A336027 The least k such that i*k + 1 is a product of i different primes for i = 1..n.

1, 10, 268, 7576, 652726, 913180816

Offset

1,2

Comments

Since for k = 5000000000420503488, i*k+1 is a product of i different primes for i = 1..7, a(7) <= 5000000000420503488.

a(n) is the least parameter k such that equation tau(x^k) = x has at least A005425(n) solutions in positive integers.

Links

Table of n, a(n) for n=1..6.

Vladimir Letsko, Problem K7 (in Russian).

Example

268 is in the sequence because 268 + 1 is prime, 2*268 + 1 is a product of 2 different primes, 3*268 + 1 is a product of 3 different primes, and 268 is the least number with such properties.

Maple

nn := 1; for kk to 6 do
n := nn; do n := nextprime(n); f := true;
for k from 2 to kk do a := k*(n-1)+1; if not IsSquareFree(a) or nops(ifactors(a)[2]) <> k then f := false; break end if end do;
if f then nn := n-1; print(nn); break end if end do end do

Prog

(PARI) isok(k, n) = {for (i=1, n, if ((omega(i*k+1) != i) || (bigomega(i*k+1) != i), return (0)); ); return (1); }
a(n) = {my(k=1); while(!isok(k, n), k++); k; } \\ Michel Marcus, Jul 15 2020

Crossrefs

Cf. A000005, A005425, A336028.

Sequence in context: A322564 A055408 A245983 * A166811 A089906 A294515

Adjacent sequences: A336024 A336025 A336026 * A336028 A336029 A336030

Keyword

nonn,more

Author

Vladimir Letsko, Jul 05 2020

Status

approved