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%I #69 Jan 14 2024 06:54:59
%S 1,2,1,2,1,3,1,2,1,2,1,4,1,2,1,2,1,3,1,2,1,2,1,4,1,2,1,2,1,3,1,2,1,2,
%T 1,4,1,2,1,2,1,3,1,2,1,2,1,4,1,2,1,2,1,3,1,2,1,2,1,6,1,2,1,2,1,3,1,2,
%U 1,2,1,4,1,2,1,2,1,3,1,2,1,2,1,4,1,2,1,2,1,3,1,2,1,2,1,4,1,2,1,2
%N a(n) = largest m such that 1, 2, ..., m divide n.
%C From _Antti Karttunen_, Nov 20 2013 & Jan 26 2014: (Start)
%C Differs from A232098 for the first time at n=840, where a(840)=8, while A232098(840)=7. A232099 gives all the differing positions. See also the comments at A055926 and A232099.
%C The positions where a(n) is an odd prime is given by A017593 up to A017593(34)=414 (so far all 3's), after which comes the first 7 at a(420). (A017593 gives the positions of 3's.)
%C (Continued on Jan 26 2014):
%C Only terms of A181062 occur as values.
%C A235921 gives such n where a(n^2) (= A235918(n)) differs from A071222(n-1) (= A053669(n)-1). (End)
%C a(n) is the largest m such that A003418(m) divides n. - _David W. Wilson_, Nov 20 2014
%C a(n) is the largest number of consecutive integers dividing n. - _David W. Wilson_, Nov 20 2014
%C A051451 gives indices where record values occur. - _Gionata Neri_, Oct 17 2015
%C Yuri Matiyasevich calls this the maximum inheritable divisor of n. - _N. J. A. Sloane_, Dec 14 2023
%H Antti Karttunen, <a href="/A055874/b055874.txt">Table of n, a(n) for n = 1..10080</a>
%H Bakir Farhi, <a href="https://www.emis.de/journals/INTEGERS/papers/j42/j42.Abstract.html">On the average asymptotic behavior of a certain type of sequences of integers</a>, Integers, Vol. 9 (2009), pp. 555-567.
%F a(n) = A007978(n) - 1. - _Antti Karttunen_, Jan 26 2014
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A064859 (Farhi, 2009). - _Amiram Eldar_, Jul 25 2022
%e a(12) = 4 because 1, 2, 3, 4 divide 12, but 5 does not.
%p N:= 1000: # to get a(1) to a(N)
%p A:= Vector(N,1);
%p for m from 2 do
%p Lm:= ilcm($1..m);
%p if Lm > N then break fi;
%p if Lm mod (m+1) = 0 then next fi;
%p for k from 1 to floor(N/Lm) do
%p A[k*Lm]:=m
%p od
%p od:
%p convert(A,list); # _Robert Israel_, Nov 28 2014
%t a[n_] := Module[{m = 1}, While[Divisible[n, m++]]; m - 2]; Array[a, 100] (* _Jean-François Alcover_, Mar 07 2016 *)
%o (Haskell)
%o a055874 n = length $ takeWhile ((== 0) . (mod n)) [1..]
%o -- _Reinhard Zumkeller_, Feb 21 2012, Dec 09 2010
%o (Scheme)
%o (define (A055874 n) (let loop ((m 1)) (if (not (zero? (modulo n m))) (- m 1) (loop (+ 1 m))))) ;; _Antti Karttunen_, Nov 18 2013
%o (PARI) a(n) = my(m = 1); while ((n % m) == 0, m++); m - 1; \\ _Michel Marcus_, Jan 17 2014
%o (Python)
%o from itertools import count
%o def A055874(n):
%o for m in count(1):
%o if n % m:
%o return m-1 # _Chai Wah Wu_, Jan 02 2022
%Y One less than A007978.
%Y Cf. also A053669, A055881, A055926, A017593, A064859, A181062, A126800, A232098, A232099, A233284, A235918, A235921.
%K easy,nonn
%O 1,2
%A _Leroy Quet_, Jul 16 2000
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