cp's OEIS Frontend

A336585 Integers m such that (m-d)*(m+d) < (m-1)^2, where d is the smallest number such that both m-d and m+d are primes.

%I A336585 #9 Oct 17 2020 02:16:21
%S A336585 22,28,32,38,46,49,55,58,68,74,82,87,94,112,121,128,130,136,146,155,
%T A336585 184,200,203,206,218,221,224,238,244,247,253,265,268,284,286,301,304,
%U A336585 306,308,316,318,320,323,326,341,344,346,362,398,412,413,428,454,466,484
%N A336585 Integers m such that (m-d)*(m+d) < (m-1)^2, where d is the smallest number such that both m-d and m+d are primes.
%C A336585 All terms in this sequence are composites since, if m is a prime, d = 0 and (m-d)*(m+d) = m^2 > (m-1)^2.
%C A336585 It seems that the number of terms in this sequence is finite, with the last term being a(1225) = 1353559. Conjecture: there exist only 1225 semiprimes of the form (m-d)*(m+d), where d is the smallest number such that (m-d)*(m+d) < (m-1)^2.
%C A336585 a(n) in this sequence is the value of n in A047160 with m > sqrt(2*n - 1).
%C A336585 All terms <= 1353559 in A335297 can be found in this sequence.
%e A336585 2 is not a term since for m = 2, d = 0 and (2-0)*(2-0) = 4 > (m-1)^2 = 1;
%e A336585 4 is not a term since for m = 4, d = 1 and (4-1)*(4+1) = 15 > (m-1)^2 = 9;
%e A336585 22 is a term since for m = 22, d = 9 and (22-9)*(22+9) = 403 < (m-1)^2 = 441;
%e A336585 1353559 is a term since for m = 1353559, d = 1722 and (1353559-1722)*(1353559+1722) = 1832119001197 < (m-1)^2 = 1832119259364.
%o A336585 (Python)
%o A336585 from sympy import isprime
%o A336585 n = 0
%o A336585 m = 2
%o A336585 while m >= 2:
%o A336585     d = 0
%o A336585     while d < m/2:
%o A336585         p = m - d
%o A336585         q = m + d
%o A336585         if isprime(p) == 1 and isprime(q) == 1:
%o A336585             if p*q < (m - 1) * (m - 1):
%o A336585                 n += 1
%o A336585                 print (m)
%o A336585             break
%o A336585         d += 1
%o A336585     m += 1
%Y A336585 Cf. A047160, A335297.
%K A336585 nonn
%O A336585 1,1
%A A336585 _Ya-Ping Lu_, Oct 04 2020