A052486 -id:A052486 - cp's OEIS frontend

A272714 Numbers n such that both n and n+1 are Achilles numbers (A052486).

5425069447, 11968683934831, 28821995554247, 48689748233307

Offset: 1

Felix Fröhlich, May 12 2016

Any term of the sequence is also a term of A227297, but the converse is not always true. The smallest term of A227297 where the converse fails is A227297(1) = 12167. Do any other such numbers exist?

			5425069447 = 7^3 * 41^2 * 97^2 and 5425069448 = 2^3 * 26041^2. Since every prime factor of 5425069447 and 5425069448 is repeated, both numbers are Achilles numbers (A052486) and since the two numbers differ by 1, i.e., the value of A247246 at the index of 5425069447 in A052486 is 1, 5425069447 is a term of the sequence.

Wikipedia, Achilles number

Cf. A052486, A227297, A247246.

is(n) = vecmin(factor(n)[, 2]) > 1 && vecmin(factor(n+1)[, 2]) > 1 && !ispower(n) && !ispower(n+1)

A358514 a(n) is the smallest number with exactly n divisors that are Achilles numbers (A052486).

Original entry on oeis.org

1, 72, 216, 432, 1296, 864, 7200, 2592, 6912, 10800, 7776, 15552, 27000, 41472, 21600, 31104, 884736, 54000, 64800, 129600, 86400, 248832, 172800, 162000, 5308416, 108000, 194400, 216000, 518400, 388800, 810000, 1323000, 1058400, 1382400, 324000, 432000, 2073600

Offset: 0

Marius A. Burtea, Dec 04 2022

nonn

			1 has no Achilles number divisors, so a(0) = 1.
72 = A052486(1), so a(1) = 72.
216 has divisors 72 = A052486(1) and 108 = A052486(2), and there are no smaller numbers that have exactly two divisors that are Achilles numbers, so a(2) = 216.

Cf. A000005, A052486.

ah:=func; a:=[]; for n in [0..37] do k:=1; while #[d:d in Divisors(k)| ah(d)] ne n do k:=k+1; end while; Append(~a,k); end for; a;

is(n) = my(f=factor(n)[, 2]); n>9 && vecmin(f)>1 && gcd(f)==1; \\ A052486
a(n) = my(k=1); while (sumdiv(k, d, is(d)) != n, k++); k; \\ Michel Marcus, Dec 13 2022

A007916 Numbers that are not perfect powers.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83

Offset: 1

R. Muller

From Gus Wiseman, Oct 23 2016: (Start)
There is a 1-to-1 correspondence between integers N >= 2 and sequences a(x_1),a(x_2),...,a(x_k) of terms from this sequence. Every N >= 2 can be written uniquely as a "power tower"
N = a(x_1)^a(x_2)^a(x_3)^...^a(x_k),
where the exponents are to be nested from the right.
Proof: If N is not a perfect power then N = a(x) for some x, and we are done. Otherwise, write N = a(x_1)^M for some M >=2, and repeat the process. QED
Of course, prime numbers also have distinct power towers (see A164336). (End)
These numbers can be computed with a modified Sieve of Eratosthenes: (1) start at n=2; (2) if n is not crossed out, then append n to the sequence and cross out all powers of n; (3) set n = n+1 and go to step 2. - Sam Alexander, Dec 15 2003
These are all numbers such that the multiplicities of the prime factors have no common divisor. The first number in the sequence whose prime multiplicities are not coprime is 180 = 2 * 2 * 3 * 3 * 5. Mathematica: CoprimeQ[2,2,1]->False. - Gus Wiseman, Jan 14 2017

			Example of the power tower factorizations for the first nine positive integers: 1=1, 2=a(1), 3=a(2), 4=a(1)^a(1), 5=a(3), 6=a(4), 7=a(5), 8=a(1)^a(2), 9=a(2)^a(1). - _Gus Wiseman_, Oct 20 2016

N. J. A. Sloane, Table of n, a(n) for n = 1..9875
Joakim Munkhammar, The Riemann zeta function as a sum of geometric series, The Mathematical Gazette (2020) Vol. 104, Issue 561, 527-530.
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564
F. Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993
Index entries for sequences generated by sieves

Complement of A001597. Union of A052485 and A052486.
Cf. A144338, A277562, A277564, A075802.
Cf. A153158 (squares of these numbers).
See A277562, A277564, A277576, A277615 for more about the power towers.
A278029 is a left inverse.
Cf. A052409.

a007916 n = a007916_list !! (n-1)
a007916_list = filter ((== 1) . foldl1 gcd . a124010_row) [2..]
-- Reinhard Zumkeller, Apr 13 2012

[n : n in [2..1000] | not IsPower(n) ];

Maple
```
See link.
```

a = {}; Do[If[Apply[GCD, Transpose[FactorInteger[n]][[2]]] == 1, a = Append[a, n]], {n, 2, 200}];
Select[Range[2,200],GCD@@FactorInteger[#][[All,-1]]===1&] (* Michael De Vlieger, Oct 21 2016. Corrected by Gus Wiseman, Jan 14 2017 *)

is(n)=!ispower(n)&&n>1 \\ Charles R Greathouse IV, Jul 01 2013

from sympy import mobius, integer_nthroot
def A007916(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 13 2024

A075802(a(n)) = 0. - Reinhard Zumkeller, Mar 19 2009
Gcd(exponents in prime factorization of a(n)) = 1, cf. A124010. - Reinhard Zumkeller, Apr 13 2012
a(n) ~ n. - Charles R Greathouse IV, Jul 01 2013
A052409(a(n)) = 1. - Ridouane Oudra, Nov 23 2024

More terms from Henry Bottomley, Sep 12 2000
Edited by Charles R Greathouse IV, Mar 18 2010
Further edited by N. J. A. Sloane, Nov 09 2016

A131605 Perfect powers of nonprimes (m^k where m is a nonprime positive integer and k >= 2).

Original entry on oeis.org

1, 36, 100, 144, 196, 216, 225, 324, 400, 441, 484, 576, 676, 784, 900, 1000, 1089, 1156, 1225, 1296, 1444, 1521, 1600, 1728, 1764, 1936, 2025, 2116, 2304, 2500, 2601, 2704, 2744, 2916, 3025, 3136, 3249, 3364, 3375, 3600, 3844, 3969, 4225, 4356, 4624

Offset: 1

Daniel Forgues, May 27 2008

nonn

Although 1 is a square, is a cube, and so on..., 1 is sometimes excluded from perfect powers since it is not a well-defined power of 1 (1 = 1^k for any k in [2, 3, 4, 5, ...])
From Michael De Vlieger, Aug 11 2025: (Start)
This sequence is A001597 \ A246547, i.e., perfect powers without proper prime powers.
Union of {1} with the intersection of A001597 and A126706, where A126706 is the sequence of numbers that are neither prime powers nor squarefree.
Union of {1} and A286708 \ A052486, i.e., powerful numbers that are not prime powers, without Achilles numbers, but including the empty product. (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (Terms a(n), n = 1..1323 from Klaus Brockhaus, and n = 1324..8649 from Daniel Forgues.)

Cf. A000961, A001597, A024619, A025475, A052486, A072102, A126706, A136141, A246547, A286708.

With[{nn = 2^20}, {1}~Join~Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}], And[Length[#2] > 1, GCD @@ #2 > 1] & @@ {#, FactorInteger[#][[;; , -1]]} &] ] (* Michael De Vlieger, Aug 11 2025 *)

isok(n) = if (n == 1, return (1), return (ispower(n, ,&np) && (! isprime(np)))); \\ Michel Marcus, Jun 12 2013

from sympy import mobius, integer_nthroot, primepi
def A131605(n):
    def f(x): return int(n-2+x+sum(mobius(k)*((a:=integer_nthroot(x,k)[0])-1)+primepi(a) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 14 2024

Sum_{n>=1} 1/a(n) = 1 + A072102 - A136141 = 1.10130769935514973882... . - Amiram Eldar, Aug 15 2025

A112526 Characteristic function for powerful numbers.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0

Offset: 1

Franklin T. Adams-Watters, Sep 09 2005

A signed multiplicative variant is defined by b(n) = a(n)*mu(n) with mu = A008683, such that b(p^e)=0 if e=1 and b(p^e)= -1 if e>1. This has Dirichlet series Sum_{n>=1} b(n)/n = A005596 and Sum_{n>=1} b(n)/n^2 = A065471. - R. J. Mathar, Apr 04 2011

			a(72) = 1 because 72 = 2^3*3^2 has all exponents > 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Powerful Number.
Index entries for characteristic functions.

Differs from characteristic function of perfect powers A075802 at Achilles numbers A052486.
Cf. A001694 (powerful numbers), A124010, A001221, A027746.
Cf. A008683, A008966, A005596, A065471, A082695, A063524.
Cf. A002117, A078434, A005361.

a112526 1 = 1
a112526 n = fromEnum $ (> 1) $ minimum $ a124010_row n
-- Reinhard Zumkeller, Jun 03 2015, Sep 16 2011

cfpn[n_]:=If[n==1||Min[Transpose[FactorInteger[n]][[2]]]>1,1,0]; Array[ cfpn,120] (* Harvey P. Dale, Jul 17 2012 *)

for(n=1, 100, print1(direuler(p=2, n, (1+X^3)/(1-X^2))[n], ", ")) \\ Vaclav Kotesovec, Jul 15 2022

a(n) = ispowerful(n); \\ Amiram Eldar, Jul 02 2025

from sympy import factorint
def A112526(n): return int(all(e>1 for e in factorint(n).values())) # Chai Wah Wu, Sep 15 2024

Multiplicative with a(p^e) = 1 - 0^(e-1), e > 0 and p prime.
Dirichlet g.f.: zeta(2*s)*zeta(3*s)/zeta(6*s), e.g., A082695 at s=1.
a(n) * A008966(n) = A063524(n). - Reinhard Zumkeller, Sep 16 2011
a(n) = {m: Min{A124010(m,k): k=1..A001221(m)} > 1}. - Reinhard Zumkeller, Jun 03 2015
Sum_{k=1..n} a(k) ~ zeta(3/2)*sqrt(n)/zeta(3) + 6*zeta(2/3)*n^(1/3)/Pi^2. - Vaclav Kotesovec, Feb 08 2019
a(n) = Sum_{d|n} A005361(d)*A008683(n/d). - Ridouane Oudra, Jul 03 2025

A143610 Numbers of the form p^2 * q^3, where p,q are distinct primes.

Original entry on oeis.org

72, 108, 200, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 2312, 2888, 3087, 3267, 4232, 4563, 5324, 6125, 6728, 7688, 7803, 8575, 8788, 9747, 10952, 11979, 13448, 14283, 14792, 15125, 17672, 19652, 19773, 21125, 22472, 22707, 25947, 27436

Offset: 1

M. F. Hasler, Aug 27 2008

Also: numbers with prime signature {3,2}.
This is a subsequence of A114128. [Hasler]
Every a(n) is an Achilles number (A052486). They are minimal, meaning no proper divisor is an Achilles number. - Antonio Roldán, Dec 27 2011

			The first three terms of this sequence are 3^2 * 2^3 = 72, 2^2 * 3^3 = 108, 5^2 * 2^3 = 200.

T. D. Noe, Table of n, a(n) for n = 1..1000
Project Euler, Problem 200: Find the 200th prime-proof sqube containing the contiguous sub-string 200.
Index to sequences related to prime signature

Cf. A114128.

Cf. A085541, A085548, A085965.

f[n_] := Sort[Last/@FactorInteger[n]] == {2, 3}; Select[Range[30000], f] (* Vladimir Joseph Stephan Orlovsky, Oct 09 2009 *)

for(n=1, 10^5, omega(n)==2 || next; vecsort(factor(n)[,2])==[2,3]~ && print1(n","))

list(lim)=my(v=List(),t);forprime(p=2, (lim\4)^(1/3), t=p^3;forprime(q=2, sqrt(lim\t), if(p==q, next);listput(v,t*q^2))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A143610(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(isqrt(x//p**3)) for p in primerange(integer_nthroot(x,3)[0]+1))+primepi(integer_nthroot(x,5)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Sum_{n>=1} 1/a(n) = P(2)*P(3) - P(5) = A085548 * A085541 - A085965 = 0.043280..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

A294068 Number of factorizations of n using perfect powers (elements of A001597) other than 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Gus Wiseman, May 05 2018

nonn

			The a(1152) = 7 factorizations are (4*4*8*9), (4*8*36), (4*9*32), (8*9*16), (8*144), (9*128), (32*36).

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of zeros are A052485.

Cf. A000961, A001055, A001222, A001597, A001694, A007716, A007916, A045778, A052409, A052410, A052486, A091050, A203025, A303707, A303710.

ispp:= proc(n) local F;
  F:= ifactors(n)[2];
  igcd(op(map(t -> t[2],F)))>1
end proc:
f:= proc(n) local F, np, Q;
  F:= map(t -> t[2], ifactors(n)[2]);
  np:= mul(ithprime(i)^F[i],i=1..nops(F));
  Q:= select(ispp, numtheory:-divisors(np));
  G(Q,np)
end proc:
G:= proc(Q,n) option remember; local q,t,k;
    if not numtheory:-factorset(n) subset `union`(seq(numtheory:-factorset(q),q=Q)) then return 0 fi;
    q:= Q[1]; t:= 0;
    for k from 0 while n mod q^k = 0 do
      t:= t + procname(Q[2..-1],n/q^k)
    od;
    t
end proc:
G({},1):= 1:
map(f, [$1..200]); # Robert Israel, May 06 2018

ppQ[n_]:=And[n>1,GCD@@FactorInteger[n][[All,2]]>1];
facsp[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsp[n/d],Min@@#>=d&]],{d,Select[Divisors[n],ppQ]}]];
Table[Length[facsp[n]],{n,100}]

A212171 Prime signature of n (nonincreasing version): row n of table lists positive exponents in canonical prime factorization of n, in nonincreasing order.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 4, 1, 2, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 3, 1

Offset: 2

Matthew Vandermast, Jun 03 2012

Length of row n equals A001221(n).
The multiset of positive exponents in n's prime factorization completely determines a(n) for a host of OEIS sequences, including several "core" sequences.  Of those not cross-referenced here or in A212172, many can be found by searching the database for A025487.
(Note: Differing opinions may exist about whether the prime signature of n should be defined as this multiset itself, or as a symbol or collection of symbols that identify or "signify" this multiset. The definition of this sequence is designed to be compatible with either view, as are the original comments. When n >= 2, the customary ways to signify the multiset of exponents in n's prime factorization are to list the constituent exponents in either nonincreasing or nondecreasing order; this table gives the nonincreasing version.)
Table lists exponents in the order in which they appear in the prime factorization of a member of A025487.  This ordering is common in database comments (e.g., A008966).
Each possible multiset of an integer's positive prime factorization exponents corresponds to a unique partition that contains the same elements (cf. A000041). This includes the multiset of 1's positive exponents, { } (the empty multiset), which corresponds to the partition of 0.
Differs from A124010 from a(23) on, corresponding to the factorization of 18 = 2^1*3^2 which is here listed as row 18 = [2, 1], but as [1, 2] (in the order of the prime factors) in A124010 and also in A118914 which lists the prime signatures in nondecreasing order (so that row 12 = 2^2*3^1 is also [1, 2]). - M. F. Hasler, Apr 08 2022

			First rows of table read:
  1;
  1;
  2;
  1;
  1,1;
  1;
  3;
  2;
  1,1;
  1;
  2,1;
  ...
The multiset of positive exponents in the prime factorization of 6 = 2*3 is {1,1} (1s are often left implicit as exponents). The prime signature of 6 is therefore {1,1}.
12 = 2^2*3 has positive exponents 2 and 1 in its prime factorization, as does 18 = 2*3^2. Rows 12 and 18 of the table both read {2,1}.

Jason Kimberley, Table of i, a(i) for i = 2..24301 (n = 2..10000)

Cf. A025487, A001221 (row lengths), A001222 (row sums). A118914 gives the nondecreasing version. A124010 lists exponents in n's prime factorization in natural order, with A124010(1) = 0.
A212172 cross-references over 20 sequences that depend solely on n's prime exponents >= 2, including the "core" sequence A000688.  Other sequences determined by the exponents in the prime factorization of n include:
Multiplicative: A000005, A007425, A008683, A008836, A034444, A037445, A181819.
Additive: A001221, A001222, A056169.
Other: A001055, A008480, A010553, A038548, A050320, A051707, A071625, A074206, A076078, A085082, A088873.
A highly incomplete selection of sequences, each definable by the set of prime signatures possessed by its members: A000040, A000290, A000578, A000583, A000961, A001248, A001358, A001597, A001694, A002808, A004709, A005117, A006881, A013929, A030059, A030229, A052486.

&cat[Reverse(Sort([pe[2]:pe in Factorisation(n)])):n in[1..76]]; // Jason Kimberley, Jun 13 2012

apply( {A212171_row(n)=vecsort(factor(n)[,2]~,,4)}, [1..40])\\ M. F. Hasler, Apr 19 2022

Row n of A118914, reversed.

Row n of A124010 for n > 1, with exponents sorted in nonincreasing order. Equivalently, row A046523(n) of A124010 for n > 1.

A072414 Non-Achilles numbers for which LCM of the exponents in the prime factorization of n is not equal to the maximum of the same exponents.

Original entry on oeis.org

360, 504, 540, 600, 756, 792, 936, 1176, 1188, 1224, 1350, 1368, 1400, 1404, 1440, 1500, 1656, 1836, 1960, 2016, 2052, 2088, 2160, 2200, 2232, 2250, 2400, 2484, 2520, 2600, 2646, 2664, 2904, 2952, 3024, 3096, 3132, 3168, 3240, 3348, 3384, 3400, 3500

Offset: 1

Labos Elemer, Jun 17 2002

nonn

Most members of this sequence fail to be Achilles numbers because they have at least one prime factor with multiplicity 1. There are also numbers in the sequence that fail to be Achilles numbers because they are perfect powers: these are precisely the proper powers of members of A072412, so the smallest such is 5184 = 2^6*3^4 = 72^2. - Franklin T. Adams-Watters, Oct 09 2006

			m = 504 = 2*2*2*3*3*7: exponent-set = E = {3,2,1}, max(E) = 3 < lcm(E) = 6, gcd(E) = min(E) = 1.

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

Cf. A005361, A051903, A051904, A052409, A072411, A072412, A052486.

Select[Range@ 3500, And[LCM @@ # != Max@ #, GCD @@ # == Min@ # == 1] &[FactorInteger[#][[All, -1]] ] &] (* Michael De Vlieger, Jul 18 2017 *)

is(n)=my(f=factor(n)[,2]); n>9 && lcm(f)!=vecmax(f) && (#f==1 || vecmin(f)<2) \\ Charles R Greathouse IV, Oct 16 2015

A051903(a(n)) is not equal A072411(a(n)) but the numbers are not in A052486.

A377854 Achilles numbers whose squarefree kernel is a primorial.

Original entry on oeis.org

72, 108, 288, 432, 648, 864, 972, 1152, 1800, 1944, 2592, 2700, 3456, 3888, 4500, 4608, 5400, 6912, 7200, 8748, 9000, 10368, 10800, 13500, 15552, 16200, 17496, 18000, 18432, 21600, 23328, 24300, 27648, 28800, 31104, 34992, 36000, 40500, 41472, 43200, 45000, 48600

Offset: 1

Michael De Vlieger, Nov 16 2024

Numbers whose squarefree kernel is a primorial that are powerful but not a perfect power.

			Prime power decomposition of the first 12 terms:
   a(1) =   72 = 2^3 * 3^2
   a(2) =  108 = 2^2 * 3^3
   a(3) =  288 = 2^5 * 3^2
   a(4) =  432 = 2^4 * 3^3
   a(5) =  648 = 2^3 * 3^4
   a(6) =  864 = 2^5 * 3^3
   a(7) =  972 = 2^2 * 3^5
   a(8) = 1152 = 2^7 * 3^2
   a(9) = 1800 = 2^3 * 3^2 * 5^2
  a(10) = 1944 = 2^3 * 3^5
  a(11) = 2592 = 2^5 * 3^4
  a(12) = 2700 = 2^2 * 3^3 * 5^2

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Fast Mathematica algorithm for A055932 (function f).

Cf. A001597, A001694, A002110, A007947, A052486, A055932, A286708, A369374, A378002.

(* First load function f in the link, then: *)
Select[Rest@ Union@ Flatten@ f[10],
 And[Divisible[#, Apply[Times, #2[[All, 1]] ]^2],
   GCD @@ #2[[All, -1]] == 1] & @@ {#, FactorInteger[#]} &]

Intersection of A286708 \ A001597 and A055932.
Intersection of A052486 and A055932.
Proper subset of A369374.
Superset of A378002.

cp's OEIS Frontend

A272714 Numbers n such that both n and n+1 are Achilles numbers (A052486).

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A358514 a(n) is the smallest number with exactly n divisors that are Achilles numbers (A052486).

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Magma

PARI

A007916 Numbers that are not perfect powers.

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Haskell

Magma

Maple

Mathematica

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A131605 Perfect powers of nonprimes (m^k where m is a nonprime positive integer and k >= 2).

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PARI

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A112526 Characteristic function for powerful numbers.

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Haskell

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PARI

PARI

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A143610 Numbers of the form p^2 * q^3, where p,q are distinct primes.

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Mathematica

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PARI

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A294068 Number of factorizations of n using perfect powers (elements of A001597) other than 1.

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