A394555 - OEIS

A394555

Powerful k such that rad(Sum({m <= k : rad(m) = rad(k)})) = rad(k), where rad = A007947.

%I #8 Apr 01 2026 23:10:40

%S 36,484,1800,7396,13689,43681,1865956,27379323,29833444,554367025,

%T 1938024529,3627816200,7635613924,7838092089,40903720200,122168424676,

%U 188731633500,564067089936,9574460449352,20812245457936,31275004868836

%N Powerful k such that rad(Sum({m <= k : rad(m) = rad(k)})) = rad(k), where rad = A007947.

%C Intersection of A001694 and A393919.

%C Most numbers k such that rad(Sum({m <= k : rad(m) = rad(k)})) = rad(k) are in A332785 (i.e., neither powerful nor squarefree). Powerful terms in A393919 are rare.

%C Does not intersect A246547 (perfect powers of primes).

%C No intersection with A383394(1)..A383394(302010) (i.e., none less than 2^46), where A383394 is the list of perfect powers of Achilles numbers (in A052486).

%C No intersection with A052486(1)^2..A052486(300000)^2.

%C For p^6*q^4 from A6881(n) = p*q, primes, no intersection for n = 1..2^20.

%C Conjecture: this sequence intersects A383394.

%e Table of n, s(i), a(n) for n = 1..21, where s = A001694:

%e n Subset a(n)

%e -----------------------------------------------------------------

%e 1 B s(9) = 36 = 2^2 * 3^2

%e 2 B s(36) = 484 = 2^2 * 11^2

%e 3 A s(74) = 1800 = 2^3 * 3^2 * 5^2

%e 4 B s(157) = 7396 = 2^2 * 43^2

%e 5 C s(220) = 13689 = 3^4 * 13^2

%e 6 B s(402) = 43681 = 11^2 * 19^2

%e 7 B s(2788) = 1865956 = 2^2 * 683^2

%e 8 A s(10922) = 27379323 = 3^3 * 19^2 * 53^2

%e 9 B s(11410) = 29833444 = 2^2 * 2731^2

%e 10 B s(49948) = 554367025 = 5^2 * 17^2 * 277^2

%e 11 B s(93822) = 1938024529 = 7^2 * 19^2 * 331^2

%e 12 A s(128613) = 3627816200 = 2^3 * 5^2 * 4259^2

%e 13 B s(186974) = 7635613924 = 2^2 * 43691^2

%e 14 C s(189440) = 7838092089 = 3^8 * 1093^2

%e 15 A s(434411) = 40903720200 = 2^3 * 3^4 * 5^2 * 7^2 * 227^2

%e 16 B s(752224) = 122168424676 = 2^2 * 174763^2

%e 17 A s(935591) = 188731633500 = 2^2 * 3^3 * 5^3 * 3739^2

%e 18 C s(1619925) = 564067089936 = 2^4 * 3^2 * 7^2 * 8941^2

%e 19 A s(6693021) = 9574460449352 = 2^3 * 709^2 * 1543^2

%e 20 C s(9873569) = 20812245457936 = 2^4 * 163^2 * 6997^2

%e 21 B s(12106840) = 31275004868836 = 2^2 * 2796203^2

%e .

%e Subset key:

%e A = Achilles A052486.

%e B = A303606, perfect powers of squarefree composite k (in A120944).

%e C = A386762, perfect powers of k that are neither squarefree nor powerful (in A332785).

%e D = A383394, perfect powers of Achilles k (in A052486) - none less than 2^46.

%t nn = 50000; rad[x_] := Times @@ FactorInteger[x][[All, 1]]; t = Rest@ Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}]; s[x_] := Module[{r}, r = rad[x]; Total@ Select[Range[x], rad[#] == r &] ]; Select[t, rad[#] == rad[s[#] ] &]

%t (* Alternative: first download the regs function from the link at A369609, then: *)

%t nn = 2^32; rad[x_] := Times @@ FactorInteger[x][[All, 1]]; s = Rest@ Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3] } ]; mm = Length[s]; Monitor[Reap[Do[If[rad[Total@ regs[-#] ] == rad[#], Print[#]; Sow[#]] &@ s[[c]], {c, Length[s] } ] ][[-1, 1]], {c, mm}]

%Y Cf. A007947, A008479, A013929, A052486, A126706, A332785, A369609, A381498, A391625, A393919.

%K nonn,hard,more

%O 1,1

%A _Michael De Vlieger_, Mar 26 2026