cp's OEIS Frontend

A380483 a(n) = least k such that A010846(6*prime(k)) = n, or -1 if no solution exists.

%I A380483 #21 Jul 21 2025 13:14:17
%S A380483 3,4,-1,-1,5,6,-1,7,8,9,-1,10,-1,12,13,14,16,17,19,21,23,24,25,29,31,
%T A380483 32,35,38,40,44,48,54,55,62,67,69,73,77,84,93,98,106,119,124,130,136,
%U A380483 151,165,173,184,191,211,219,232,243,270,296,310,328,343,378,399,422
%N A380483 a(n) = least k such that A010846(6*prime(k)) = n, or -1 if no solution exists.
%C A380483 Conjecture: n = 30 is the largest number for which a(n) = -1.
%H A380483 Michael De Vlieger, <a href="/A380483/b380483.txt">Table of n, a(n) for n = 18..280</a>
%H A380483 Michael De Vlieger, <a href="/A380483/a380483.png">Plot of terms k = p^a*q^b*r^c, primes p < q < r, in row a(n) of A162306</a>, with n = 18..58 and a(n) > -1, at (x,y,z) = (a,b,c). For a(n) there are n blocks in each diagram.
%e A380483 Let f = A010846.
%e A380483 For k >= 3 and m = 6*prime(k), f(k) = [k=3] + f(6) + Sum_{j=0..floor(log_3 m)} floor(log_2 m/3^j), with Iverson brackets. (This, since for k = 3, prime(k)^2 < 2*3*prime(k) < prime(k)^3, but for k > 3, 2*3*prime(k) < prime(k)^2.)
%e A380483 a(18) = 3 since 2*3*prime(3) = 30 is the smallest sphenic number k, and f(30) = A363061(3) = 18. This is to say that row 30 of A162306 = {m : rad(m) | 30} = {1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,27,30} has cardinality 18.
%e A380483 a(19) = 4 since 2*3*prime(4) = 42 and f(42) = 19. This is to say that row 30 of A162306 = {m : rad(m) | 42} = {1,2,3,4,6,7,8,9,12,14,16,18,21,24,27,28,32,36,42} has cardinality 19.
%e A380483 a(m) = -1, m = 20..21, since for k >= 3, there exists no solution to f(6*prime(k)) = m.
%e A380483 a(22) = 5 since 2*3*prime(5) = 66 and f(66) = 22.
%e A380483 a(23) = 6 since 2*3*prime(6) = 78 and f(78) = 23.
%e A380483 a(24) = = -1 since for k >= 3, there exists no solution to f(6*prime(k)) = 24.
%e A380483 a(25) = 7 since 2*3*prime(7) = 102 and f(102) = 25.
%e A380483 a(26) = 8 since 2*3*prime(8) = 114 and f(114) = 26.
%e A380483 a(27) = 9 since 2*3*prime(9) = 138 and f(138) = 27.
%e A380483 a(28) = = -1 since for k >= 3, there exists no solution to f(6*prime(k)) = 28.
%e A380483 a(29) = 10 since 2*3*prime(10) = 174 and f(174) = 29.
%e A380483 a(30) = = -1 since for k >= 3, there exists no solution to f(6*prime(k)) = 30.
%e A380483 Note: 2*3*prime(11) = 186; f(186) = f(174) = 29.
%e A380483 a(31) = 12 since 2*3*prime(12) = 222 and f(222) = 31, etc.
%t A380483 k = 3; Table[While[Set[t, Boole[k == 3] + 5 + Sum[Floor[1 + Log2[#/3^j]], {j, 0, Floor[Log[3, #]]}] &[6*Prime[k] ] ] < n, k++]; If[t == n, k, -1], {n, 18, 30}]
%Y A380483 Cf. A007304, A010846, A363061.
%K A380483 sign
%O A380483 18,1
%A A380483 _Michael De Vlieger_, Jul 08 2025