A393914 Numbers k such that A008472(k) is coprime to A001414(k).

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 72, 75, 76, 80, 88, 90, 92, 96, 98, 99, 104, 108, 116, 117, 124, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 198, 200, 207, 212, 224, 232, 236, 242, 244, 245, 248, 250, 252

Offset

1,1

Comments

Numbers k such that the sum of distinct prime factors of k is coprime to the sum of prime factors with multiplicity.

Let s = A008472 and let t = A001414.

By convention, 1 is not in this sequence.

Proper subset of A013929, since squarefree k implies s(k) = t(k).

Proper subset of A126706, since for p^m, m > 1, s(p^m) = p and t(p^m) = m*p = m*s(p^m).

Does not meet A303606, since for k^m, m > 1 for squarefree composite k (in A120944), s(k^m) = s(k) and t(k^m) = t(m*k) = m*s(k). Therefore this sequence is a proper subset of A059404.

Does not include all Achilles numbers (in A052486); for example, for 1568 = 2^5*7^2, s(1578) = 9 and t(1578) = 24; gcd(9,24) = 3.

This sequence is the same if we redefine s = A088865, since A088865(k) = A008472(k)^A001222(k), i.e., A088865(k) is a perfect power of A008472(k), thus both have the same squarefree kernel.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Example

Table of n, a(n) = k, s(k), and t(k) for select n:
     n    a(n) = k                      s(k)  t(k)
  ------------------------------------------------
     1     12 = 2^2 * 3                   5     7
     2     18 = 2   * 3^2                 5     8
     3     20 = 2^2 * 5                   7     9
     4     24 = 2^3 * 3                   5     9
     5     28 = 2^2 * 7                   9    11
     8     45 = 3^2 * 5                   8    11
    16     72 = 2^3 * 3^2                 5    12
    21     90 = 2   * 3^2 * 5            10    13
    32    144 = 2^4 * 3^2                 5    14
    99    420 = 2^2 * 3   * 5 * 7        17    19
  1157   5184 = 2^6 * 3^4                 5    24
  1549   6930 = 2   * 3^2 * 5 * 7 * 11   28    31

Mathematica

r = Select[Range[2^8], Nor[SquareFreeQ[#], PrimePowerQ[#] ] &]; s[x_] := If[x == 1, 0, Total[FactorInteger[x][[All, 1]] ] ]; t[x_] := If[x == 1, 0, Total[Join @@ ConstantArray @@@ FactorInteger[x] ] ]; Select[r, CoprimeQ[s[#], t[#] ] &]

Prog

(PARI) isok(k) = my(f=factor(k)); gcd(vecsum(f[, 1]), f[, 1]~*f[, 2]) == 1; \\ Michel Marcus, Apr 22 2026

Crossrefs

Cf. A001414, A007947, A008472, A013929, A027746, A027748, A052486, A059404, A088865, A126706, A303606.

Sequence in context: A375934 A332785 A393946 * A383113 A367589 A177425

Adjacent sequences: A393911 A393912 A393913 * A393915 A393917 A393918

Keyword

nonn,easy

Author

Michael De Vlieger, Apr 20 2026

Status

approved