A085779 Smallest m such that the triangular number A000217(n) divides m!.

1, 3, 3, 5, 5, 7, 7, 6, 6, 11, 11, 13, 13, 7, 5, 17, 17, 19, 19, 7, 11, 23, 23, 10, 13, 13, 9, 29, 29, 31, 31, 11, 17, 17, 7, 37, 37, 19, 13, 41, 41, 43, 43, 11, 23, 47, 47, 14, 14, 17, 17, 53, 53, 11, 11, 19, 29, 59, 59, 61, 61, 31, 8, 13, 13, 67, 67, 23, 23, 71, 71, 73, 73, 37, 19, 19

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A002034(A000217(n)).

Example

t(4) = 10 and the first factorial divisible by 10 is 5!, so a(4) = 5.

Mathematica

nn=80; With[{f=Range[nn]!, trs=Accumulate[Range[nn]]}, Flatten[ Table[ Position[f, _?(Divisible[#, trs[[n]]]&), {1}, 1], {n, nn}]]] (* Harvey P. Dale, Nov 18 2013 *)

Prog

(PARI) t(n)=n*(n+1)/2;
for (n=1, 50, c=1; while (c!%t(n)!=0, c++); print1(c", "));
(Python)
from itertools import count
from collections import Counter
from sympy import factorint, multiplicity
def A085779(n):
    m = 1
    f = Counter(factorint(n))+Counter(factorint(n+1))
    f[2] -= 1
    for p, e in f.items():
        if e<=p:
            m = max(m, e*p)
        else:
            c, a = 0, max(p, (e*(p-1)//p**2)*p)
            b = a-p
            while b>0:
                c += (b:=b//p)
            for k in count(a, p):
                c += multiplicity(p, k) if p>2 else (~k&k-1).bit_length()
                if c+k+p > e:
                    m = max(m, p*(max(k, e-c) if e>=p else e+1-c))
                    break
    return m # Chai Wah Wu, Feb 26 2026

Crossrefs

Cf. A000217, A002034.

Sequence in context: A141424 A069902 A335568 * A078936 A293702 A061794

Adjacent sequences: A085776 A085777 A085778 * A085780 A085781 A085782

Keyword

nonn

Author

Jon Perry, Jul 23 2003

Extensions

Extended by Ray Chandler, Jun 06 2008

Status

approved