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A289548 The lesser of two semiprime brothers. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

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)

EXAMPLE

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.

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

PROG

(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

CROSSREFS

Cf. A000040, A001358.

Sequence in context: A186778 A070552 A272141 * A001198 A151915 A100263

Adjacent sequences:  A289545 A289546 A289547 * A289549 A289550 A289551

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified July 17 02:56 EDT 2019. Contains 325092 sequences. (Running on oeis4.)