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.

12, 36, 60, 84, 108, 120, 132, 156, 180, 204, 228, 240, 252, 276, 300, 324, 348, 360, 372, 396, 420, 444, 468, 480, 492, 516, 540, 564, 588, 600, 612, 636, 660, 684, 708, 732, 756, 780, 804, 828, 840, 852, 876, 900, 924, 948, 960, 972, 996, 1020, 1044, 1068

Offset

1,1

Comments

From Antti Karttunen, Nov 20 - Dec 06 2013: (Start)

This sequence has several interpretations:

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.]

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).

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).

Please see my attached notes for the proof of the equivalence of these interpretations.

Additional implications based on that proof:

A232099 is a subset of this sequence.

A055881(a(n))+1 is always composite. In the range n = 1..17712, only values 4, 6, 8, 9 and 10 occur.

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.

(End)

Links

Antti Karttunen, Table of n, a(n) for n = 1..17712

Antti Karttunen, A proof for the equivalence of three alternative definitions of A055926

Wikipedia, Wilson's theorem (See the section "Composite modulus")

Example

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.
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.

Prog

(Scheme, with Antti Karttunen's IntSeq-library)
(define A055926 (MATCHING-POS 1 1 (lambda (n) (not (= (A055874 n) (A055881 n)))))) ;; Antti Karttunen, Nov 18 2013
(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

Crossrefs

Union of A073762 and A232743. Equivalently, setwise difference of A232742 and A017593. Subset: A232099.

Cf. A055874, A055881, A072668, A232098, A232100, A232741, A232744, A232745.

Sequence in context: A322411 A339425 A063298 * A073762 A211609 A338393

Adjacent sequences: A055923 A055924 A055925 * A055927 A055928 A055929

Keyword

easy,nonn

Author

Leroy Quet, Jul 16 2000

Extensions

More terms from Antti Karttunen, Dec 01 2013

Status

approved