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%I #17 Aug 29 2021 11:59:36
%S 1,60,210,572,910,935,936,1155,1197,1309,1820,1848,2030,2090,2142,
%T 2145,2730,2871,2964,3315,3400,3857,3927,3978,4028,4080,4185,4199,
%U 4329,4550,4669,4675,4845,4884,5320,5423,5681,5742,5950,5985,6006,6032,6235,6426
%N Numbers m that divide binomial(m*(m+1), m+1)/m^2.
%C Equivalently, numbers m such that m divides binomial(m*(m+1),m). - _David Wasserman_, Sep 13 2004
%C 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
%H Robert Israel, <a href="/A082529/b082529.txt">Table of n, a(n) for n = 1..10000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Kummer%27s_theorem">Kummer's theorem</a>
%e m = 2: C(6, 3)/4 = 5, which is not divisible by 2, so 2 is not in the sequence.
%e m = 60: C(60*61, 61)/3600 has 130 digits and is divisible by 60, so 60 is in the sequence.
%p filter:= proc(n) local F,i,j, p,A,B,c,t, target;
%p F:= ifactors(n)[2];
%p for i from 1 to nops(F) do
%p p:= F[i][1];
%p target:= F[i][2];
%p t:= 0;
%p A:= convert(n,base,p);
%p B:= convert(n^2,base,p);
%p c:= 0;
%p for j from 1 to nops(A) while t < target do
%p if c + A[j] + B[j] >= p then
%p t:= t+1;
%p c:= 1;
%p else c:= 0
%p fi;
%p od;
%p if t >= target then next fi;
%p for j from nops(A)+1 to nops(B) while t < target do
%p if c + B[j] >= p then
%p t:= t+1;
%p c:= 1;
%p else break
%p fi;
%p od;
%p if t < target then return false fi
%p od;
%p true
%p end proc:
%p select(filter, [$1..10000]); # _Robert Israel_, May 10 2020
%o (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))
%K nonn
%O 1,2
%A _Benoit Cloitre_, Apr 30 2003
%E More terms from _David Wasserman_, Sep 13 2004
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