A289548 The lesser of two semiprime brothers.

9, 14, 21, 26, 403, 12367, 41303, 66893, 68297, 73147, 111607, 116813, 118003, 130133, 146873, 222757, 260497, 418307, 429491, 439097, 478061, 559003, 628241, 729007, 822397, 1116707, 1239869, 1595683, 1887239, 2148589, 2225669, 2481463, 2502977, 2539553

Offset: 1

Author

Jonathan Vos Post and Robert G. Wilson v, Jul 07 2017

Keywords

• nonn

Comments

Semiprime brothers are two consecutive semiprimes (A001358) whose prime factors are consecutive primes (A000040).

The first several examples of semiprime brothers are {9, 10}, {14, 15}, {21, 22}, {26, 33} & {403, 407}.

The only square term is 9 and the only even terms are 14 and 26.

Obviously the difference between the primepi of the factors of the two consecutive semiprimes is either {-1, 1} or {1, -1}.

Number of terms < 10^n: 1, 4, 5, 5, 10, 25, 62, 143, 319, 761, 2010, 5275, etc.

Only the first three terms have as the next semiprime the next integer making them twins. - Robert G. Wilson v, Jun 21 2018

Examples

			26 is in the sequence because 26 = 2*13 and the next semiprime is 33 = 3*11 with 2 & 3 consecutive primes and 11 & 13 consecutive primes.
403 is in the sequence because 403 = 13*31 and the next semiprime is 407 = 11*37 with 11 & 13 and 31 & 37 being consecutive primes.
		

Links

• Jonathan Vos Post, Robert G. Wilson v, and Giovanni Resta, Table of n, a(n) for n = 1..5275 (terms < 10^12, terms > 10^10 from G. Resta)

Crossrefs

Cf. A000040, A001358.

Programs

Mathematica

p = q = 4; fp = fq = {1, 1}; lst = {}; While[p < 26000000, While[fq = Flatten[ Table[#1, {#2}] & @@@ FactorInteger@ q]; Length@ fq != 2, q++]; d = Sort[{fp, fq}]; If[ NextPrime[ d[[1, 1]]] == d[[2, 1]] && NextPrime[ d[[2, 2]]] == d[[1, 2]], AppendTo[lst, p]]; p = q; fp = fq; q++]; lst

PARI

isok(p, q) = (nextprime(p+1) == q) || (nextprime(q+1) == p);
pairp(n) = if (issquare(n), vector(2, k, sqrtint(n)), (factor(n)[,1])~);
lista(nn) = {na = 2; while (na < nn, if (bigomega(na) != 2, na++, nb = na + 1; while (bigomega(nb) != 2, nb++); fpa = pairp(na); fpb = pairp(nb); if (isok(fpa[1], fpb[1]) && isok(fpa[2], fpb[2]), print1(na, ", ")); na = nb;););} \\ Michel Marcus, Jul 11 2017

Python

from sympy import factorint, nextprime
def is_semiprime(n):
    return sum(e for e in factorint(n).values()) == 2
def next_semiprime(n):
    nxt = n + 1
    while not is_semiprime(nxt): nxt += 1
    return nxt
def are_consecutive(p, q):
    return max(p, q) == nextprime(min(p, q))
def ok(n):
    if not is_semiprime(n): return False
    nextsp = next_semiprime(n)
    fn, fm = factorint(n, multiple=True), factorint(nextsp, multiple=True)
    return are_consecutive(fn[0], fm[0]) and are_consecutive(fn[1], fm[1])
print(list(filter(ok, range(150000)))) # Michael S. Branicky, Sep 14 2021