A396380 Number of integers m in {n, n+1, n+2, ..., n+A396377(n)} so that m divides the product of the other integers.

1, 1, 2, 2, 1, 1, 2, 2, 1, 3, 1, 1, 3, 3, 2, 1, 2, 2, 1, 1, 3, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1

Offset

1,3

Comments

From a problem proposed by Paul Erdős (see link).

The smallest of these integers m is A396378(n).

Are there integers n with a(n) = 4, 5, ... or a(n) always <= 3 ?

a(n) = 3 for 7 consecutive values of n : from 3030 up to 3036.

The least integer with a(n) = 4 is n = 407049; a(n) < 5 for all n <= 5*10^8. - David Radcliffe, Jun 10 2026

Links

Table of n, a(n) for n=1..90.

Paul Erdős, Quelques problèmes de Théorie des Nombres, Monographies de l'Enseignement Mathématique, No. 6, pp. 81-135, L'Enseignement Mathématique, Université, Geneva, 1963, Problem 15, p. 90.

Example

a(6) = 1 because A396377(6) = 3 and in {6,7,8,9}, only 6 | 7*8*9.
a(8) = 2 because A396377(8) = 4 and in {8,9,10,11,12}, we get 8 | 9*10*11*12 and also 12 | 8*9*10*11.
a(10) = 3 because A396377(10) = 5 and in {10,11,12,13,14,15}, we get 10 | 11*12*13*14*15, then 12 | 10*11*13*14*15 and also 15 | 10*11*12*13*14.

Maple

c := proc(n)
local k, L, i, j, t, g, C;
for k from 1 do
    L := [seq(n+r, r=0..k)];
    C := 0;
    for i from 1 to k+1 do
        t := L[i];
        for j from 1 to k+1 do
            if j <> i then
                g := gcd(t, L[j]);
                t := iquo(t, g);
            end if;
        end do;
        if t = 1 then
            C := C+1;
        end if;
    end do;
    if C > 0 then
        return C;
    end if;
end do;
end proc:
seq(c(n), n=1..50);

Prog

(Python)
def A396380(n: int) -> int:
    k = 1
    p = n * (n + 1)
    while (count := sum(p % (j*j) == 0 for j in range(n, n+k+1))) == 0:
        k += 1
        p *= n + k
    return count # David Radcliffe, Jun 10 2026

Crossrefs

Cf. A396377, A396378, A396379, A396950.

Sequence in context: A372673 A271381 A393576 * A086247 A116452 A219966

Adjacent sequences: A396377 A396378 A396379 * A396381 A396382 A396383

Keyword

nonn

Author

Bernard Schott, Jun 04 2026

Status

approved