%I
%S 12,36,60,84,108,120,132,156,180,204,228,240,252,276,300,324,348,360,
%T 372,396,420,444,468,480,492,516,540,564,588,600,612,636,660,684,708,
%U 732,756,780,804,828,840,852,876,900,924,948,960,972,996,1020,1044,1068
%N Numbers n such that {largest m such that 1, 2, ..., m divide n} is different from {largest m such that m! divides n}; numbers n which are either odd multiples of 12 or the largest m such that (m1)! divides n is a composite number > 5.
%C Contribution from _Antti Karttunen_, Nov 20  Dec 06 2013: (Start)
%C This sequence has several interpretations:
%C Numbers n such that A055874(n) differs from A055881(n). [_Leroy Quet_'s original definition of the sequence. Note that A055874(n) >= A055881(n) for all n].
%C Numbers n such that {largest m such that m! divides n^2} is different from {largest m such that m! divides n}, i.e. numbers n for which A232098(n) > A055881(n).
%C Numbers n which are either 12 times an odd number (A073762) or the largest m such that (m1)! divides n is a composite number > 5 (A232743).
%C Please see my attached notes for the proof of the equivalence of these interpretations.
%C Additional implications based on that proof:
%C A232099 is a subset of this sequence.
%C A055881(a(n))+1 is always composite. In range n=1..17712, only values 4, 6, 8, 9 and 10 occur.
%C The new definition can be also rephrased by saying that the sequence contains all the natural numbers n whose factorial base representation of (A007623(n)) either ends as '...200' (in which case n is an odd multiple of 12, 12 = '200', 36 = '1200', 60 = '2200', ...) or the number of trailing zeros + 2 in that representation is a composite number greater than or equal to 6 , e.g. 120 = '10000' (in other words, A055881(n) is one of the terms of A072668 after the initial 3). Together these conditions also imply that all the terms are divisible by 12.
%C (End)
%H Antti Karttunen, <a href="/A055926/b055926.txt">Table of n, a(n) for n = 1..17712</a>
%H Antti Karttunen, <a href="/A055926/a055926.txt">A proof for the equivalence of three alternative definitions of A055926</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Wilsons_theorem">Wilson's theorem</a> (See the section "Composite modulus")
%e 12 is included because 3! is the highest factorial to divide 12, but 1, 2, 3 and 4 all divide 12. Equally, 12 is included because it is one of the terms of A073762, or equally, because its factorial base representation ends with digits '...200': A007623(12) = 200.
%e 840 (= 3*5*7*8) is included because the highest factorial which divides 840 is 5! (840 = 7*120), but all natural numbers up to 8 divide 840. Equally, 840 is included because it is one of the terms of A232743 as 5+1 = 6 is a composite number larger than 5. Note that A007623(840) = 110000.
%o (Scheme, with _Antti Karttunen_'s IntSeqlibrary)
%o (define A055926 (MATCHINGPOS 1 1 (lambda (n) (not (= (A055874 n) (A055881 n)))))) ;; _Antti Karttunen_, Nov 18 2013
%o (define A055926 (MATCHINGPOS 1 1 (lambda (n) (cond ((and (integer? (/ n 12)) (odd? (/ n 12)))) ((A055881 n) => (lambda (k) (and (> k 4) (not (prime? (+ k 1)))))))))) ;; _Antti Karttunen_, Dec 01 2013
%Y Union of A073762 and A232743. Equivalently, setwise difference of A232742 and A017593. Subset: A232099.
%Y Cf. A055874, A055881, A072668, A232098, A232100, A232741, A232744, A232745.
%K easy,nonn
%O 1,1
%A _Leroy Quet_, Jul 16 2000
%E More terms from _Antti Karttunen_, Dec 01 2013
