A241102 Semiprimes of the form prime(n+1)^3 - prime(n)^3.

218, 866, 345602, 477146, 726626, 1280666, 2291546, 3936602, 4113506, 6242402, 7154786, 13177946, 22395746, 26158466, 26763266, 30862946, 43352066, 52925402, 68952602, 74680706, 87646106, 96962402, 109499906, 112909466, 181632026, 192077786, 205335002, 257572226

Offset: 1

Author

K. D. Bajpai, Apr 16 2014

Keywords

• nonn

Comments

All the terms in the sequence are even semiprimes.

All the terms in the sequence are congruent to 2 mod 3.

Examples

			a(1) = 201658 = 59^3 - 61^2: Also 201658 = 2*100829. Hence 201658 is semiprime.
a(2) = 563866 = 83^3 - 89^2: Also 563866 = 2*281933. Hence 563866 is semiprime.
		

Links

• K. D. Bajpai, Table of n, a(n) for n = 1..4048

Crossrefs

Cf. A001358 (semiprimes: product of two primes).

Cf. A046388 (odd numbers: p*q ( p and q are primes)).

Cf. A046315 (odd semiprimes: divisible by exactly 2 primes).

Cf. A240859 (cubes k^3: k^3 + (k+1)^3 are semiprimes).

Cf. A240884 (semiprimes: n-th cube + n-th triangular numbers).

Programs

Maple

with(numtheory):KD:= proc() local a,b; a:=ithprime(n)^3 - ithprime(n+1)^2;b:=bigomega(a); if b=2 then RETURN (a); fi; end: seq(KD(), n=1..800);

Mathematica

KD = {}; Do[t = Prime[n]^3 - Prime[n + 1]^2; If[PrimeOmega[t] == 2, AppendTo[KD, t]], {n, 500}]; KD
n = 0; Do[t = Prime[k]^3 - Prime[k + 1]^2; If[PrimeOmega[t] == 2, n = n + 1; Print[n, " ", t]], {k, 1, 500000}] (* b- file *)
Select[#[[2]]^3-#[[1]]^3&/@Partition[Prime[Range[1500]],2,1], PrimeOmega[ #] == 2&] (* Harvey P. Dale, Jul 01 2015 *)
Select[Differences[Prime[Range[1500]]^3],PrimeOmega[#]==2&] (* Harvey P. Dale, May 26 2025 *)

PARI

s=[]; for(n=1, 4000, t=prime(n+1)^3-prime(n)^3; if(bigomega(t)==2, s=concat(s, t))); s \\ Colin Barker, Apr 16 2014

Python

from itertools import islice
from sympy import isprime, nextprime
def A241102_gen(): # generator of terms
    p, q = 3**3, 5
    while True:
        if isprime((m:=q**3)-p>>1):
            yield m-p
        p, q = m, nextprime(q)
A241102_list = list(islice(A241102_gen(),10)) # Chai Wah Wu, Feb 27 2023