A128556 -id:A128556 - cp's OEIS frontend

A354960 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that is a multiple of the number of proper divisors of a(n-1).

1, 2, 3, 4, 6, 9, 8, 12, 5, 7, 10, 15, 18, 20, 25, 14, 21, 24, 28, 30, 35, 27, 33, 36, 16, 32, 40, 42, 49, 22, 39, 45, 50, 55, 48, 54, 56, 63, 60, 11, 13, 17, 19, 23, 26, 51, 57, 66, 70, 77, 69, 72, 44, 65, 75, 80, 81, 52, 85, 78, 84, 88, 91, 87, 90, 99, 95, 93, 96, 110, 98, 100, 64, 102, 105, 112

Offset: 1

Scott R. Shannon, Jul 23 2022

The terms are concentrated along numerous lines, some of which curve upward while others curve downward. See the first linked image. Surprisingly these lines are not shared by terms which are a multiple of a given proper divisor count, but predominantly by terms sharing a certain prime factor. See the second linked image.
The sequence is conjectured to be a permutation of the positive integers although it may take an extremely large number of terms for the primes to appear; e.g., 263 has not occurred after 500000 terms. Also although the vast majority of primes will appear in their natural order, some may not; e.g., a(455) = 840, which has 31 proper divisors, so a(456) = 31, and then a(457) = 29.
In the first 500000 terms the only fixed points beyond the first two are 3, 4, 1159, 1207. It is possible that no more exist, although this is unknown.

			a(3) = 3 as a(2) = 2 which has one proper divisor, and 2 is the smallest unused multiple of 1.
a(5) = 6 as a(4) = 4 which has two proper divisors, and 6 is the smallest unused multiple of 2.
a(9) = 5 as a(8) = 12 which has five proper divisors, and 5 is the smallest unused multiple of 5.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log-log scatterplot of a(n), n - 1..2^16, showing records in red, local minima in blue, fixed points in gold, primes in green, and prime powers in cyan.
Michael De Vlieger, Log-log scatterplot of a(n) n = 1..2^12, with a color function indicating tau(a(n-1))-1.
Scott R. Shannon, Image of the first 50000 terms. The green line is y = n.
Scott R. Shannon, Image of the first 50000 terms with color. Terms containing prime factors 17, 13, 11, 7, 5, 3, 2 are colored purple, blue, green, yellow, orange, red, and white respectively. All other terms are colored gray.
Scott R. Shannon, Image of the first 500000 terms.

Cf. A128556 (all divisors), A339793, A000120, A032741, A005179, A027751, A335372.

nn = 76; c[] = False; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; u = 2; Do[k = DivisorSigma[0, a[n - 1]] - 1; m = 1 + Floor[u/k]; While[c[m k], m++]; a[n] = m k; c[m k] = True; If[a[n] == u, While[c[u], u++]], {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger, Jul 24 2022 *)

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.

Original entry on oeis.org

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

Leroy Quet, Mar 10 2007

nonn

This sequence is a permutation of the positive integers.

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

			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.

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.

Cf. A000005, A128556, A358820 (inverse).

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

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 *)

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

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

cp's OEIS Frontend

A354960 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that is a multiple of the number of proper divisors of a(n-1).

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