A256017 Least integer k > n such that all divisors of n are exactly the first divisors of k in increasing order.

2, 4, 9, 8, 25, 18, 49, 16, 27, 50, 121, 156, 169, 98, 75, 32, 289, 54, 361, 100, 147, 242, 529, 696, 125, 338, 81, 196, 841, 930, 961, 64, 363, 578, 245, 1332, 1369, 722, 507, 1640, 1681, 294, 1849, 484, 2115, 1058, 2209, 2544, 343, 250, 867, 676, 2809, 162

Offset

1,1

Comments

Gcd(n,k) = n and lcm(n,k) = k; a(n)/n is a prime number.

a(n) is alternatively even and odd for n >= 2. A001248 (squares of primes) is a subsequence.

Links

Michel Lagneau and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 600 terms from Michel Lagneau)

Formula

For p prime, a(p) = p^2. - Michel Marcus, Jun 02 2015

Example

a(6)=18 because the divisors of 6 = {1,2,3,6} are the first divisors of 18 = {1,2,3,6,9,18}.

Maple

with(numtheory):nn:=100:
for n from 1 to nn do:
lst0:={}:x:=divisors(n):n0:=nops(x):ii:=0:
  for k from 1 to 10^8 while (ii=0) do:
   y:=divisors(k):n1:=nops(y):lst1:={}:
    if n<>k and n1>=n0 then
    for i from 1 to n0 do
    lst0:=lst0 union {x[i]}:
    od:
     for j from 1 to n0 do
     lst1:=lst1 union {y[j]}:
     od:
     if lst0=lst1
     then
     printf(`%d, `, k):ii:=1:
     else fi:fi:
     od:
    od:

Mathematica

a[n_]:=If[n==1, 2, Block[{k= 2*n, f, d, m}, f = FactorInteger @n; If[1 == Length@f, f[[1, 1]]^(1 + f[[1, 2]]), d = Divisors@ n; m = Length@ d; While[ Take[ Divisors@ k, m] != d, k += n]; k]]]; Array[a, 1000] (* Giovanni Resta, Jun 01 2015 *)
adn[n_]:=Module[{d=Divisors[n], k=n+1, len}, len=Length[d]; While[Length[ Divisors[ k]]<len||Take[Divisors[k], len]!=d, k++]; k]; Array[adn, 60] (* Harvey P. Dale, Mar 05 2020 *)

Prog

(PARI) okd(k, d) = {my(dk = divisors(k)); (#dk >= #d) && (d == vector(#d, i, dk[i])); }
a(n) = {my(k = n+1); my(d = divisors(n)); while (! okd(k, d), k++); k; } \\ Michel Marcus, Jun 02 2015

Crossrefs

Cf. A001248.

Sequence in context: A033149 A131094 A129598 * A125752 A103147 A340169

Adjacent sequences: A256014 A256015 A256016 * A256018 A256019 A256020

Keyword

nonn

Author

Michel Lagneau, Jun 01 2015

Status

approved