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Smallest number having exactly n partitions into products of two successive primes (A006094), or -1 if no such number exists.
6

%I #17 Sep 12 2024 07:49:27

%S 1,6,30,60,90,105,120,135,143,158,155,167,173,182,185,207,197,203,212,

%T 215,221,231,227,233,239,242,256,245,251,261,257,260,263,266,282,272,

%U 275,278,281,291,-1,287,290,293,296,309,312,302,305,319,308,314,-1,317,322,320

%N Smallest number having exactly n partitions into products of two successive primes (A006094), or -1 if no such number exists.

%C If a(n) <> -1: A116357(a(n))=n and A116357(m)<>n for m<n.

%C From _David A. Corneth_, Sep 11 2024: (Start)

%C To prove a value -1 we need two facts:

%C 1. For some k we have A116357(k), A116357(k+1), A116357(k+2), A116357(k+3), A116357(k+4), A116357(k+5) > n as A116357(k + 6) >= A116357(k) for all k.

%C 2. A116357(m) != n for 1 <= m < k. (End)

%H David A. Corneth, <a href="/A116360/b116360.txt">Table of n, a(n) for n = 0..10000</a>

%e Without proof: a(40) = -1 and a(52) = -1.

%e a(40) = -1 as A116357(296) through A116357(296+5) are larger than 40 and for 1 <= m < 296 we have A116357(m) != 40. - _David A. Corneth_, Sep 11 2024

%Y Cf. A006094, A116357.

%K sign

%O 0,2

%A _Reinhard Zumkeller_, Feb 12 2006

%E Edited by _D. S. McNeil_, Sep 06 2010