A028388 -id:A028388 - cp's OEIS frontend

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

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

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

A172166 Partial sums of A028388 good primes (version 2).

Original entry on oeis.org

5, 16, 33, 62, 99, 140, 193, 252, 319, 390, 487, 588, 715, 864, 1043, 1234, 1457, 1684, 1935, 2192, 2461, 2768, 3079, 3410, 3757, 4176, 4607, 5148, 5705, 6268, 6837, 7424, 8017, 8616, 9257, 9984, 10717, 11456, 12265, 13086, 13939, 14868, 15805, 16772

Offset: 1

Jonathan Vos Post, Jan 27 2010

Prime partial sums of good primes begin a(1) = 5, a(7) = 193, a(11) = 487, a(23) = 3079. Good prime partial sums of good primes begin a(1) = 5, and what are the next of that subsequence?

			a(20) = 5 + 11 + 17 + 29 + 37 + 41 + 53 + 59 + 67 + 71 + 97 + 101 + 127 + 149 + 179 + 191 + 223 + 227 + 251 + 257 = 2192.

Cf. A000040, A028388.

a(n) = SIM[i=1..n] {p_n such that (p_n)^2 > p_{n-i}p_{n+i} for all 1 <= i <= n-1}.

More terms from R. J. Mathar, Jan 30 2010

A046869 Good primes (version 1): prime(n)^2 > prime(n-1)*prime(n+1).

Original entry on oeis.org

5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 157, 163, 173, 179, 191, 197, 211, 223, 227, 239, 251, 257, 263, 269, 277, 281, 307, 311, 331, 347, 367, 373, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499, 521, 541

Offset: 1

N. J. A. Sloane

nonn

Also called geometrically strong primes. - Amarnath Murthy, Mar 08 2002
The idea can be extended by defining a geometrically strong prime of order k to be a prime that is greater than the geometric mean of r neighbors on both sides for all r = 1 to k but not for r = k+1. Similar generalizations can be applied to the sequence A051634. - Amarnath Murthy, Mar 08 2002
It appears that a(n) ~ 2*prime(n). - Thomas Ordowski, Jul 25 2012
Conjecture: primes p(n) such that 2*p(n) >= p(n-1) + p(n+1). - Thomas Ordowski, Jul 25 2012
Probably {3,7,23} U {good primes} = {primes p(n) > 2/(1/p(n-1) + 1/p(n+1))}. - Thomas Ordowski, Jul 27 2012
Except for A001359(1), A001359 is a subsequence. - Chai Wah Wu, Sep 10 2019

			37 is a member as 37^2 = 1369 > 31*41 = 1271.

R. K. Guy, Unsolved Problems in Number Theory, Section A14.

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

Cf. A001359, A006562, A028388, A046868, A051634, A051635, A068828.

[NthPrime(n): n in [2..100] | NthPrime(n)^2 gt NthPrime(n-1)*NthPrime(n+1)]; // Bruno Berselli, Oct 23 2012

with(numtheory): a := [ ]: P := [ ]: M := 300: for i from 2 to M do if p(i)^2>p(i-1)*p(i+1) then a := [ op(a),i ]; P := [ op(P),p(i) ]; fi; od: a; P;

Do[ If[ Prime[n]^2 > Prime[n - 1]*Prime[n + 1], Print[ Prime[n] ] ], {n, 2, 100} ]
Transpose[Select[Partition[Prime[Range[300]],3,1],#[[2]]^2>#[[1]]#[[3]]&]][[2]] (* Harvey P. Dale, May 13 2012 *)
Select[Prime[Range[2, 100]], #^2 > NextPrime[#]*NextPrime[#, -1] &] (* Jayanta Basu, Jun 29 2013 *)

forprime(n=o=p=3,999,o+0<(o=p)^2/(p=n) & print1(o", "))
isA046869(p)={ isprime(p) & p^2>precprime(p-1)*nextprime(p+1) } \\ M. F. Hasler, Jun 15 2011

Corrected and extended by Robert G. Wilson v, Dec 06 2000

Edited by N. J. A. Sloane at the suggestion of Giovanni Resta, Aug 20 2007

A197297 The Riemann primes of the theta type and index 1.

Original entry on oeis.org

2, 5, 7, 11, 17, 29, 37, 41, 53, 59, 97, 127, 137, 149, 191, 223, 307, 331, 337, 347, 419, 541, 557, 809, 967, 1009, 1213, 1277, 1399, 1409, 1423, 1973, 2203, 2237, 2591, 2609, 2617, 2633, 2647, 2657, 3163, 3299, 4861, 4871, 4889, 4903, 4931, 5381, 7411

Offset: 1

Michel Planat, Oct 13 2011

nonn

The sequence consists of the prime numbers p that are champions (left to right maxima) of the function |theta(p)-p|, where theta(p) is the Chebyshev theta function.

Dana Jacobsen, Table of n, a(n) for n = 1..744
Michel Planat and Patrick Solé, Efficient prime counting and the Chebyshev primes, arXiv:1109.6489 [math.NT], 2011.
L. Schoenfeld, Sharper bounds for the Chebyshev functions theta(x) and psi(x). II, Math. Comp. 30 (1976) 337-360.

Cf. A028388, A196667, A197185, A215013.

Equivalent sequences for other indices: A197298(2), A197299(3), A197300(4).

use ntheory ":all"; my($max,$f)=(0); forprimes { $f=abs(chebyshev_theta($)-$); if ($f > $max) { say; $max=$f; } } 10000; # Dana Jacobsen, Dec 29 2015

A021005 Let q_k=p(p+2) be product of k-th pair of twin primes; sequence gives values of p such that (q_k)^2 > q_{k-i}q_{k+i} for all 1 <= i <= k-1.

Original entry on oeis.org

3, 11, 29, 59, 101, 137, 179, 191, 227, 419, 521, 569, 599, 809, 821, 1019, 1229, 1277, 1289, 1607, 1667, 1871, 2081

Offset: 1

Naohiro Nomoto

nonn

			E.g. (11*13)^2 > (5*7)*(17*19): (11*13)^2 > (3*5)*(29*31).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A028388, A021007.

twins=List(); p=3;forprime(q=5,1e5,if(q-p==2,listput(twins,q)); p=q); for(k=1,(#twins+1)\2, for(i=1,k-1,if(twins[k]^2 < twins[k-i]*twins[k+i],next(2))); print1(twins[k]-2", ")) \\ Charles R Greathouse IV, Apr 02 2014

a(1) inserted by Charles R Greathouse IV, Apr 02 2014

A021007 Let q_k = p(p+2) be product of k-th pair of twin primes; sequence gives values of p+2 such that (q_k)^2 > q_{k-i}q_{k+i} for all 1 <= i <= k-1.

Original entry on oeis.org

5, 13, 31, 61, 103, 139, 181, 193, 229, 421, 523, 571, 601, 811, 823, 1021, 1231, 1279, 1291, 1609, 1669, 1873, 2083, 2551, 2659, 2689, 2971, 3121, 3253, 3331, 3361, 3769, 3823, 3919, 4003, 5233, 5419, 5479, 6091, 6271, 6553, 6661, 6691, 8221, 8821, 8971

Offset: 1

Naohiro Nomoto

nonn

Even if there are infinitely many twin primes, it is not clear that this sequence is infinite. The Hardy-Littlewood conjecture implies that there are infinitely many twin primes where p+2 is not in the sequence. - Robert Israel, Apr 02 2014

			(11*13)^2 > (5*7)*(17*19): (11*13)^2 > (3*5)*(29*31).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A028388, A021005.

N:= 20000:
Primes:= [seq(ithprime(i),i=1..N)]:
Twink:= select(t-> (Primes[t+1]=Primes[t]+2),[$1..N-1]):
Qk:= [seq(Primes[i]*Primes[i+1],i=Twink)]:
filter:= proc(k)
   local T,i;
   T:= Qk[k]^2;
   for i from 1 to k-1 do
     if Qk[k-i]*Qk[k+i]>=T then return false fi
   od;
   true
end;
R:= select(filter,[$1 .. floor(nops(Twink)/2)]):
A021007:= map(k -> Primes[Twink[k]+1],R); # Robert Israel, Apr 02 2014

twins=List(); p=3;forprime(q=5,1e5,if(q-p==2,listput(twins,q)); p=q); for(k=1,(#twins+1)\2, for(i=1,k-1,if(twins[k]^2 < twins[k-i]*twins[k+i],next(2))); print1(twins[k]", ")) \\ Charles R Greathouse IV, Apr 02 2014

a(1) inserted by Robert Israel, Apr 02 2014

A096473 Palindromic good primes.

Original entry on oeis.org

5, 11, 101, 191, 727, 929, 30803, 74047, 77477, 1123211, 1150511, 1338331, 1444441, 1684861, 1761671, 3065603, 3392933, 3503053, 3541453, 9779779, 9845489, 9926299, 9927299, 9932399, 112959211, 113030311, 114535411, 119676911

Offset: 1

Farideh Firoozbakht, Jun 28 2004

p is in the sequence iff p is in the sequences A028388 and A002385.

Thus, version 2 (A028388) of the definition of "good primes" is used here. - Harvey P. Dale, May 13 2012

			11 is in the sequence because 11 is a palindromic number which is a good prime (11^2>7*13, 11^2>5*17, 11^2>3*19 & 11^2>2*23).

Carlos Rivera, Puzzle 273. Consecutive 'good' primes, The Prime Puzzles & Problems Connection.
Eric Weisstein's World of Mathematics, Good Prime.

Cf. A028388, A002385.

b[n_]:=(For[m=1, mPrime[n-m]Prime[n+m], m++ ];m); v={};Do[If[IntegerDigits[Prime[n]]==Reverse[IntegerDigits[Prime [n]]]&& b[n]==n, v=Append[v, Prime[n]];Print[v]], {n, 6986301}]

A127925 Primes p such that 2p < prime(k-i) + prime(k+i) for i=1..k-1, where p=prime(k).

Original entry on oeis.org

3, 7, 19, 23, 43, 47, 73, 109, 113, 199, 283, 293, 313, 317, 463, 467, 503, 509, 523, 619, 661, 683, 691, 887, 1063, 1069, 1109, 1129, 1303, 1307, 1321, 1327, 1613, 1621, 1627, 1637, 1669, 1789, 2143, 2161, 2383, 2393, 2399, 2477, 2731, 2753, 2803, 2861, 2971

Offset: 1

T. D. Noe, Feb 06 2007

nonn

One of several sets of "good primes" in section A14 of Guy.

McNew calls these numbers "midpoint convex primes". - Peter Munn, Jul 04 2025

R. K. Guy, Unsolved Problems in Number Theory, 3rd ed. Springer, 2004.

T. D. Noe, Table of n, a(n) for n=1..10001
Nathan McNew, Popular values of the largest prime divisor function (corrected version), page 16, November 2015.

Cf. A028388.
A246033 is a subset.
Subset of A124661, A178954.

t={}; n=1; While[Length[t]<100, n++; p=Prime[n]; i=1; While[i

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

Original entry on oeis.org

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

cp's OEIS Frontend

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

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

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A172166 Partial sums of A028388 good primes (version 2).

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A046869 Good primes (version 1): prime(n)^2 > prime(n-1)*prime(n+1).

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Magma

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A197297 The Riemann primes of the theta type and index 1.

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Perl

A021005 Let q_k=p(p+2) be product of k-th pair of twin primes; sequence gives values of p such that (q_k)^2 > q_{k-i}q_{k+i} for all 1 <= i <= k-1.

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A021007 Let q_k = p*(p+2) be product of k-th pair of twin primes; sequence gives values of p+2 such that (q_k)^2 > q_{k-i}*q_{k+i} for all 1 <= i <= k-1.

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Maple

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A096473 Palindromic good primes.

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A021007 Let q_k = p(p+2) be product of k-th pair of twin primes; sequence gives values of p+2 such that (q_k)^2 > q_{k-i}q_{k+i} for all 1 <= i <= k-1.