 103903

Offset: 0

Henry Bottomley, Apr 10 2000

nonn

Cf. A051634, A054800-A054840.

A054817 Fifth term of strong prime sextets: p(m-3)-p(m-4) > p(m-2)-p(m-3) > p(m-1)-p(m-2) > p(m)-p(m-1) > p(m+1)-p(m).

Original entry on oeis.org

1871, 2539, 3461, 9619, 21491, 23017, 25243, 26111, 32189, 33409, 33937, 39103, 40819, 41231, 44267, 47293, 48821, 55901, 57791, 67043, 70121, 74293, 74713, 74717, 75161, 75167, 82997, 84317, 87877, 88423, 88661, 103811, 103913

Offset: 0

Henry Bottomley, Apr 10 2000

nonn

Cf. A051634, A054800-A054840.

A155189 Square-weak primes.

Original entry on oeis.org

3, 5, 7, 13, 19, 23, 31, 43, 47, 53, 61, 73, 83, 89, 103, 109, 113, 131, 139, 151, 157, 167, 173, 181, 193, 199, 211, 229, 233, 241, 257, 263, 271, 283, 293, 313, 317, 337, 349, 353, 359, 373, 383, 389, 401, 409, 421, 433, 443, 449, 463, 467, 491, 503, 509, 523

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

5^2 = 25 < 29 = (3^2+7^2)/2, ...

Cf. A045536, A005384, A051634, A006562, A051635, A155188

lst={};Do[p0=Prime[n];p1=Prime[n+1];p2=Prime[n+2];If[p1^2<(p0^2+p2^2)/2,AppendTo[lst,p1]],{n,5!}];lst
Select[Partition[Prime[Range[100]],3,1],#[[2]]^2<(#[[1]]^2+#[[3]]^2)/2&][[All,2]] (* Harvey P. Dale, May 01 2021 *)

A235874 First term of the earliest sequence of n consecutive strong primes.

Original entry on oeis.org

11, 37, 1657, 1847, 74687, 322193, 5051341, 11938853, 245333213, 397597169, 130272314657, 1273135176871

Offset: 1

Giovanni Resta, Jan 16 2014

nonn

A strong prime is a prime p(n) such that p(n) > (p(n-1) + p(n+1))/2.

			a(2) = 37 because the two consecutive primes 37 and 41 are both strong and are the first such pair.

Cf. A051634, A229832.

cp's OEIS Frontend

A109270 Numbers k such that k^2 > (1/2)*(prevprime(k^2) + nextprime(k^2)).

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A159686 Sum of strong primes < 10^n.

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A159687 Number of strong primes < 10^n.

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PARI

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A175145 Primes associated with A175102.

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A362017 a(n) is the leading prime in the n-th prime sublist defined in A348168.

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Python

A364778 Products of two distinct strong primes.

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Mathematica

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A054816 Fourth term of strong prime sextets: p(m-2)-p(m-3) > p(m-1)-p(m-2) > p(m)-p(m-1) > p(m+1)-p(m) > p(m+2)-p(m+1).

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A054817 Fifth term of strong prime sextets: p(m-3)-p(m-4) > p(m-2)-p(m-3) > p(m-1)-p(m-2) > p(m)-p(m-1) > p(m+1)-p(m).

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A155189 Square-weak primes.

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