A046869 -id:A046869 - cp's OEIS frontend

A095924 a(n) is the smallest prime of earliest set of at least n consecutive good primes version 1 (see A046869).

5, 37, 211, 251, 32467, 96377, 96377, 5647409, 12285587, 202924901, 3916407479, 108233238469, 1279155333257

Offset: 1

Enoch Haga and Farideh Firoozbakht, Jul 12 2004

			A good prime (version 1) is a prime p = prime(n) such that prime(n)^2 > prime(n-1)*prime(n+1), so 5 is a good prime because 5 = prime(3); prime(2) = 3; prime(4) = 7 and 5^2 > 3*7.
a(11) = 3916407479 because the 11 consecutive primes 3916407479, 3916407527, 3916407569, 3916407611, 3916407653, 3916407679, 3916407697, 3916407713, 3916407727, 3916407739 and 3916407751 are good primes and 3916407479 is the smallest prime with this property.

Carlos Rivera, Puzzle 273. Consecutive 'good' primes, The Prime Puzzles & Problem Connection.

Cf. A046869, A028388, A096473.

lista(pmax) = {my(c = 0, cmax = 0, p1 = 2, p2 = 3, p); forprime(p3 = 5, pmax, if(p2^2 > p1*p3, c++, if(c > cmax, p = p1; for(i = 1, c-1, p = precprime(p-1)); for(i = 1, c-cmax, print1(p, ", ")); cmax = c); c = 0); p1 = p2; p2 = p3);} \\ Amiram Eldar, Apr 29 2024

Name clarified and a(12)-a(13) added by Amiram Eldar, Apr 29 2024

A068829 Erroneous version of A046869.

Original entry on oeis.org

5, 17, 29, 37, 41, 53, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 157, 163, 173, 179

Offset: 1

dead

A006562 Balanced primes (of order one): primes which are the average of the previous prime and the following prime.

Original entry on oeis.org

5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903, 2963, 3307, 3313, 3637, 3733, 4013, 4409, 4457, 4597, 4657, 4691, 4993, 5107, 5113, 5303, 5387, 5393

Offset: 1

N. J. A. Sloane and Robert G. Wilson v

Subsequence of A075540. - Franklin T. Adams-Watters, Jan 11 2006
This subsequence of A125830 and of A162174 gives primes of level (1,1): More generally, the i-th prime p(i) is of level (1,k) if and only if it has level 1 in A117563 and 2 p(i) - p(i+1) = p(i-k). - Rémi Eismann, Feb 15 2007
Note the similarity between plots of A006562 and A013916. - Bill McEachen, Sep 07 2009
Balanced primes U strong primes = good primes. Or, A006562 U A051634 = A046869. - Juri-Stepan Gerasimov, Mar 01 2010
Primes prime(n) such that A001223(n-1) = A001223(n). - Irina Gerasimova, Jul 11 2013
Numbers m such that A346399(m) is odd and >= 3. - Ya-Ping Lu, Dec 26 2021 and May 07 2024
"Balanced" means that the next and preceding gap are of the same size, i.e., the second difference A036263 vanishes; so these are the primes whose indices are 1 more than indices of zeros in A036263, listed in A064113. - M. F. Hasler, Oct 15 2024
Primes which are the average of three consecutive primes. - Peter Schorn, Apr 30 2025

			5 belongs to the sequence because 5 = (3 + 7)/2. Likewise 53 = (47 + 59)/2.
5 belongs to the sequence because it is a term, but not first or last, of the AP of consecutive primes (3, 5, 7).
53 belongs to the sequence because it is a term, but not first or last, of the AP of consecutive primes (47, 53, 59).
257 and 263 belong to the sequence because they are terms, but not first or last, of the AP of consecutive primes (251, 257, 263, 269).

A. Murthy, Smarandache Notions Journal, Vol. 11 N. 1-2-3 Spring 2000.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 134.

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 870.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Shubhankar Paul, Ten Problems of Number Theory, International Journal of Engineering and Technical Research (IJETR), ISSN: 2321-0869, Volume-1, Issue-9, November 2013.
Shubhankar Paul, Legendre, Grimm, Balanced Prime, Prime triple, Polignac's conjecture, a problem and 17 tips with proof to solve problems on number theory, International Journal of Engineering and Technical Research (IJETR), ISSN: 2321-0869, Volume-1, Issue-10, December 2013.
Wikipedia, Balanced prime.

Primes A000040 whose indices are 1 more than A064113, indices of zeros in A036263 (second differences of the primes).
Cf. A082077, A082078, A082079, A096697, A096698, A096699, A096700, A096701, A096702, A096703, A096704, A096693, A051634, A051635, A054342, A117078, A117563, A125830, A117876, A125576, A046869, A173891, A173892, A173893, A006560, A075540.
Cf. A225494 (multiplicative closure); complement of A178943 with respect to A000040.
Cf. A055380, A051795, A081415, A096710 for other balanced prime sequences.

a006562 n = a006562_list !! (n-1)
a006562_list = filter ((== 1) . a010051) a075540_list
-- Reinhard Zumkeller, Jan 20 2012

a006562 n = a006562_list !! (n-1)
a006562_list = h a000040_list where
   h (p:qs@(q:r:ps)) = if 2 * q == (p + r) then q : h qs else h qs
-- Reinhard Zumkeller, May 09 2013

[a: n in [1..1000] | IsPrime(a) where a is NthPrime(n)-NthPrime(n+1)+NthPrime(n+2)]; // Vincenzo Librandi, Jun 23 2016

Transpose[ Select[ Partition[ Prime[ Range[1000]], 3, 1], #[[2]] ==(#[[1]] + #[[3]])/2 &]][[2]]
p=Prime[Range[1000]]; p[[Flatten[1+Position[Differences[p, 2], 0]]]]
Prime[#]&/@SequencePosition[Differences[Prime[Range[800]]],{x_,x_}][[All,2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 31 2019 *)

betwixtpr(n) = { local(c1,c2,x,y); for(x=2,n, c1=c2=0; for(y=prime(x-1)+1,prime(x)-1, if(!isprime(y),c1++); ); for(y=prime(x)+1,prime(x+1)-1, if(!isprime(y),c2++); ); if(c1==c2,print1(prime(x)",")) ) } \\ Cino Hilliard, Jan 25 2005

forprime(p=1,999, p-precprime(n-1)==nextprime(p+1)-p && print1(p",")) \\ M. F. Hasler, Jun 01 2013

is(n)=n-precprime(n-1)==nextprime(n+1)-n && isprime(n) \\ Charles R Greathouse IV, Apr 07 2016

from sympy import nextprime; p, q, r = 2, 3, 5
while q < 6000:
    if 2*q == p + r: print(q, end = ", ")
    p, q, r = q, r, nextprime(r) # Ya-Ping Lu, Dec 23 2021

2*p_n = p_(n-1) + p_(n+1).
Equals { p = prime(k) | A118534(k) = prime(k-1) }. - Rémi Eismann, Nov 30 2009
a(n) = A000040(A064113(n) + 1) = (A122535(n) + A181424(n)) / 2. - Reinhard Zumkeller, Jan 20 2012
a(n) = A122535(n) + A117217(n). - Zak Seidov, Feb 14 2013
Equals A145025 intersect A000040 = A145025 \ A024675. - M. F. Hasler, Jun 01 2013
Conjecture: Limit_{n->oo} n*(log(a(n)))^2 / a(n) = 1/2. - Alain Rocchelli, Mar 21 2024
Conjecture: The asymptotic limit of the average of a(n+1)-a(n) is equivalent to 2*(log(a(n)))^2. Otherwise formulated: 2 * Sum_{n=1..N} (log(a(n)))^2 ~ a(N). - Alain Rocchelli, Mar 23 2024

Reworded comment and added formula from R. Eismann. - M. F. Hasler, Nov 30 2009

Edited by Daniel Forgues, Jan 15 2011

A028388 Good primes (version 2): prime(n) such that prime(n)^2 > prime(n-i)*prime(n+i) for all 1 <= i <= n-1.

Original entry on oeis.org

5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307, 311, 331, 347, 419, 431, 541, 557, 563, 569, 587, 593, 599, 641, 727, 733, 739, 809, 821, 853, 929, 937, 967, 1009, 1031, 1087, 1151, 1213, 1277

Offset: 1

Eric W. Weisstein

nonn

Selfridge conjectured, and Pomerance proved, that there are infinitely many numbers in this sequence. Pomerance asks if the sequence has density 0. - Charles R Greathouse IV, Apr 14 2011

Guy, R. K. `Good' Primes and the Prime Number Graph. A14 in Unsolved Problems in Number Theory, 2nd ed. Springer-Verlag, pp. 32-33, 1994.

T. D. Noe, Table of n, a(n) for n = 1..10000
Carl Pomerance, The prime number graph, Mathematics of Computation 33:145 (1979), pp. 399-408.
Eric Weisstein's World of Mathematics, Good Prime
Eric Weisstein's World of Mathematics, Selfridge's Conjecture

Cf. A046869.

[NthPrime(n): n in [2..220] | forall{i: i in [1..n-1] | NthPrime(n)^2 gt NthPrime(n-i)*NthPrime(n+i)}]; // Bruno Berselli, Oct 23 2012

Module[{nn=300,prs},prs=Prime[Range[2nn]];qprQ[n_]:=Module[{pi= PrimePi[n]}, n^2>Max[Times@@@Thread[{Take[prs,pi-1],Reverse[Take[ prs,{pi+1,2 pi-1}]]}]]];Select[Take[prs,nn],qprQ]] (* Harvey P. Dale, May 13 2012 *)

is(n)=if(!isprime(n),return(0));my(p=n,q=n,n2=n^2);while(p>2, p=precprime(p-1); q=nextprime(q+1); if(n22 \\ Charles R Greathouse IV, Jul 02 2013

A046868 Numbers n such that prime(n)^2 > prime(n-1)*prime(n+1).

Original entry on oeis.org

3, 5, 7, 10, 12, 13, 16, 17, 19, 20, 22, 25, 26, 28, 31, 33, 35, 37, 38, 40, 41, 43, 45, 47, 48, 49, 52, 54, 55, 56, 57, 59, 60, 63, 64, 67, 69, 73, 74, 75, 78, 81, 83, 85, 88, 89, 92, 93, 95, 98, 100, 102, 103, 104, 107, 108, 109, 111, 112, 113, 115, 116, 119, 120, 122

Offset: 1

N. J. A. Sloane

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A046869.

Select[Range[2, 122], Prime[#]^2 > Prime[# - 1]*Prime[# + 1] &] (* Jayanta Basu, Jun 29 2013 *)

Extended by Don Reble, Nov 19 2006

A068828 Geometrically weak primes: primes that are smaller than the geometric mean of their neighbors (2 is included by convention).

Original entry on oeis.org

2, 3, 7, 13, 19, 23, 31, 43, 47, 61, 73, 83, 89, 103, 109, 113, 131, 139, 151, 167, 181, 193, 199, 229, 233, 241, 271, 283, 293, 313, 317, 337, 349, 353, 359, 383, 389, 401, 409, 421, 433, 443, 449, 463, 467, 491, 503, 509, 523, 547, 571, 577, 601, 619, 643

Offset: 1

Amarnath Murthy, Mar 08 2002

Or, bad primes (version 1): primes not in A046869. - Jonathan Vos Post, Aug 20 2007
The idea can be extended by defining a geometrically weak prime of order k to be a prime which is less than the geometric mean of r neighbors on both sides for all r = 1 to k and not true for r = k+1. A similar extension could be defined for the sequence A051635.
It is easy to show that, except for the twin prime pair (3,5), the larger prime of every twin prime pair is in this sequence. The smaller prime of the pair is always in A046869. - T. D. Noe, Feb 19 2008

			23 belongs to this sequence as 23^2 = 529 < 19*29 = 551.

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

Cf. A051634, A051635, A006562, A000040, A046869.

Join[{2}, Prime[Select[Range[2, 120], Prime[ # ]^2 <= Prime[ # - 1]*Prime[ # + 1]&]]] (* Stefan Steinerberger, Aug 21 2007 *)
Join[{2},Transpose[Select[Partition[Prime[Range[500]],3,1],#[[2]]< GeometricMean[ {#[[1]],#[[3]]}]&]][[2]]] (* Harvey P. Dale, Apr 05 2014 *)

prime(k)^2 <= prime(k-1)*prime(k+1).

Corrected and extended by Stefan Steinerberger, Aug 21 2007

Edited by N. J. A. Sloane, Feb 19 2008

A130903 Bad primes (version 2). Primes not in A028388.

Original entry on oeis.org

2, 3, 7, 13, 19, 23, 31, 43, 47, 61, 73, 79, 83, 89, 103, 107, 109, 113, 131, 137, 139, 151, 157, 163, 167, 173, 181, 193, 197, 199, 211, 229, 233, 239, 241, 263, 271, 277, 281, 283, 293, 313, 317, 337, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 421

Offset: 1

Jonathan Vos Post, Aug 22 2007

Cf. A000040, A028388, A046869, A068828.

isA028388 := proc(p) local a,n,pmin,pplu; if isprime(p) then a := true ; if p < 5 then RETURN(false) ; fi ; pmin := p ; pplu := p ; while pmin > 2 do pmin := prevprime(pmin) ; pplu := nextprime(pplu) ; if p^2 <= pmin*pplu then a := false ; break ; fi ; od: RETURN(a) ; else RETURN(false) ; fi ; end: isA130903 := proc(p) if not isprime(p) then RETURN(false) ; else RETURN(not isA028388(p)) ; fi ; end: for i from 1 to 100 do p := ithprime(i) ; if isA130903(p) then printf("%d,",p) ; fi ; od: # R. J. Mathar, Sep 02 2007

{p in A000040 and p not in A028388}.

Corrected and extended by R. J. Mathar, Sep 02 2007

A349793 Primes which are the nearest integer to the harmonic mean of the previous prime and the following prime.

Original entry on oeis.org

3, 7, 13, 23, 47, 89, 157, 173, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903, 2963, 3307, 3313, 3637, 3733, 4013, 4409, 4457, 4597, 4657, 4691, 4993, 5107, 5113, 5303, 5387, 5393

Offset: 1

Hugo Pfoertner, Nov 30 2021

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A006562, A046869, A051635, A075540, A216124.

Select[Partition[Prime[Range[750]],3,1],Round[HarmonicMean[{#[[1]],#[[3]]}]]==#[[2]]&][[;;,2]] (* Harvey P. Dale, Dec 29 2024 *)

a349793(limit) = {my(p1=2,p2=3); forprime(p3=5, limit, my(hm=round((2*p1*p3)/(p1+p3))); if(p2==hm, print1(p2,", ")); p1=p2;p2=p3)};
a349793(5500)

A095957 a(n) is the smallest prime of earlier set of n consecutive good primes version 2 (see A028388).

Original entry on oeis.org

5, 37, 557, 1847, 216703, 6929381, 134193727, 15118087477

Offset: 1

Carlos Rivera and Farideh Firoozbakht, Jul 14 2004

Puzzle 273 of The Prime Puzzles & Problem Connection (www.primepuzzles.net).

Carlos Rivera, Consecutive 'good' primes

Cf. A028388, A095924, A046869, A096473.

a(7) found by Jim Fougeron (jfoug(AT)kdsi.net), Aug 25 2004

A139312 Characteristic function of the good primes (version 1).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0

Offset: 2

Roger L. Bagula, Jun 07 2008

a(n)=1 if prime(n)^2 - prime(n-1)*prime(n+1) >=0, else a(n)=0.

Cf. A056221, A046869, A068828.

f[n_] := If[ Prime[n]^2 - Prime[n - 1]*Prime[n + 1] > 0, 1, 0]; Array[f, 105, 2] (*alternative formula: derived*) Solve[x^2 - (x - a)*(x + b) == 0, x]; a = -Prime[n - 1] + Prime[n]; b = -Prime[n] + Prime[n + 1]; f[n_] = If[-Prime[-1 + n] + 2 Prime[n] - Prime[1 + n] == 0, 0, a*b/(b - a)]; Table[ If[ f[n] > 0, 0, 1], {n, 2, 106}]
If[#[[2]]^2-(#[[1]]#[[3]])>=0,1,0]&/@Partition[Prime[Range[110]],3,1] (* Harvey P. Dale, Jan 25 2015 *)

a(n)=my(p=prime(n));p^2>=precprime(p-1)*nextprime(p+1) \\ Charles R Greathouse IV, Jun 24 2011

a(n) = 1 if A056221(n-1)<=0, else a(n)=0.

All entries corrected. - R. J. Mathar, Charles R Greathouse IV Robert G. Wilson v, Jun 16 2011

cp's OEIS Frontend

A095924 a(n) is the smallest prime of earliest set of at least n consecutive good primes version 1 (see A046869).

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A068829 Erroneous version of A046869.

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A006562 Balanced primes (of order one): primes which are the average of the previous prime and the following prime.

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A028388 Good primes (version 2): prime(n) such that prime(n)^2 > prime(n-i)*prime(n+i) for all 1 <= i <= n-1.

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A046868 Numbers n such that prime(n)^2 > prime(n-1)*prime(n+1).

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A068828 Geometrically weak primes: primes that are smaller than the geometric mean of their neighbors (2 is included by convention).

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A130903 Bad primes (version 2). Primes not in A028388.

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A349793 Primes which are the nearest integer to the harmonic mean of the previous prime and the following prime.

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