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

1,2

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

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.

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A000005, A128556, A358820 (inverse).

Sequence in context: A355435 A155487 A356430 * A039874 A193991 A368523

Adjacent sequences: A128552 A128553 A128554 * A128556 A128557 A128558

Keyword

nonn

Author

Leroy Quet, Mar 10 2007

Extensions

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

Status

approved