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Primordial LB numbers: LB numbers (A267856) that are not of the form 10*n where n is also an LB number.
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%I #20 Feb 11 2016 22:55:46

%S 250,375,648,972,2430,6750,9375,36450,60750,84672,546750,8346672,

%T 12605250,18907875,26406250,31513125,39609375,44118375,53466750,

%U 69328875,81934125,107144625,119749875,144960375,182776125,195381375,233197125,555644448,579296448,774927552,833466672

%N Primordial LB numbers: LB numbers (A267856) that are not of the form 10*n where n is also an LB number.

%C An LB number is a positive integer of the form n = a*10^k+b (with a > 0 and b < 10^k) satisfying two properties: 1) the set of prime factors of n is the union of the sets of prime factors of a and b; and 2) A001222(n) = A001222(a) + A001222(b) where A001222(n) = Bigomega(n) gives the number of primes divisors of n counted with multiplicity.

%C This sequence is an infinite subsequence of A267856.

%C a(13) > 10^7.

%H Robert O. Stanton, <a href="http://www.jstor.org/stable/2690478">The Remarkable Number 648</a>, Mathematics Magazine, Vol. 66, No. 1 (Feb., 1993), pp. 48-55.

%e 972 is a term since 972 is an LB number (see A267856 for the reason) and 972 is not 10*k where k is an LB number.

%e 2500 is an LB number but is not a term of this sequence since 2500 = 10*20 and 250 is an LB number.

%o (PARI) already(n, v) = {for (k=1, #v, q = n/v[k]; if (denominator(q) == 1, e = valuation(q, 10); if (q == 10^e, return (1));););}

%o isok(n) = {nb = #Str(n); spf = Set(factor(n)[,1]~); nbpfr = bigomega(n); for (k=1, nb-1, a = n\10^k; b = n - 10^k*a; if (b && (bigomega(a)+ bigomega(b) == nbpfr) && (setunion(factor(a)[,1]~, factor(b)[,1]~) == spf), return (1)););}

%o lista(nn) = {my(v = []); for (n=1, nn, if (isok(n) && ! already(n, v), print1(n, ", "); v = concat(v, n);););} \\ _Michel Marcus_, Jan 31 2016

%Y Cf. A001222, A267856.

%K nonn,base

%O 1,1

%A _Tom Edgar_, Jan 29 2016

%E a(13)-a(31) from _Michel Marcus_, Jan 30 2016