A394753 - OEIS

%I #15 May 13 2026 20:10:55

%S 1,2,3,5,4,8,5,10,11,18,6,14,15,16,26,7,18,32,20,21,36,8,23,41,24,44,

%T 25,68,26,49,9,53,29,55,30,58,31,96,32,64,65,33,10,69,35,71,115,36,76,

%U 78,38,81,131,39,83,84,40,11,145,88,42,91,156,43,96,97,44

%N Number of k <= m such that rad(k) | m, where rad = A007947 and m = A055932(n).

%C Analogous to A364225, which instead regards A025487, a proper subset of A055932.

%H Michael De Vlieger, <a href="/A394753/b394753.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A394753/a394753.png">Log log scatterplot of a(n)</a>, n = 1..2^14, with a color function showing primes in red, proper prime powers in gold, composite primorials in green, numbers neither squarefree nor powerful in blue, and powerful numbers that are not prime powers in purple.

%F a(n) = A010846(A055932(n)).

%F Length of row A055932(n) of A162306.

%F For numbers k in A055932 such that omega(k) = 1 (i.e., 2^m), a(k) = m, where omega = A001221.

%F For numbers k in A055932 such that omega(k) = 2 (i.e., k = A003586(i), 3-smooth k), a(k) = i.

%F For numbers k in A055932 such that omega(k) = 3 (i.e., k = A051037(j), 5-smooth k), a(k) = j, etc.

%t fQ[x_] := Or[IntegerQ[Log2[x]], And[EvenQ[x], Union[Differences[PrimePi[FactorInteger[x][[All, 1]] ] ] ] == {1}]]; a010846[x_] := Count[Range[x], _?(Divisible[x, Times @@ FactorInteger[#][[All, 1]] ] &) ]; Map[a010846, Select[Range[1600], fQ] ]

%t (* Alternative: load "theta" program from link at A369609, and function f from link at A376690 *)

%t Map[theta, Union@ Flatten@ f[6] ]

%Y Cf. A000010, A000079, A002110, A002473, A003586, A007947, A010846, A051037, A055932, A162306, A364225.

%K nonn

%O 1,2

%A _Michael De Vlieger_, May 06 2026