A350150 a(1)=1; thereafter a(n+1) is the smallest unused number k such that d(k) and d(a(n)) are coprime, where d is the divisor counting function A000005.

1, 2, 4, 3, 9, 5, 16, 6, 25, 7, 36, 8, 49, 10, 64, 11, 81, 12, 625, 13, 100, 14, 121, 15, 144, 17, 169, 19, 196, 21, 225, 22, 256, 23, 289, 24, 324, 26, 361, 27, 400, 29, 441, 30, 484, 31, 529, 33, 576, 34, 676, 35, 729, 18, 1024, 20, 1296, 28, 2401, 32, 4096

Offset: 1

Author

David James Sycamore, Dec 16 2021

Keywords

• nonn

Comments

A permutation of the positive integers. Identical to A137442 until a(18), with some terms in common thereafter. Numbers with the same number of divisors appear in their natural order, e.g. primes; d(p)=2, odd squarefree semiprimes; d(p*q) = 4, etc.

From Michael De Vlieger, Dec 16 2021: (Start)

a(2n+1) is square and d(a(2n+1)) odd. Let a(2n+1) constitute an "alpha" ray in scatterplot.

d(a(2n)) is even. Let a(2n) constitute a "beta" ray in scatterplot.

The occasion of d(a(2n)) = 6 induces a "flare" phase in the sequence, evident in scatterplot. The following term a(2n+1) is forced to have d(a(2n+1)) congruent to 1 or 5 (mod 6).

There are 5 flare-phases in the scatterplot associated with the occasion of d(a(2n)) = 6:

(I) a(18) = 12, a(19) = 625. The latter term interrupts what had been thereto and thereafter a series of square a(2n+1).

(II) a(54..61);

(III) 144..169 where a(k) with k in {152, 158, 164, 166} have d(a(k)) =/= 6, a characteristic common to subsequent phases;

(IV) 686..849;

(V) 11664..15515. (End)

Examples

			a(2) = 2 because d(1) = 1, d(2) = 2 and gcd(1,2) = 1.
a(3) cannot be 3 since d(2) = d(3) = 2, but gcd(d(2),d(4)) = gcd(2,3) = 1, so a(3) = 4.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log-log scatterplot of a(n) for n = 1..10^5 showing maxima in red and minima in blue. The golden line indicates the smallest missing number. We label terms that begin and end a flare phase among squares in alpha that comprise many records, and same in beta that comprise nearly all local minima.
• Michael De Vlieger, Log-log scatterplot of a(n) for n = 1..512 annotating a(n) in black above and d(a(n)) in red below the point. The plot illustrates bifurcation of the sequence, with a(2n) such that d(a(2n)) is even in trajectory beta below, while square a(2n+1) such that d(a(2n+1)) is odd in trajectory alpha above. Terms a(2n) such that d(a(2n)) = 6 appear in blue, while a(2n+1) such that d(a(2n)) = 6 appear in red. The faint green line below represents d(a(n)) for reference.

Crossrefs

Cf. A000005, A000027, A000040, A000290, A030515, A046388, A048691, A137442.

Programs

Mathematica

a[1] = 1; a[n_] := a[n] = Module[{k = 2, s = Array[a, n - 1], d = DivisorSigma[0, a[n - 1]]}, While[MemberQ[s, k] || ! CoprimeQ[d, DivisorSigma[0, k]], k++]; k]; Array[a, 100] (* Amiram Eldar, Dec 16 2021 *)

Extensions

More terms from Amiram Eldar, Dec 16 2021