A082529 Numbers m that divide binomial(m*(m+1), m+1)/m^2.

1, 60, 210, 572, 910, 935, 936, 1155, 1197, 1309, 1820, 1848, 2030, 2090, 2142, 2145, 2730, 2871, 2964, 3315, 3400, 3857, 3927, 3978, 4028, 4080, 4185, 4199, 4329, 4550, 4669, 4675, 4845, 4884, 5320, 5423, 5681, 5742, 5950, 5985, 6006, 6032, 6235, 6426

Offset

1,2

Comments

Equivalently, numbers m such that m divides binomial(m*(m+1),m). - David Wasserman, Sep 13 2004

Numbers m such that, whenever m is divisible by p^k for prime p, there are at least k carries in the base-p addition of m and m^2. - Robert Israel, May 10 2020

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Wikipedia, Kummer's theorem

Example

m = 2: C(6, 3)/4 = 5, which is not divisible by 2, so 2 is not in the sequence.
m = 60: C(60*61, 61)/3600 has 130 digits and is divisible by 60, so 60 is in the sequence.

Maple

filter:= proc(n) local F, i, j, p, A, B, c, t, target;
   F:= ifactors(n)[2];
   for i from 1 to nops(F) do
     p:= F[i][1];
     target:= F[i][2];
     t:= 0;
     A:= convert(n, base, p);
     B:= convert(n^2, base, p);
     c:= 0;
     for j from 1 to nops(A) while t < target do
       if c + A[j] + B[j] >= p then
         t:= t+1;
         c:= 1;
       else c:= 0
       fi;
     od;
     if t >= target then next fi;
     for j from nops(A)+1 to nops(B) while t < target do
       if c + B[j] >= p then
         t:= t+1;
         c:= 1;
       else break
       fi;
     od;
     if t < target then return false fi
   od;
   true
end proc:
select(filter, [$1..10000]); # Robert Israel, May 10 2020

Prog

(PARI) count = 0; n = 0; while (count < 50, n = n + 1; works = 1; f = factor(n); for (k = 1, matsize(f)[1], p = f[k, 1]; pow = 0; for (i = 1, n, num = n*n + i; while (num%p == 0, pow = pow + 1; num = num/p); num = i; while (num%p == 0, pow = pow - 1; num = num/p)); if (pow < f[k, 2], works = 0)); if (works, print(n); count = count + 1))

Crossrefs

Sequence in context: A292223 A112827 A181333 * A126248 A224335 A068628

Adjacent sequences: A082526 A082527 A082528 * A082530 A082531 A082532

Keyword

nonn

Author

Benoit Cloitre, Apr 30 2003

Extensions

More terms from David Wasserman, Sep 13 2004

Status

approved