A394555 Powerful k such that rad(Sum({m <= k : rad(m) = rad(k)})) = rad(k), where rad = A007947.

36, 484, 1800, 7396, 13689, 43681, 1865956, 27379323, 29833444, 554367025, 1938024529, 3627816200, 7635613924, 7838092089, 40903720200, 122168424676, 188731633500, 564067089936, 9574460449352, 20812245457936, 31275004868836

Offset

1,1

Comments

Intersection of A001694 and A393919.

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.

Does not intersect A246547 (perfect powers of primes).

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).

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

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

Conjecture: this sequence intersects A383394.

Links

Table of n, a(n) for n=1..21.

Example

Table of n, s(i), a(n) for n = 1..21, where s = A001694:
   n Subset                     a(n)
  -----------------------------------------------------------------
   1 B        s(9) =             36 = 2^2 * 3^2
   2 B       s(36) =            484 = 2^2 * 11^2
   3 A       s(74) =           1800 = 2^3 * 3^2 * 5^2
   4 B      s(157) =           7396 = 2^2 * 43^2
   5 C      s(220) =          13689 = 3^4 * 13^2
   6 B      s(402) =          43681 = 11^2 * 19^2
   7 B     s(2788) =        1865956 = 2^2 * 683^2
   8 A    s(10922) =       27379323 = 3^3 * 19^2 * 53^2
   9 B    s(11410) =       29833444 = 2^2 * 2731^2
  10 B    s(49948) =      554367025 = 5^2 * 17^2 * 277^2
  11 B    s(93822) =     1938024529 = 7^2 * 19^2 * 331^2
  12 A   s(128613) =     3627816200 = 2^3 * 5^2 * 4259^2
  13 B   s(186974) =     7635613924 = 2^2 * 43691^2
  14 C   s(189440) =     7838092089 = 3^8 * 1093^2
  15 A   s(434411) =    40903720200 = 2^3 * 3^4 * 5^2 * 7^2 * 227^2
  16 B   s(752224) =   122168424676 = 2^2 * 174763^2
  17 A   s(935591) =   188731633500 = 2^2 * 3^3 * 5^3 * 3739^2
  18 C  s(1619925) =   564067089936 = 2^4 * 3^2 * 7^2 * 8941^2
  19 A  s(6693021) =  9574460449352 = 2^3 * 709^2 * 1543^2
  20 C  s(9873569) = 20812245457936 = 2^4 * 163^2 * 6997^2
  21 B s(12106840) = 31275004868836 = 2^2 * 2796203^2
.
Subset key:
A = Achilles A052486.
B = A303606, perfect powers of squarefree composite k (in A120944).
C = A386762, perfect powers of k that are neither squarefree nor powerful (in A332785).
D = A383394, perfect powers of Achilles k (in A052486) - none less than 2^46.

Mathematica

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[#] ] &]
(* Alternative: first download the regs function from the link at A369609, then: *)
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}]

Crossrefs

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

Sequence in context: A104671 A383773 A323549 * A128986 A281371 A090408

Adjacent sequences: A394552 A394553 A394554 * A394556 A394557 A394558

Keyword

nonn,hard,more

Author

Michael De Vlieger, Mar 26 2026

Status

approved