A128555 a(n) = the smallest positive multiple of d(n) that does not occur earlier in the sequence, where d(n) is the number of positive divisors of n.

1, 2, 4, 3, 6, 8, 10, 12, 9, 16, 14, 18, 20, 24, 28, 5, 22, 30, 26, 36, 32, 40, 34, 48, 15, 44, 52, 42, 38, 56, 46, 54, 60, 64, 68, 27, 50, 72, 76, 80, 58, 88, 62, 66, 78, 84, 70, 90, 21, 96, 92, 102, 74, 104, 100, 112, 108, 116, 82, 120, 86, 124, 114, 7, 128, 136, 94, 126

Offset: 1

Author

Leroy Quet, Mar 10 2007

Keywords

• nonn

Comments

This sequence is a permutation of the positive integers.

a(2^(p+1)) = p, where p is prime. - Michael De Vlieger, Dec 07 2022

Examples

			8 has 4 positive divisors. So a(8) is the smallest positive multiple of 4 that has yet to appear in the sequence. 4 and 8 occur among the first 7 terms of the sequence, but 12 does not. So a(8) = 12.
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, composite prime powers (in A246547) in gold, squarefree composites (A120944) in green, numbers neither prime power nor squarefree (A126706) in blue, with numbers in A286708 in large light blue. Highlighted in light green are squarefree composites divisible by 6.

Crossrefs

Cf. A000005, A128556, A358820 (inverse).

Programs

Maple

A128555 := proc(nmin) local a,n,d,k ; a := [1,2] ; while nops(a) < nmin do n := nops(a)+1 ; d := numtheory[tau](n) ; k := 1; while k*d in a do k := k+1 ; od; a := [op(a),k*d] ; od: RETURN(a) ; end: A128555(80) ; # R. J. Mathar, Oct 09 2007

Mathematica

a = {1}; Do[AppendTo[a, Min[Complement[Range[Max[a] + 1]*DivisorSigma[0,n], a]]], {n, 2, 68}]; a (* Ivan Neretin, May 03 2015 *)
nn = 120; c[] = False; q[] = 1; Do[d = DivisorSigma[0, n]; m = q[d]; While[c[m d], m++]; If[m == q[d], While[c[m d], m++]; q[d] = m]; Set[{a[n], c[m d]}, {m d, True}], {n, nn}]; Array[a, nn] (* Michael De Vlieger, Dec 07 2022 *)

Python

from itertools import count, islice
from sympy import divisor_count as d
def agen():
    seen = set()
    for n in count(1):
        dn = d(n)
        m = dn
        while m in seen: m += dn
        yield m
        seen.add(m)
print(list(islice(agen(), 68))) # Michael S. Branicky, Dec 08 2022

Extensions

More terms from R. J. Mathar, Oct 09 2007