OFFSET
1,1
COMMENTS
Take the family of correlated prime-indexed conjectures appearing in A343249 - A343253, in which an alternative formula for the p-adic order of positive integers is proposed. There, the general p-indexed conjecture says that v_p(n), the p-adic order of n, is given by the formula: v_p(n) = log_p(n / L_p(k0, n)), where L_p(k0, n) is the lowest common denominator of the elements of the set S_p(k0, n) = {(1/n)*binomial(n, k), with 0 < k <= k0 such that k is not divisible by p}. Evidence suggests that the primality of p is a necessary condition in this general conjecture. So, if a composite number q is used instead of a prime p in the proposed formula for the p-adic (now, q-adic) order of n, the first counterexample (failure) is expected to occur for n = q * a(i), where i is the index of the smallest prime that divides q.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
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.
FORMULA
EXAMPLE
a(4) = 8 because the fourth prime number is 7, and the least power of a prime which is greater than 7 is 2^3 = 8.
MAPLE
f:= proc(n) local p, x;
p:= ithprime(n);
for x from p+1 do
if nops(numtheory:-factorset(x)) = 1 then return x fi
od
end proc:
map(f, [$1..100]); # Robert Israel, Aug 25 2024
MATHEMATICA
a[i_]:= Module[{j, k, N = 0, tab={}}, tab = Sort[Drop[DeleteDuplicates[Flatten[Table[ If[Prime[j]^k > Prime[i], Prime[j]^k], {j, 1, i+1}, {k, 1, Floor[Log[Prime[j], Prime[i+1]]]}]]], 1]]; N = Take[tab, 1][[1]]; N];
tabseq = Table[a[i], {i, 1, 100}];
PROG
(PARI)
A000015(n) = for(k=n, oo, if((1==k)||isprimepower(k), return(k)));
CROSSREFS
KEYWORD
nonn
AUTHOR
Dario T. de Castro, Jun 20 2021
STATUS
approved