OFFSET
1,3
COMMENTS
Conjecture: a(n) is the greatest power of a prime different from 2 that divides n.
LINKS
Dario T. de Castro, Table of n, a(n) for n = 1..1000
Dario T. de Castro, P-adic Order of Positive Integers via Binomial Coefficients, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 22, Paper A61, 2022.
EXAMPLE
For n = 15, a(15) = 5. To understand this result, consider the largest set S_2, which is the S_2(k0=15, 15). According to the definition, S_2(n, n) is the set of elements of the form (1/n)*binomial(n, k), where k goes from 1 to n, skipping the multiples of 2. The elements of S_2(15, 15) are: {1, 0, 91/3, 0, 1001/5, 0, 429, 0, 1001/3, 0, 91, 0, 7, 0, 1/15}, where the zeros were put pedagogically to identify the skipped terms, i.e., when k is divisible by 2. At this point we verify which of the nested subsets {1}, {1, 0}, {1, 0, 91/3}, {1, 0, 91/3, 0}, {1, 0, 91/3, 0, 1001/5},... will match for the first time the p-adic order’s formula. If k vary from 1 to 5 (instead of 15) we see that the lowest common denominator of the set S_2(5, 15) will be 15. So, L_2(5, 15) = 15 and the equation v_2(15) = log_2(15/15) yields a True result. Then we may say that a(15) = 5 specifically because 5 was the least k0.
MATHEMATICA
j = 1;
Nmax = 250;
Array[val, Nmax];
Do[val[i] = 0, {i, 1, Nmax}];
Do[flag = 0;
Do[If[(flag == 0 &&
Prime[j]^IntegerExponent[n, Prime[j]] ==
n/LCM[Table[
If[Divisible[k, Prime[j]], 1,
Denominator[(1/n) Binomial[n, k]]], {k, 1, k}] /.
List -> Sequence]), val[n] = k; flag = 1; , Continue], {k, 1,
n, 1}], {n, 1, Nmax}];
tabseq = Table[val[i], {i, 1, Nmax}];
PROG
(PARI) Lp(k, n, p) = {my(list = List()); for (i=1, k, if (i%p, listput(list, binomial(n, i)/n)); ); lcm(apply(denominator, Vec(list))); }
isok(k, n, v, p) = p^v == n/Lp(k, n, p);
a(n, p=2) = {my(k=1, v=valuation(n, p)); for (k=1, n, if (isok(k, n, v, p), return(k)); ); n; } \\ Michel Marcus, Apr 22 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Dario T. de Castro, Apr 09 2021
STATUS
approved