A134170 a(n) = the smallest positive integer which, expressed in the form d*q + r, satisfies q >= r for every d from 1 to n. In other words, when a(n) is divided by the numbers from 1 to n, the remainder is never more than the quotient.

1, 2, 3, 4, 10, 12, 21, 24, 36, 40, 60, 60, 84, 84, 112, 112, 144, 144, 180, 180, 240, 252, 308, 336, 336, 400, 432, 432, 504, 540, 540, 651, 660, 660, 792, 792, 792, 936, 936, 936, 1080, 1092, 1092, 1260, 1260, 1260, 1440, 1440, 1440, 1680

Offset

1,2

Comments

If a prospective term is at least k(k-1) for a fixed value k, then the criterion will be satisfied for all d less than or equal to k.

Note that a(n) >= n, otherwise quotient for n would be 0 and so condition on remainder would not be satisfied. - Michel Marcus, Sep 04 2013

Links

Table of n, a(n) for n=1..50.

Example

a(7)=21 because division by d=1 to 7 gives 21 r0, 10 r1, 7 r0, 5 r1, 4 r1, 3 r3 and 3 r0, respectively.

Maple

A134170 := proc(n)
    local a, wrks, d;
    for a from 1 do
        wrks := true;
        for d from 1 to n do
            if modp(a, d) > floor(a/d) then
                wrks := false;
                break;
            end if;
        end do:
        if wrks then
            return a;
        end if;
    end do:
end proc: # R. J. Mathar, Sep 04 2013

Prog

(PARI) isok(m, n) = {for (d = 1, n, if (m\d < m%d, return (0)); ); return (1); }
a(n) = {m = 1; while (! isok(m, n), m++); m; } \\ Michel Marcus, Sep 03 2013
(Sage)
def FindM(n):
    m=n-1
    test=False
    while not test:
        test=True
        m+=1
        for d in [1..n]:
            if Integer((m//d))<m.mod(d):
                test=False
    return m
[FindM(k) for k in [1..50]] # Tom Edgar, Sep 03 2013

Crossrefs

Sequence in context: A076079 A229593 A196007 * A276560 A339577 A049548

Adjacent sequences: A134167 A134168 A134169 * A134171 A134172 A134173

Keyword

easy,nonn

Author

Bryce Herdt (mathidentity(AT)yahoo.com), Jan 12 2008

Extensions

More terms added and incorrect conjecture removed by Michel Marcus, Sep 03 2013

Status

approved