cp's OEIS Frontend

A055926 Numbers k such that {largest m such that 1, 2, ..., m divide k} is different from {largest m such that m! divides k}; numbers k which are either odd multiples of 12 or the largest m such that (m-1)! divides k is a composite number > 5.

%I A055926 #69 May 08 2021 22:57:48
%S A055926 12,36,60,84,108,120,132,156,180,204,228,240,252,276,300,324,348,360,
%T A055926 372,396,420,444,468,480,492,516,540,564,588,600,612,636,660,684,708,
%U A055926 732,756,780,804,828,840,852,876,900,924,948,960,972,996,1020,1044,1068
%N A055926 Numbers k such that {largest m such that 1, 2, ..., m divide k} is different from {largest m such that m! divides k}; numbers k which are either odd multiples of 12 or the largest m such that (m-1)! divides k is a composite number > 5.
%C A055926 From _Antti Karttunen_, Nov 20 - Dec 06 2013: (Start)
%C A055926 This sequence has several interpretations:
%C A055926 Numbers k such that A055874(k) differs from A055881(k). [_Leroy Quet_'s original definition of the sequence. Note that A055874(k) >= A055881(k) for all k.]
%C A055926 Numbers k such that {largest m such that m! divides k^2} is different from {largest m such that m! divides k}, i.e., numbers k for which A232098(k) > A055881(k).
%C A055926 Numbers k which are either 12 times an odd number (A073762) or the largest m such that (m-1)! divides k is a composite number > 5 (A232743).
%C A055926 Please see my attached notes for the proof of the equivalence of these interpretations.
%C A055926 Additional implications based on that proof:
%C A055926 A232099 is a subset of this sequence.
%C A055926 A055881(a(n))+1 is always composite. In the range n = 1..17712, only values 4, 6, 8, 9 and 10 occur.
%C A055926 The new definition can be also rephrased by saying that the sequence contains all the positive integers k whose factorial base representation of (A007623(k)) either ends as '...200' (in which case k 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(k) 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 A055926 (End)
%H A055926 Antti Karttunen, <a href="/A055926/b055926.txt">Table of n, a(n) for n = 1..17712</a>
%H A055926 Antti Karttunen, <a href="/A055926/a055926.txt">A proof for the equivalence of three alternative definitions of A055926</a>
%H A055926 Wikipedia, <a href="http://en.wikipedia.org/wiki/Wilsons_theorem">Wilson's theorem</a> (See the section "Composite modulus")
%e A055926 12 is included because 3! is the largest 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 A055926 840 (= 3*5*7*8) is included because the largest factorial which divides 840 is 5! (840 = 7*120), but all positive integers 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 A055926 (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o A055926 (define A055926 (MATCHING-POS 1 1 (lambda (n) (not (= (A055874 n) (A055881 n)))))) ;; _Antti Karttunen_, Nov 18 2013
%o A055926 (define A055926 (MATCHING-POS 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 A055926 Union of A073762 and A232743. Equivalently, setwise difference of A232742 and A017593. Subset: A232099.
%Y A055926 Cf. A055874, A055881, A072668, A232098, A232100, A232741, A232744, A232745.
%K A055926 easy,nonn
%O A055926 1,1
%A A055926 _Leroy Quet_, Jul 16 2000
%E A055926 More terms from _Antti Karttunen_, Dec 01 2013