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

Author

Michel Lagneau, Jun 01 2015

Keywords

• nonn

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.

Examples

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

Links

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

Crossrefs

Cf. A001248.

Programs

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]]Harvey P. Dale, Mar 05 2020 *)

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

Formula

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