cp's OEIS Frontend

A357601 For n a power of 2, a(n) = n; otherwise, if 2^m is the greatest power of 2 not exceeding n and if k = n-2^m, then a(n) is the smallest number having d(a(k))+1 divisors which has not occurred earlier (d is the divisor counting function A000005).

%I A357601 #22 Jun 29 2025 16:43:34
%S A357601 1,2,3,4,5,9,25,8,7,49,121,6,169,10,14,16,11,289,361,15,529,21,22,81,
%T A357601 841,26,27,625,33,2401,14641,32,13,961,1369,34,1681,35,38,28561,1849,
%U A357601 39,46,83521,51,130321,279841,12,2209,55,57,707281,58,923521,1874161,18
%N A357601 For n a power of 2, a(n) = n; otherwise, if 2^m is the greatest power of 2 not exceeding n and if k = n-2^m, then a(n) is the smallest number having d(a(k))+1 divisors which has not occurred earlier (d is the divisor counting function A000005).
%C A357601 Based on a similar recursion to that which produces the Doudna sequence, A005940.  Conjectured to be permutation of the positive integers in which the primes appear in natural order.
%H A357601 Rémy Sigrist, <a href="/A357601/b357601.txt">Table of n, a(n) for n = 1..10000</a>
%H A357601 Michael De Vlieger, <a href="/A357601/a357601.png">Fan style binary tree</a> showing a(n), n = 1..2^13, with a color function representing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and magenta, where the latter also represents powerful numbers that are not prime powers.
%H A357601 Michael De Vlieger, <a href="/A357601/a357601_1.png">Log log scatterplot of a(n)</a>, n = 1..10000, using the same color function as immediately above.
%H A357601 Rémy Sigrist, <a href="/A357601/a357601.gp.txt">PARI program</a>
%F A357601 a(2^n + 1) = prime(n + 1); n >= 0
%F A357601 A000005(a(n)) = A063787(n). - _Rémy Sigrist_, Oct 06 2022
%e A357601 a(9)=7 because k=1, and a(1)=1, which has 1 divisor, so we are looking for the smallest number not yet seen which has 2 divisors. This must be 7 because 2,3,5 have occurred already.
%t A357601 nn = 70; kk = 2^20; c[_] = False; to = Map[DivisorSigma[0, #] &, Range[kk]^2]; t = DivisorSigma[0, Range[kk]]; Do[Set[{m, k}, {1, n - 2^Floor[Log2[n]]}]; If[k == 0, Set[{a[n], c[n]}, {n, True}], d = 1 + DivisorSigma[0, a[k]]; If[OddQ[d], While[Nand[! c[m^2], to[[m]] == d], m++]; Set[{a[n], c[#]}, {#, True}] &[m^2], While[Nand[! c[m], t[[m]] == d], m++]; Set[{a[n], c[m]}, {m, True}]] ], {n, nn}]; Array[a, nn] (* _Michael De Vlieger_, Oct 05 2022 *)
%o A357601 (PARI) \\ See Links section.
%Y A357601 Cf. A000005, A005940, A063787.
%K A357601 nonn
%O A357601 1,2
%A A357601 _David James Sycamore_, Oct 05 2022