A292815 Nonsquarefree numbers n = p_1^s_1...p_m^s_m (m > 1) such that (p_i^s_i - 1) | n-1 for all 0 < i <= m.

12025, 13833, 35425, 54145, 89425, 187461, 203841, 321201, 499681, 501025, 566401, 595441, 717025, 784225, 856801, 877825, 965497, 1080801, 1165537, 1299961, 1335961, 1439425, 1566891, 1658385, 1935025, 2058049, 2514337, 2668225, 2817001, 3078361

Offset

1,1

Comments

If squarefree numbers were accepted, then Carmichael numbers would be included. - Michel Marcus, Mar 13 2018

Links

Table of n, a(n) for n=1..30.

Maple

isA292815 := proc(n)
    local pf, pfs;
    pfs := ifactors(n)[2] ;
    if nops(pfs) = 1 or issqrfree(n) then
        return false;
    end if;
    for pf in pfs do
        if modp(n-1, op(1, pf)^op(2, pf)-1) > 0 then
            return false;
        end if;
    end do:
    true ;
end proc:
for n from 1 do
    if isA292815(n) then
        print(n) ;
    end if;
end do: # R. J. Mathar, May 02 2018

Mathematica

fa[n_] := fa[n] = FactorInteger[n];
free[n_] := Union[Table[fa[n]〚i, 2〛, {i, Length[fa[n]]}]] == {1}
tes1[n_] := Union@Table[IntegerQ[(n - 1)/(fa[n][[i, 1]]^fa[n][[i, 2]] - 1)], {i, Length[fa[n]]}] == {True};
Select[1 + Range[3300200], ! free[#] && Length@fa[#] > 1 && tes1[#] &]

Prog

(PARI) isok(n) = {my(f = factor(n)); if ((#f~ > 1) && ! issquarefree(n), for (k=1, #f~, if ((n-1) % (f[k, 1]^f[k, 2] -1), return (0)); ); return (1); ); return (0); } \\ Michel Marcus, Mar 05 2018

Crossrefs

Cf. A013929, A087442, A002997 (Carmichael numbers).

Sequence in context: A287046 A235308 A064966 * A252304 A251680 A247688

Adjacent sequences: A292812 A292813 A292814 * A292816 A292817 A292818

Keyword

nonn

Author

José María Grau Ribas, Sep 24 2017

Status

approved