1,4

Is this sequence finite; or is there always a divisor of n! where the sum of the first n terms of the sequence divides n!, for every positive integer n?

R. G. Wilson v, Table of n, a(n) for n=1,...,250

For n = 5 we check the divisors of 5!=120, from the largest downward: a(1)+a(2)+a(3)+a(4) + 120 = 126, which is not a divisor of 120. 1+1+1+3 + 60 = 66, which is not a divisor of 120. 1+1+1+3 + 40 = 46, which is not a divisor of 120. 1+1+1+3 + 30 = 36, which is not a divisor of 120. But 1+1+1+3 + 24 = 30, which is a divisor of 120. So, a(5) = 24 = the largest divisor of 5! such that a(1)+a(2)+a(3)+a(4)+a(5) also divides 5!.

A156832 := proc(n) local dvs, i, largd ; option remember; if n = 1 then 1; else dvs := sort(convert(numtheory[divisors](n!), list)) ; for i from 1 to nops(dvs) do largd := op(-i, dvs) ; if largd+add( procname(i), i=1..n-1) in dvs then RETURN(largd) ; fi; od: error(n) ; fi; end: for n from 1 do printf("%d, \n", A156832(n)) ; od; [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 20 2009]

f[n_] := f[n] = Block[{d = 1, s = Sum[ f@i, {i, n - 1}]}, While[ Mod[n!, d] > 0 || Mod[n!, n!/d + s] > 0, d++ ]; n!/d]; Array[f, 23] (Robert G. Wilson, v Feb 16 2009).

For n!/a(n) see A145500.

Cf. A145499.

Sequence in context: A092468 A027158 A117642 this_sequence A092780 A004282 A164938

Adjacent sequences: A156829 A156830 A156831 this_sequence A156833 A156834 A156835

nonn

Leroy Quet (q1qq2qqq3qqqq(AT)yahoo.com), Feb 16 2009

More terms from Robert G. Wilson v, Joshua Zucker and R. J. Mathar, Feb 16 2009