OFFSET
0,2
COMMENTS
The sequence is well defined as 18*4^m has exactly m Achilles divisors. - David A. Corneth, Mar 17 2026
LINKS
David A. Corneth, Table of n, a(n) for n = 0..10000 (first 847 terms from Michael De Vlieger)
David A. Corneth, PARI program.
EXAMPLE
1 has no Achilles number divisors, so a(0) = 1.
72 = A052486(1), so a(1) = 72.
216 has divisors 72 = A052486(1) and 108 = A052486(2), and there are no smaller numbers that are divisible by exactly two Achilles numbers, so a(2) = 216.
From Michael De Vlieger, Mar 16 2026: (Start)
Table of n, a(n) for select n:
n a(n)
--------------------------------------------------------------------
1 s(12) = t(15) = 72 = 2^3 * 3^2
2 s(23) = t(23) = 216 = 2^3 * 3^3
3 s(34) = t(30) = 432 = 2^4 * 3^3
4 s(61) = t(44) = 1296 = 2^4 * 3^4
5 s(49) = t(37) = 864 = 2^5 * 3^3
6 s(155) = t(76) = 7200 = 2^5 * 3^2 * 5^2
16 s(1903) = t(277) = 884736 = 2^15 * 3^3
31 s(2336) = t(307) = 1323000 = 2^3 * 3^3 * 5^3 * 7^2
49 s(3690) = t(381) = 3240000 = 2^6 * 3^4 * 5^4
141 s(76156) = t(1337) = 1280664000 = 2^6 * 3^3 * 5^3 * 7^2 * 11^2
172 s(547790) = t(2744) = 2^5 * 3^4 * 5^2 * 7^2 * 11^2 * 13^2
348 t(6966) = 2^5 * 3^4 * 5^2 * 7^2 * 11^2 * 13^2 * 17^2 (End)
MATHEMATICA
Block[{a, r, s, fQ}, a[_] := 0; r = 0; s = With[{nn = 2^24}, Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3] } ] ]; fQ[x_] := And[Min[#] > 1, GCD @@ # == 1] &[FactorInteger[x][[;; , -1]] ]; Monitor[Do[If[a[#2] == 0, Set[a[#2], #1]; If[#2 > r, Set[r, #2]]] & @@ {#, DivisorSum[#, 1 &, fQ]} &[s[[n] ] ], {n, Length[s]}], n]; {1}~Join~TakeWhile[Array[a, r], # > 0 &] ] (* Michael De Vlieger, Mar 16 2026 *)
PROG
(Magma) ah:=func<n|n ne 1 and forall{i:i in [1..#Factorisation(n)] |Factorisation(n)[i][2] gt 1} and Gcd([Factorisation(n)[i][2] :i in [1..#Factorisation(n)]]) eq 1>; a:=[]; for n in [0..37] do k:=1; while #[d:d in Divisors(k)| ah(d)] ne n do k:=k+1; end while; Append(~a, k); end for; a;
(PARI) is(n) = my(f=factor(n)[, 2]); n>9 && vecmin(f)>1 && gcd(f)==1; \\ A052486
a(n) = my(k=1); while (sumdiv(k, d, is(d)) != n, k++); k; \\ Michel Marcus, Dec 13 2022
(PARI) \\ See PARI link \\ David A. Corneth, Mar 17 2026
CROSSREFS
KEYWORD
nonn
AUTHOR
Marius A. Burtea, Dec 04 2022
EXTENSIONS
Name edited by Peter Munn, Mar 16 2026
STATUS
approved