OFFSET
1,2
LINKS
EXAMPLE
If n = 2 and k = 2 then pLow(2,2) = 3*5*7=105 and pHigh(2,2)=11*13=143 so pHigh(2,2) > pLow(2,2). On the other hand, we have pLow(2,3) = 5*7*11 = 385 and pHigh(2,3) = 13 * 17 = 221 so pHigh(2,3) < pLow(2,3). The conjecture is that pHigh(2,i) < pLow(2,i) for all i >=3 so a(2) = 3.
For another example, pLow(4,5)=11*13*17*19*23=1062347 and pHigh(4,5)=29*31*37*41=1363783 so plow(4,5)<pHigh(4,5) but pLow(4,6)=13*17*19*23*29=2800733 and pHigh(4,6)=31*37*41*43=2022161 so pLow(4,5) < pHigh(4,5) while pLow(4,6) > pHigh(4,6) and the conjecture says that pLow(4,i) > pHigh(4,i) for all i >= 6 so a(4)=6.
MATHEMATICA
products[m_, n_, i_] := {
Times @@ Table[Prime[j], {j, i, i + m - 1}],
Times @@ Table[Prime[j], {j, i + m, i + m + n - 1}]};
lowLarger[i_, m_, n_] :=
Module[{p = products[m, n, i]}, p[[1]] > p[[2]]];
breakPosition[n_] :=
Module[{i = 0},
For[i = 1, ! lowLarger[i, n + 1, n], i++, ]; i];
breakPosition /@ Range[1, 30]
PROG
(PARI) a(n) = my(k=1); while (prod(i=k, n+k, prime(i)) < prod(i=n+k+1, k+2*n, prime(i)), k++); k; \\ Michel Marcus, Jan 27 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Darrell Plank, Apr 25 2021
STATUS
approved