A128555 - OEIS

%I #19 Dec 08 2022 11:03:52

%S 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,

%T 44,52,42,38,56,46,54,60,64,68,27,50,72,76,80,58,88,62,66,78,84,70,90,

%U 21,96,92,102,74,104,100,112,108,116,82,120,86,124,114,7,128,136,94,126

%N 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.

%C This sequence is a permutation of the positive integers.

%C a(2^(p+1)) = p, where p is prime. - _Michael De Vlieger_, Dec 07 2022

%H Ivan Neretin, <a href="/A128555/b128555.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A128555/a128555.png">Log log scatterplot of a(n)</a>, 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.

%e 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.

%p 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

%t a = {1}; Do[AppendTo[a, Min[Complement[Range[Max[a] + 1]*DivisorSigma[0,n], a]]], {n, 2, 68}]; a (* _Ivan Neretin_, May 03 2015 *)

%t 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 *)

%o (Python)

%o from itertools import count, islice

%o from sympy import divisor_count as d

%o def agen():

%o     seen = set()

%o     for n in count(1):

%o         dn = d(n)

%o         m = dn

%o         while m in seen: m += dn

%o         yield m

%o         seen.add(m)

%o print(list(islice(agen(), 68))) # _Michael S. Branicky_, Dec 08 2022

%Y Cf. A000005, A128556, A358820 (inverse).

%K nonn

%O 1,2

%A _Leroy Quet_, Mar 10 2007

%E More terms from _R. J. Mathar_, Oct 09 2007