A083550 -id:A083550 - cp's OEIS frontend

A383652 Primes p preceded and followed by gaps whose product is less than (log(p))^2.

17, 19, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 139, 149, 151, 163, 167, 179, 181, 191, 193, 197, 199, 227, 229, 233, 239, 241, 269, 271, 277, 281, 283, 311, 313, 347, 349, 353, 379, 383, 397, 401, 419, 421, 431, 433, 439, 443, 457, 461, 463, 487, 491, 499, 503, 521, 523, 563, 569, 571, 593, 599

Offset: 1

Alain Rocchelli, May 04 2025

nonn

Since the geometric mean is never greater than the arithmetic mean: A381850 is a subsequence.

			17 is a term because (17-13)*(19-17)=8 is less than (log(17))^2=8.0271.
19 is a term because (19-17)*(23-19)=8 is less than (log(19))^2=8.6697.
29 is not a term because(29-23)*(31-29)=12 is greater than (log(29))^2=11.3387.

A288907 and A381850 are subsequences.

Cf. A083550.

Select[Range[2, 110] // Prime, (# - NextPrime[#, -1])(NextPrime[#] - #) < Log[#]^2 &] (* Stefano Spezia, May 04 2025 *)

forprime(P=3, 600, my(M=P-precprime(P-1), Q=nextprime(P+1)-P, AR=M*Q, AR0=(log(P))^2); if(AR

Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 0.720268...

A083549 Quotient if least common multiple (lcm) of cototient values of consecutive integers is divided by the greatest common divisor (gcd) of the same pair of consecutive numbers.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 4, 12, 2, 6, 8, 8, 8, 56, 56, 8, 12, 12, 12, 12, 12, 12, 16, 80, 70, 126, 144, 16, 22, 22, 16, 208, 234, 198, 264, 24, 20, 12, 40, 24, 30, 30, 24, 56, 56, 24, 32, 224, 210, 570, 532, 28, 36, 60, 480, 672, 70, 30, 44, 44, 32, 864, 864, 544, 782, 46, 36, 900

Offset: 1

Labos Elemer, May 22 2003

nonn

			n=33: cototient(33) = 33-20 = 13, cototient(34) = 34-16 = 18;
lcm(13,18) = 234, gcd(13,18) = 1, so a(34) = 234.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A051953, A083538, A083539, A083540, A083541, A083542, A083543, A083544, A083545, A083546, A083547, A083548, A083549, A083550, A083551, A083552, A083553, A083554, A083555, A049586.

f[x_] := x-EulerPhi[x]; Table[LCM[f[w+1], f[w]]/GCD[f[w+1], f[w]], {w, 69}]
(* Second program: *)
Map[Apply[LCM, #]/Apply[GCD, #] &@ Map[# - EulerPhi@ # &, #] &, Partition[Range[69], 2, 1]] (* Michael De Vlieger, Mar 17 2018 *)

a(n) = lcm(A051953(n), A051952(n+1))/gcd(A051953(n), A051952(n+1)) = lcm(cototient(n+1), cototient(n))/A049586(n).

A375009 a(n) = smallest prime Q of a consecutive prime triple {P, Q, R} such that floor( (R-Q) * (Q-P) / 8 ) = n.

Original entry on oeis.org

7, 139, 23, 53, 1151, 89, 113, 10007, 509, 331, 91079, 479, 541, 79699, 631, 1129, 293, 211, 5557, 265621, 2633, 1259, 1599709, 3659, 1327, 2127269, 4703, 1847, 1349533, 4201, 7621, 16519, 2579, 41333, 10343761, 4621, 4327, 8039, 16729, 3433, 166209301, 3271, 44351

Offset: 1

Carl R. White, Jul 27 2024

nonn

Without taking the floor there is no such Q where (R-Q)*(Q-P)/8 = 2.

Dickson's conjecture implies that if n == 0 or 1 (mod 3) there are infinitely many primes Q such that Q - 2 is the previous prime and Q + 4 n is the next prime, and that if n == 2 (mod 3) there are infinitely many primes Q such that Q - 2 is the previous prime and Q + 4 n + 2 is the next prime. - Robert Israel, Sep 24 2024

Cf. A083550, A001223.

N:= 50: # for a(1) .. a(N)
V:= Vector(N): count:= 0:
q:= 2: r:= 3:
while count < N do
  p:= q; q:= r; r:= nextprime(r);
  v:= floor((r-q)*(q-p)/8);
  if v >= 1 and v <= N and V[v] = 0 then
    V[v]:= q; count:= count+1
  fi;
od:
convert(V,list); # Robert Israel, Sep 24 2024

With[{p = Prime[Range[10^6]]}, r = (Floor[(#[[3]] - #[[2]])*(#[[2]] - #[[1]])/8]) & /@ Partition[p, 3, 1]; p[[1 + TakeWhile[FirstPosition[r, #] & /@ Range[Max[r]], ! MissingQ[#] &] // Flatten]]] (* Amiram Eldar, Sep 24 2024 *)

lista(len) = {my(c = 0, v = vector(len), p1 = 2, p2 = 3, i); forprime(p3 = 5, , i = floor((p3-p2)*(p2-p1)/8); if(i > 0 && i <= len && v[i] == 0, c++; v[i] = p2; if(c==len, break)); p1 = p2; p2 = p3); v;} \\ Amiram Eldar, Sep 24 2024

from sympy import nextprime
def A375009(n):
    p,q = 2,3
    while True:
        r = nextprime(q)
        if (r-q)*(q-p)>>3==n:
            return q
        p,q = q,r # Chai Wah Wu, Oct 21 2024

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A383652 Primes p preceded and followed by gaps whose product is less than (log(p))^2.

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A083549 Quotient if least common multiple (lcm) of cototient values of consecutive integers is divided by the greatest common divisor (gcd) of the same pair of consecutive numbers.

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A375009 a(n) = smallest prime Q of a consecutive prime triple {P, Q, R} such that floor( (R-Q) * (Q-P) / 8 ) = n.

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