A341340 a(n) is the least k such that the product of n+1 consecutive primes starting with the k-th prime is greater than or equal to the product of the following n primes.

1, 3, 4, 6, 8, 10, 12, 14, 17, 22, 24, 28, 31, 35, 40, 46, 51, 55, 59, 64, 72, 79, 83, 88, 94, 103, 113, 120, 126, 131, 138, 146, 158, 170, 180, 186, 190, 196, 204, 216, 240, 250, 258, 265, 272, 283, 293, 304, 318, 326, 333, 342, 375, 396, 403, 409, 416, 424

Offset

1,2

Links

Table of n, a(n) for n=1..58.

Darrell Plank, Products of Consecutive Primes (PDF)

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

Sequence in context: A204662 A135667 A156624 * A242932 A351863 A025201

Adjacent sequences: A341337 A341338 A341339 * A341341 A341342 A341343

Keyword

nonn

Author

Darrell Plank, Apr 25 2021

Status

approved