Midpoint convex primes defined generally by the pi function

[Tomasz Ordowski]

Hello again to all interested! 

Good primes of the basic type A127925, known as "midpoint convex primes".  

(*) Primes prime(k) such that 2 prime(k) < prime(k-j) + prime(k+j) for 0 < j < k.

3, 7, 19, 23, 43, 47, 73, 109, 113, 199, 283, 293, 313, 317, 463, 467, ... 

(**) I noticed (easily provable) that they can be defined equivalently as:

Numbers n > 1 such that 2 pi(n) > pi(n-m) + pi(n+m) for 0 < m < n. 

(***) Conjecture: these are numbers n > 1 for which 

2 pi(n) = pi(n-m) + pi(n+m) has no solution 0 < m < n. 

Can someone test and (dis)prove my bold conjecture? 

The conjecture is true if and only if for every composite n 

this equation has such a solution m. 

Now the task is easier.

Good luck!

Thomas Ordowski 

______________________

(***) It should be noted that (contrary to appearances) 

these are not A178954, but their proper subset 

(without 2, 79, 149, 163, ...).  

[Tomasz Ordowski]

PS. Regarding all primes, I put forward Conjecture: 

2/pi(n) < 1/pi(n-m) + 1/pi(n+m) for 0 < m < n-1 

if and only if n is a prime number.  

[Tomasz Ordowski]

By the last Conjecture:  
if p_k = (p_m + p_n)/2, 

then 2/k < 1/m + 1/n. 

It seems provable.   

[M F Hasler]

On Tue, Jun 16, 2026, 16:38 Tomasz Ordowski <tomaszo...@gmail.com> wrote:

By the last Conjecture:  
if p_k = (p_m + p_n)/2, 

then 2/k < 1/m + 1/n. 

It seems provable.

Nope. Counter-example :

prime(14) = 43

prime(11) = 31

prime(12) = 37

 1/11+1/14 - 2/12 = -1/231

(PARI)

for(m=1,999, pm=prime(m); 

for(n=2,m-2, is prime(pk=(pm+prime(n))/2) && 2/primepi(pk)<1/m+1/n && return([m,n,pk]))) 

M.

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[Tomasz Ordowski]

Yes, thanks! 

There are (infinitely) many counterexamples that refute this more general conjecture as well. 

However, it can probably be proven, in particular, that there exists N such that 

if |m - n| >= N and 2*p_k = p_m + p_n, then 2/k < 1/m + 1/n.

Will there be counterexamples at N = 10? 

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[M F Hasler]

On Wed, Jun 17, 2026 at 5:54 AM Tomasz Ordowski <tomaszo...@gmail.com> wrote:

Yes, thanks! 

There are (infinitely) many counterexamples that refute this more general conjecture as well. 

However, it can probably be proven, in particular, that there exists N such that 

if |m - n| >= N and 2*p_k = p_m + p_n, then 2/k < 1/m + 1/n.

Will there be counterexamples at N = 10? 

{for(m=1,99, pm=prime(m); for(n=2,m-10, is prime(pk=(pm+prime(n))/2) && 2/primepi(pk) > 1/m+1/n && return([m,n,pk]))) }

immediately returns [55, 45, 227]

with prime(55) = 257, prime(45) = 197,  primepi(227) = 49

1/55+1/45 - 2/49 = -2/4851

- Maximilian

[M F Hasler]

PS: The following code produces counter-examples for increasingly large m-n,

it could be used for some new OEIS sequences :

f(d) = Least m / prime(m) / ... such that there is n<k<m with m-n >= d such that 

(prime(n)+prime(m))/2 = prime(k) and 1/n + 1/m < 2/k.

- Maximilian

a(n) = {for(m=1,oo, pm=prime(m); for(n=2,m-n, is prime(pk=(pm+prime(n))/2) && 2/primepi(pk) > 1/m+1/n && return([m,n]))) }

{show(mn)=my(m,n,k,p,q,pk); printf("((prime(m=%d) = %d) + (prime(%d) = %d))/2 = %d = prime(k=%d), 1/m+1/n - 2/k = %s\n", m=mn[1], p=prime(m), n=mn[2], q=prime(n=mn[2]), pk=(p+q)/2, k=primepi(pk),1/m+1/n-2/k)}

t=0; for(n=1, 99, cmp(t, t=f(n)) && show(t)) \\ only show new counter-examples

((prime(m=14) = 43) + (prime(11) = 31))/2 = 37 = prime(k=12), 1/m+1/n - 2/k = -1/231
((prime(m=20) = 71) + (prime(15) = 47))/2 = 59 = prime(k=17), 1/m+1/n - 2/k = -1/1020
((prime(m=30) = 113) + (prime(24) = 89))/2 = 101 = prime(k=26), 1/m+1/n - 2/k = -1/520
((prime(m=39) = 167) + (prime(32) = 131))/2 = 149 = prime(k=35), 1/m+1/n - 2/k = -11/43680
((prime(m=55) = 257) + (prime(45) = 197))/2 = 227 = prime(k=49), 1/m+1/n - 2/k = -2/4851
((prime(m=75) = 379) + (prime(61) = 283))/2 = 331 = prime(k=67), 1/m+1/n - 2/k = -38/306525
((prime(m=79) = 401) + (prime(62) = 293))/2 = 347 = prime(k=69), 1/m+1/n - 2/k = -67/337962
((prime(m=111) = 607) + (prime(93) = 487))/2 = 547 = prime(k=101), 1/m+1/n - 2/k = -14/347541
((prime(m=114) = 619) + (prime(90) = 463))/2 = 541 = prime(k=100), 1/m+1/n - 2/k = -1/8550
((prime(m=118) = 647) + (prime(91) = 467))/2 = 557 = prime(k=102), 1/m+1/n - 2/k = -79/547638
((prime(m=120) = 659) + (prime(91) = 467))/2 = 563 = prime(k=103), 1/m+1/n - 2/k = -107/1124760
((prime(m=243) = 1543) + (prime(213) = 1303))/2 = 1423 = prime(k=224), 1/m+1/n - 2/k = -229/1932336
((prime(m=244) = 1549) + (prime(204) = 1249))/2 = 1399 = prime(k=222), 1/m+1/n - 2/k = -1/115107
((prime(m=246) = 1559) + (prime(205) = 1259))/2 = 1409 = prime(k=223), 1/m+1/n - 2/k = -7/274290
((prime(m=247) = 1567) + (prime(202) = 1231))/2 = 1399 = prime(k=222), 1/m+1/n - 2/k = -55/5538234
((prime(m=251) = 1597) + (prime(204) = 1249))/2 = 1423 = prime(k=224), 1/m+1/n - 2/k = -61/1433712
((prime(m=254) = 1609) + (prime(203) = 1237))/2 = 1423 = prime(k=224), 1/m+1/n - 2/k = -27/412496
((prime(m=257) = 1621) + (prime(203) = 1237))/2 = 1429 = prime(k=226), 1/m+1/n - 2/k = -191/5895323
((prime(m=258) = 1627) + (prime(202) = 1231))/2 = 1429 = prime(k=226), 1/m+1/n - 2/k = -34/1472277
((prime(m=259) = 1637) + (prime(199) = 1217))/2 = 1427 = prime(k=225), 1/m+1/n - 2/k = -32/11596725
((prime(m=415) = 2857) + (prime(352) = 2377))/2 = 2617 = prime(k=380), 1/m+1/n - 2/k = -7/555104
((prime(m=416) = 2861) + (prime(350) = 2357))/2 = 2609 = prime(k=379), 1/m+1/n - 2/k = -443/27591200
((prime(m=421) = 2909) + (prime(350) = 2357))/2 = 2633 = prime(k=382), 1/m+1/n - 2/k = -89/28143850
((prime(m=426) = 2957) + (prime(350) = 2357))/2 = 2657 = prime(k=384), 1/m+1/n - 2/k = -3/795200
((prime(m=427) = 2963) + (prime(349) = 2351))/2 = 2657 = prime(k=384), 1/m+1/n - 2/k = -31/28612416
((prime(m=503) = 3593) + (prime(421) = 2909))/2 = 3251 = prime(k=458), 1/m+1/n - 2/k = -167/48493727
((prime(m=511) = 3659) + (prime(424) = 2939))/2 = 3299 = prime(k=463), 1/m+1/n - 2/k = -423/100315432
((prime(m=512) = 3671) + (prime(423) = 2927))/2 = 3299 = prime(k=463), 1/m+1/n - 2/k = -247/100274688
((prime(m=761) = 5801) + (prime(666) = 4973))/2 = 5387 = prime(k=710), 1/m+1/n - 2/k = -241/179923230
((prime(m=763) = 5813) + (prime(666) = 4973))/2 = 5393 = prime(k=711), 1/m+1/n - 2/k = -11/13381494
((prime(m=777) = 5903) + (prime(675) = 5039))/2 = 5471 = prime(k=722), 1/m+1/n - 2/k = -101/63111825

[Tomasz Ordowski]

Well, my conjecture is beyond saving, but maybe it will leave behind a new OEIS sequence. Thanks!

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