A052486 -id:A052486 - cp's OEIS frontend

A383394 Perfect powers of Achilles numbers.

5184, 11664, 40000, 82944, 153664, 186624, 250000, 373248, 419904, 455625, 640000, 746496, 937024, 944784, 1259712, 1265625, 1327104, 1750329, 1827904, 1882384, 2458624, 3240000, 3779136, 4000000, 5345344, 6718464, 7290000, 8000000, 8340544, 9529569, 10240000

Offset: 1

Michael De Vlieger, Aug 01 2025

Proper subset of A131605, where A286708 is the union of A131605 and A052486. Therefore these are both powerful numbers and perfect powers, unlike Achilles numbers themselves.
Proper subset of A366854.
This sequence does not intersect A303606, also a proper subset of A131605.

			Table of n, a(n) for n = 1..12:
 n      a(n)
--------------------------------
 1     5184 =  72^2 = 2^6  * 3^4
 2    11664 = 108^2 = 2^4  * 3^6
 3    40000 = 200^2 = 2^6  * 5^4
 4    82944 = 288^2 = 2^10 * 3^4
 5   153664 = 392^2 = 2^6  * 7^4
 6   186624 = 432^2 = 2^8  * 3^6
 7   250000 = 500^2 = 2^4  * 5^6
 8   373248 =  72^3 = 2^9  * 3^6
 9   419904 = 648^2 = 2^6  * 3^8
10   455625 = 675^2 = 3^6  * 5^4
11   640000 = 800^2 = 2^10 * 5^4
12   746496 = 864^2 = 2^10 * 3^6

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

Cf. A001597, A052486, A126708, A131605, A286708, A303606, A366854, A386762.

nn = 2^24; mm = Sqrt[nn]; i = 1; k = 2; MapIndexed[Set[S[First[#2]], #1] &, Rest@ Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[mm, 3]}, {a, Sqrt[mm/b^3]}], GCD @@ FactorInteger[#][[;; , -1]] == 1 &]]; Union@ Reap[While[j = 2; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 2, k++; i++] ][[-1, 1]]

from math import isqrt
from sympy import integer_nthroot, mobius
def A383394(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while f(kmin) < kmin: kmin >>= 1		
        kmin = max(kmin,kmax >> 1)
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x):
        c, l = squarefreepi(integer_nthroot(x,3)[0])+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length()))-1, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c += j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c-l
    def f(x): return n+x-sum(g(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length()))
    return bisection(f,n,n) # Chai Wah Wu, Aug 11 2025

A194085 Strong Achilles numbers: Achilles numbers m such that phi(m) is also an Achilles number, where phi(m) denotes Euler's totient function of m.

Original entry on oeis.org

500, 864, 1944, 2000, 2592, 3456, 5000, 10125, 10368, 12348, 12500, 16875, 19652, 19773, 30375, 31104, 32000, 33275, 37044, 40500, 49392, 50000, 52488, 55296, 61731, 64827, 67500, 69984, 78608, 80000, 81000, 83349, 84375, 93312, 108000

Offset: 1

Kausthub Gudipati, Aug 14 2011

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..656 from Ray Chandler)
Project Euler, Problem 302: Strong Achilles Numbers

Cf. A052486 (Achilles numbers)

achillesQ[n_] := With[{ee = FactorInteger[n][[All, 2]]}, Min[ee] > 1 && GCD @@ ee == 1];
Select[Range[10^6], achillesQ[#] && achillesQ[EulerPhi[#]]&] (* Jean-François Alcover, Sep 26 2020 *)

isA(n)=!ispower(n) && ispowerful(n)
is(n)=isA(n) && isA(eulerphi(n)) \\ Charles R Greathouse IV, Dec 19 2013

# uses program in A052486
from itertools import count, islice
from math import gcd
from sympy import factorint, totient
def A194085_gen(): # generator of terms
    return map(lambda x:x[0],filter(lambda x:all(d>1 for d in x[1]) and gcd(*x[1])==1,map(lambda x: (x,factorint(totient(x)).values()),(A052486(i) for i in count(1)))))
A194085_list = list(islice(A194085_gen(),20)) # Chai Wah Wu, Sep 10 2024

Extended by Ray Chandler, Aug 16 2011

A203663 Achilles number whose double is also an Achilles number.

Original entry on oeis.org

432, 972, 1944, 2000, 2700, 3456, 4500, 5292, 5400, 5488, 8748, 9000, 10584, 10800, 12348, 12500, 13068, 15552, 16000, 17496, 18000, 18252, 21168, 21296, 21600, 24300, 24500, 24696, 25000, 26136

Offset: 1

Antonio Roldán, Jan 04 2012

nonn

Every term is a multiple of 4.

			15552 is in the sequence because 15552 = 2^6*3^5 (Achilles number) and 15552*2 = 2^7*3^5 is also an Achilles number.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..4148 from Robert Israel)

Cf. A052486, A143610, A203662.

filter:= proc(n) local e2,F;
  e2:= padic:-ordp(n,2);
  if e2 < 2 then return false fi;
  F:= map(t -> t[2], ifactors(n/2^e2)[2]);
  min(F) > 1 and igcd(e2,op(F))=1 and igcd(e2+1,op(F))=1
end proc:
select(filter, [seq(i,i=4..10^5,4)]); # Robert Israel, Jan 16 2018

achillesQ[n_] := With[{ee = FactorInteger[n][[All, 2]]}, Min[ee] > 1 && GCD@@ee == 1];
Select[Range[4, 10^5, 4], achillesQ[#] && achillesQ[2#]&] (* Jean-François Alcover, Sep 25 2020 *)

achilles(n) = { n>1 & vecmin(factor(n)[, 2])>1 & !ispower(n) } \\ M. F. Hasler, 2010
{ for (n=1, 10^6, if (achilles(n)==1 && achilles(2*n)==1, print1(n,", "))); } \\ Antonio Roldán, Oct 07 2012

# uses program in A052486
from itertools import count, islice
from math import gcd
from sympy import factorint
def A203663_gen(): # generator of terms
    return map(lambda x:x[0],filter(lambda x:all(d>1 for d in x[1]) and gcd(*x[1])==1,map(lambda x: (x,factorint(x<<1).values()),(A052486(i) for i in count(1)))))
A203663_list = list(islice(A203663_gen(),30)) # Chai Wah Wu, Sep 10 2024

A248022 Achilles numbers which are coprime to the sum of their divisors.

Original entry on oeis.org

392, 800, 968, 1352, 2312, 2888, 3087, 3267, 3872, 4232, 5408, 6075, 6125, 6272, 6728, 7688, 7803, 9248, 10952, 11552, 12800, 13448, 14283, 14792, 15125, 15488, 16928, 17672, 19208, 20000, 21632, 22472, 22707, 25088, 26912, 27783, 27848, 29403, 29768, 30752, 33275

Offset: 1

Robert G. Wilson v, Sep 29 2014

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A052486 and A003624. - Michel Marcus, Sep 30 2014

achillesQ[n_] := Block[{ls = Last /@ FactorInteger@ n}, Min@ ls > 1 == GCD @@ ls]; Select[ Range@ 35000, achillesQ[ #] && GCD[#, DivisorSigma[1, #]] == 1 &]

A378629 Powerful numbers k such that both k-1 and k+1 are in A126706.

Original entry on oeis.org

49, 125, 243, 343, 1681, 1849, 3249, 4913, 6724, 6859, 8649, 9801, 11449, 13689, 13924, 17576, 20449, 24389, 24649, 28125, 28224, 29791, 31212, 36125, 37249, 40328, 42849, 45125, 57121, 59049, 63001, 66049, 68921, 79507, 83349, 85849, 94249, 99127, 106929, 110224

Offset: 1

Michael De Vlieger, Dec 03 2024

Contains certain powerful k in A246547 (perfect powers of primes) or in A286708 (powerful numbers that are not prime powers).

Contains certain Achilles numbers (in A052486); a(20) = 28125 = 3^2 * 5^5.

			Let S = A126706, the sequence of k that are neither squarefree nor prime powers.
{1, 4, 8, 9} are not in the sequence since S(1) = 12.
a(1) = 49 = 7^2 since both 48 = 2^3 * 3 and 50 = 2 * 5^2 are in S.
64 is not in the sequence since 65 is squarefree.
a(2) = 125 = 5^3 since both 124 = 2^2 * 41 and 126 = 2 * 3^2 * 7 are in S.
128 is not in the sequence since 127 is prime.
a(3) = 243 = 3^5 since both 242 = 2 * 11^2 and 244 = 2^2 * 61 are in S.
a(7) = 3249 = 3^2 * 19^2, since both 3248 = 2^4 * 7 * 29 and 3250 = 2 * 5^3 * 13 are in S, etc.

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

Cf. A001694, A013929, A024619, A052486, A126706, A246547, A286708.

With[{nn = 2^30}, Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}], AllTrue[# + {-1, 1}, Nor[SquareFreeQ[#], PrimePowerQ[#] ] &] &] ]

A378875 Number of Achilles numbers k between consecutive perfect powers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 2, 0, 0, 2, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 3, 0, 0

Offset: 1

Michael De Vlieger, Dec 09 2024

Within the sequence S = A001694 of powerful numbers, we have either perfect powers k (in A001597) and numbers m that are not perfect powers, i.e., Achilles numbers (in A052486). This sequence is the number of m between k.

			We partition S = A001694 by numbers k in A001597 (in brackets) and derive the following irregular table:
     [1];              hence a(1) = 0,
     [4];                    a(2) = 0,
     [8];                    a(3) = 0,
     ...
    [64], 72;               a(11) = 1,
    [81];                   a(12) = 0,
   [100], 108;              a(13) = 1,
     ...
   [625], 648, 675;         a(34) = 2,
     ...
  [4489], 4500, 4563, 4608; a(85) = 3, etc.

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

Cf. A001597, A001694, A052486.

nn = 2^12;
s = Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}];
-1 + Length /@ TakeList[s,
  {1}~Join~Differences@
  Position[s, _?(GCD @@ FactorInteger[#][[All, -1]] > 1 &),
  Heads -> False][[All, 1]] ]

A382770 Number of powerful k < n such that k and n are coprime.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 3, 2, 4, 1, 4, 2, 3, 2, 5, 1, 5, 2, 4, 2, 5, 1, 5, 3, 5, 4, 7, 1, 7, 4, 6, 4, 7, 2, 9, 4, 6, 3, 9, 2, 9, 4, 5, 4, 9, 2, 9, 4, 7, 5, 10, 3, 9, 4, 7, 5, 10, 2, 10, 5, 6, 5, 10, 3, 11, 5, 8, 3, 11, 3, 12, 5, 7, 5, 11, 3, 12, 4, 8, 6, 13, 2

Offset: 1

Michael De Vlieger, Apr 05 2025

First differs from A304574 at n = 73: a(73) = 12, A304574(n) = 11, because 72 = 2^3 * 3^2 is powerful but not a perfect power (i.e., an Achilles number in A052486).

			Let s = A001694, the sequence of powerful numbers.
a(1) = 0 since the smallest powerful number is 1 itself.
a(2) = 1 since s(1) = 1 is smaller than and coprime to 2.
a(3) = 1 since s(1) = 1 is smaller than and coprime to 3.
a(4) = 1 since s(1..2) = {1, 4}; 1 is smaller than and coprime to 4, but 4 = 4.
a(5) = 2 since s(1..2) = {1, 4}, both smaller than and coprime to 5.
a(6) = 1 since s(1..2) = {1, 4}; 1 is smaller than and coprime to 6, but gcd(4,6) > 1.
a(8) = 1 since s(1..3) = {1, 4, 8}; 1 and 4 are both smaller than and coprime to 8, but 8 = 8.
a(9) = 3 since s(1..3) = {1, 4, 8} are all smaller than and coprime to 9.
a(73) = 12 since s(1..12) = {1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72}, all coprime to prime 73. All except 72 are perfect powers, thus A304574(73) = 11, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, where red represents prime n, gold represents proper prime power n, green represents squarefree composite n, and blue and purple represent n that is neither squarefree nor a prime power, where purple represents a powerful n that is not a prime power. Large green dots represent composite primorials n.

Cf. A001694, A304574.

nn = 120; q = 1; rad[x_] := Times @@ FactorInteger[x][[All, 1]]; {0}~Join~Rest@ Table[Set[{c, i}, {0, 1}]; If[Divisible[n, rad[n]^2], t[q] = n; q++]; While[i < q, If[CoprimeQ[t[i], n], c++]; i++]; c, {n, nn}]

a(n) = #select(x->(ispowerful(x) && gcd(x,n)==1), [1..n-1]); \\ Michel Marcus, Apr 11 2025

a(n) > 0 for n > 1, since 1 is powerful, smaller than n > 1, and coprime to n >= 1.

A386433 Achilles numbers with a primorial squarefree kernel that are not products of primorials.

Original entry on oeis.org

108, 648, 972, 1944, 2700, 3888, 4500, 8748, 9000, 13500, 16200, 17496, 18000, 23328, 24300, 34992, 36000, 40500, 45000, 48600, 52488, 67500, 69984, 72000, 78732, 81000, 97200, 112500, 121500, 132300, 135000, 139968, 144000, 145800, 180000, 209952, 218700, 220500

Offset: 1

Michael De Vlieger, Jul 21 2025

nonn

			Table of n, a(n), and A053669(a(n)) for n = 1..12.
 n     a(n)                    A053669(a(n))
--------------------------------------------
 1     108 = 2^2 * 3^3         5
 2     648 = 2^3 * 3^4         5
 3     972 = 2^2 * 3^5         5
 4    1944 = 2^3 * 3^5         5
 5    2700 = 2^2 * 3^3 * 5^2   7
 6    3888 = 2^4 * 3^5         5
 7    4500 = 2^2 * 3^2 * 5^3   7
 8    8748 = 2^2 * 3^7         5
 9    9000 = 2^3 * 3^2 * 5^3   7
10   13500 = 2^2 * 3^3 * 5^3   7
11   16200 = 2^3 * 3^4 * 5^2   7
12   17496 = 2^3 * 3^7         5
Let s = A052486.
The number 12 is not a term since it is not powerful (i.e., not in A001694).
The number 36, though powerful, is not a term since it is a perfect square.
s(1) = 72 is not in this sequence since rad(72) = P(2) = 6 and 72 = 2*6*6 = P(1)*P(2)*P(2).
s(2) = 108 = 3*6*6 is in the sequence since it is not a product of primorials.
The number 144, though powerful, is not a term because it is a perfect square.
s(3) = 200 is not a term because rad(200) = 10 = 2*5 is not also divisible by A053669(200) = 3.
s(4) = 288 is not in this sequence since rad(288) = P(2) = 6 and 288 = 2*2*2*6*6 = P(1)*P(1)*P(1)*P(2)*P(2), etc.

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

Cf. A001221, A001597, A001694, A002110, A007947, A025487, A055932, A056808, A377854, A378002.

(* Load Fast Mathematica algorithm for A055932 linked at A377854, then: *)
Select[Union@ Flatten[f[7][[3 ;; -1, 2 ;; -1]] ], And[Divisible[#1, Apply[Times, #2[[All, 1]] ]^2], GCD @@ #2[[;; , -1]] == 1, Max@ Differences[#2[[All, -1]] ] > 0] & @@ {#, FactorInteger[#]} &]

Let rad = A007947, omega = A001221, and P = A002110.
rad(a(n)) = P(omega(a(n))).
Intersection of A052486 and A056808 = A377854 \ A378002.

A386434 Achilles numbers k such that A053669(k) < A006530(k).

Original entry on oeis.org

200, 392, 500, 675, 800, 968, 1125, 1323, 1352, 1372, 1568, 2000, 2312, 2888, 3087, 3200, 3267, 3528, 3872, 4000, 4232, 4563, 5000, 5292, 5324, 5408, 5488, 6075, 6125, 6272, 6728, 7688, 7803, 8575, 8712, 8788, 9248, 9747, 9800, 10125, 10584, 10952, 10976, 11552

Offset: 1

Michael De Vlieger, Jul 21 2025

nonn

			Let s = A052486, q = A053669, and gpf = A006530.
Table of n, a(n), and q(a(n)) for n = 1..12:
 n    a(n)               q(a(n))
--------------------------------
 1    200 = 2^3 * 5^2    3
 2    392 = 2^3 * 7^2    3
 3    500 = 2^2 * 5^3    3
 4    675 = 3^3 * 5^2    2
 5    800 = 2^5 * 5^2    3
 6    968 = 2^3 * 11^2   3
 7   1125 = 3^2 * 5^3    2
 8   1323 = 3^3 * 7^2    2
 9   1352 = 2^3 * 13^2   3
10   1372 = 2^2 * 7^3    3
11   1568 = 2^5 * 7^2    3
12   2000 = 2^4 * 5^3    3
The number 12 is not a term since it is not powerful (i.e., not in A001694).
The number 36, though powerful, is not a term since it is a perfect square.
s(1) = 72 is not in the sequence since q(72) > gpf(72), i.e., 5 > 3.
s(2) = 108 is not in the sequence since q(108) > gpf(108), i.e., 5 > 3.
a(1) = s(3) = 200 because q(200) < gpf(200), i.e., 3 < 5.
a(2) = s(4) = 392 because q(392) < gpf(392), i.e., 3 < 7, etc.

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

Cf. A001597, A001694, A006530, A052486, A053669, A052486, A080259, A377854.

(* Load Fast Mathematica algorithm for A055932 linked at A377854, then: *)
nn = 6; mm = Times @@ Prime@ Range[nn]; Complement[Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[mm, 3]}, {a, Sqrt[mm/b^3]}], And[Length[#2] > 1, GCD @@ #2 == 1] & @@ {#, FactorInteger[#][[;; , -1]]} &], Union@ Flatten[f[nn][[3 ;; -1, 2 ;; -1]] ] ]

Intersection of A052486 and A080259 = A052486 \ A377854.

A379164 Achilles numbers that are deficient.

Original entry on oeis.org

675, 1125, 1323, 2312, 2888, 3087, 3267, 4232, 4563, 5324, 6075, 6125, 6728, 7688, 7803, 8575, 8788, 9747, 10125, 10952, 11907, 11979, 13448, 14283, 14792, 15125, 16875, 17672, 19652, 19773, 21125, 22472, 22707, 25947, 27436, 27783, 27848, 28125, 29403, 29768, 30375, 33275, 35912, 36125, 36963, 40328

Offset: 1

Massimo Kofler, Dec 17 2024

nonn

			675=3^3*5^2 is a term because it is an Achilles number (powerful but imperfect, see A052486) and it is larger than the sum of its proper divisors (1+3+5+9+15+25+27+45+75+135+225=565).
1125=3^2*5^3 is a term because it is an Achilles number (powerful but imperfect, see A052486) and it is larger than the sum of its proper divisors (1+3+5+9+15+25+45+75+125+225+ 375=903).

Intersection of A005100 and A052486.

Cf. A378859.

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; AllTrue[e, # > 1 &] && GCD @@ e == 1 && Times @@ ((p - 1/p^e)/(p - 1)) < 2]; Select[Range[42000], q] (* Amiram Eldar, Dec 17 2024 *)

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A383394 Perfect powers of Achilles numbers.

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A194085 Strong Achilles numbers: Achilles numbers m such that phi(m) is also an Achilles number, where phi(m) denotes Euler's totient function of m.

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A203663 Achilles number whose double is also an Achilles number.

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A248022 Achilles numbers which are coprime to the sum of their divisors.

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A378629 Powerful numbers k such that both k-1 and k+1 are in A126706.

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A378875 Number of Achilles numbers k between consecutive perfect powers.

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A382770 Number of powerful k < n such that k and n are coprime.

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A386433 Achilles numbers with a primorial squarefree kernel that are not products of primorials.

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