A358820 -id:A358820 - cp's OEIS frontend

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

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

A358916 a(1) = 1. Thereafter a(n) is the least novel k != n such that A007947(k)|n.

Original entry on oeis.org

1, 4, 9, 2, 25, 3, 49, 16, 27, 5, 121, 6, 169, 7, 45, 8, 289, 12, 361, 10, 63, 11, 529, 18, 125, 13, 81, 14, 841, 15, 961, 64, 99, 17, 175, 24, 1369, 19, 117, 20, 1681, 21, 1849, 22, 75, 23, 2209, 32, 343, 40, 153, 26, 2809, 36, 275, 28, 171, 29, 3481, 30, 3721

Offset: 1

David James Sycamore, Dec 05 2022

nonn

In other words, a(1) = 1, then for n > 1, a(n) is the least number k, not occurring earlier, whose squarefree kernel (rad(k)) is a divisor of n.
A permutation of the positive integers. - Robert Israel, Dec 11 2022
From Michael De Vlieger, Dec 06 2022, corrected by Robert Israel, Dec 11 2022: (Start)
Some consequences of definition:
Prime n = p implies a(p) = p^2, comprising maxima.
n = 2p implies a(2p) = p, n = 4p implies a(4p) = 2p.
n = 2^e with e >= 1 implies a(2^e) = 2^(e+1) if e is odd, 2^(e-1) if e is even.
n = p^e with e >= 1 and p an odd prime implies a(n) = p^(e+1).
Composite squarefree 2n implies a(2n) = n, comprising minima.
gcd(n, n +/- 1) = 1 implies gcd(a(n), a(n +/- 1)) = 1.
Let K = rad(n); a(n) is an element of R_K, the list of K-regular numbers, 1 and those whose prime divisors are restricted to p | K. For example, if K = 6, then a(n) != n is in A003586, and if K = 10, then a(n) != n is in A003592. (End)

			a(5) = 25 because rad(25) = 5  and there is no smaller number not equal to 5 which has this property.

Robert Israel, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log log scatterplot of a(n), n = 1..2^12, highlighting and labeling primes in red, composite prime powers in gold and labeling in orange, composite squarefree in green, numbers that are products of composite prime powers in large light blue and labeling in blue, and all other numbers in dark blue. The first 36 terms are simply labeled in black.

Cf. A000005, A000040, A001248, A005117, A007947, A032741, A358820, A358971.

N:= 100: # for a(1)..a(N)
R:= map(NumberTheory:-Radical, [$1..N^2]):
A[1]:= 1:
Agenda:= [$2..N^2]:
for n from 2 to N do
  if isprime(R[n]) then
    if R[n] = 2 and padic:-ordp(n,2)::even then A[n]:= n/2
    else A[n]:= R[n]*n
    fi;
    if A[n] <= N then Agenda:= subs(A[n]=NULL,Agenda) fi;
    next
  fi;
  found:= false;
  for j from 1 to nops(Agenda) do
    x:= Agenda[j];
    if x <> n  and n mod R[x] = 0 then
      A[n]:= x; Agenda:= subsop(j=NULL,Agenda); found:= true; break
    fi
  od;
  if not found then break fi;
od:
convert(A,list); # Robert Israel, Dec 11 2022

nn = 120; c[] = False; f[n] := f[n] = Times @@ FactorInteger[n][[All, 1]]; a[1] = 1; c[1] = True; u = 2; Do[Which[PrimeQ[n], k = n^2, PrimePowerQ[n], Set[{p, k}, {f[n], 1}]; While[Nand[! c[p^k], p^k != n], k++]; k = p^k, True, k = u; While[Nand[! c[k], k != n, Divisible[n, f[k]]], k++]]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Dec 06 2022 *)

For n = p^k, where p is prime and k >= 1, a(n) = p^(k+1). In particular, a(p) = p^2 (records).

More terms from Michael De Vlieger, Dec 07 2022

A359036 a(1) = 1. Thereafter a(n) is the least unused k distinct from n such that d(k) = d(n), where d is the divisor counting function, A000005.

Original entry on oeis.org

1, 3, 2, 9, 7, 8, 5, 6, 4, 14, 13, 18, 11, 10, 21, 81, 19, 12, 17, 28, 15, 26, 29, 30, 49, 22, 33, 20, 23, 24, 37, 44, 27, 35, 34, 100, 31, 39, 38, 42, 43, 40, 41, 32, 50, 51, 53, 80, 25, 45, 46, 63, 47, 56, 57, 54, 55, 62, 61, 72, 59, 58, 52, 729, 69, 70, 71, 75

Offset: 1

David James Sycamore, Dec 13 2022

nonn

A self-inverse permutation of the natural numbers.

			a(16) = 81 because this is the smallest unused k != 16, having the same number (5) of divisors as 16.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Scatterplot of a(n), n = 1..1024, showing primes in red, composite prime powers in gold, squarefree composites in green, products of composite prime powers in magenta, and other numbers in blue. Powerful numbers are labeled unless they are less than 32.
Index entries for sequences that are permutations of the natural numbers

Cf. A000005, A005179, A358820.

Block[{a, c, k, t, u, nn}, nn = 68; c[] = False; a[1] = 1; c[1] = True; u = 2; Do[Set[{k, t}, {u, DivisorSigma[0, n]}]; While[Or[c[k], k == n, t != DivisorSigma[0, k]], k++]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] ] (* _Michael De Vlieger, Dec 20 2022 *)

from functools import lru_cache
from sympy import divisor_count
from itertools import count, islice
@lru_cache(maxsize=None)
def d(n): return divisor_count(n)
def agen():
    mink, seen = 2, {1}
    yield 1
    for n in count(2):
        k = mink
        while k == n or k in seen or d(k) != d(n): k += 1
        while mink in seen: mink += 1
        yield k
        seen.add(k)
print(list(islice(agen(), 68))) # Michael S. Branicky, Dec 13 2022

a(a(n)) = n.

a(prime(k)^(p-1)) = prime(k-1)^(p-1) for even k and prime p else prime(k+1)^(p-1). - Michael De Vlieger, Dec 20 2022

More terms from Michael S. Branicky, Dec 13 2022

cp's OEIS Frontend

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.

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A358916 a(1) = 1. Thereafter a(n) is the least novel k != n such that A007947(k)|n.

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A359036 a(1) = 1. Thereafter a(n) is the least unused k distinct from n such that d(k) = d(n), where d is the divisor counting function, A000005.

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