A052486 -id:A052486 - cp's OEIS frontend

A072413 Numbers k such that the LCM of exponents in the prime factorization of k does not equal the product of the exponents.

36, 100, 144, 180, 196, 216, 225, 252, 300, 324, 396, 400, 441, 450, 468, 484, 576, 588, 612, 676, 684, 700, 720, 784, 828, 882, 900, 980, 1000, 1008, 1044, 1080, 1089, 1100, 1116, 1156, 1200, 1225, 1260, 1296, 1300, 1332, 1444, 1452, 1476, 1512, 1521

Offset: 1

Labos Elemer, Jun 17 2002

nonn

The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 2, 29, 348, 3548, 35761, 358258, 3583892, 35843109, 358440763, ... . Apparently, the asymptotic density of this sequence exists and equals 0.03584... . - Amiram Eldar, Sep 09 2022

			k = 36 = 2*2*3*3; exponent set = {2,2}; LCM = 2, product = 4.

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

Cf. A005361, A051903, A051904, A052409, A052486, A072411-A072414.

Select[Range@ 1600, LCM @@ # != Times @@ # &@ Map[Last, FactorInteger@ #] &] (* Michael De Vlieger, May 15 2016 *)

is(n)=my(f=factor(n)[,2]); n>9 && lcm(f)!=factorback(f) \\ Charles R Greathouse IV, Jan 14 2017

A005361(a(n)) != A072411(a(n)).

A304328 a(n) = n/(largest perfect power divisor of n).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 1, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 3, 1, 26, 1, 7, 29, 30, 31, 1, 33, 34, 35, 1, 37, 38, 39, 5, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 2, 55, 7, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 2

Offset: 1

Gus Wiseman, May 10 2018

nonn

Not all terms are squarefree numbers; for example, a(500) = 4.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000961, A001055, A001222, A001597, A001694, A005117, A007716, A007916, A052485, A052486, A059404, A066636, A091050, A160400, A203025, A294068, A303707, A304327, A304339.

Table[n/Last[Select[Divisors[n],#===1||GCD@@FactorInteger[#][[All,2]]>1&]],{n,100}]

a(n)={my(m=1); fordiv(n, d, if(ispower(d), m=max(m,d))); n/m} \\ Andrew Howroyd, Aug 26 2018

a(n) * A203025(n) = n.

A072412 Numbers k such that the LCM of exponents in the prime factorization of k does not equal the largest exponent.

Original entry on oeis.org

72, 108, 200, 288, 360, 392, 432, 500, 504, 540, 600, 648, 675, 756, 792, 800, 864, 936, 968, 972, 1125, 1152, 1176, 1188, 1224, 1323, 1350, 1352, 1368, 1372, 1400, 1404, 1440, 1500, 1568, 1656, 1800, 1836, 1944, 1960, 2000, 2016, 2052, 2088, 2160

Offset: 1

Labos Elemer, Jun 17 2002

nonn

This sequence differs from the Achilles numbers (A052486).

			k = 360 = 2*2*2*3*3*5, exponent set = {3,2,1}; LCM=6, max=3.

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

Cf. A005361, A051903, A051904, A052409, A052486, A072411, A072413, A072414.

Select[Range[2000], LCM @@ (e = FactorInteger[#][[;; , 2]]) != Max[e] &] (* Amiram Eldar, Jul 30 2022 *)

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

A051903(a(n)) != A072411(a(n)).

A378002 Achilles numbers that are products of primorials.

Original entry on oeis.org

72, 288, 432, 864, 1152, 1800, 2592, 3456, 4608, 5400, 6912, 7200, 10368, 10800, 15552, 18432, 21600, 27648, 28800, 31104, 41472, 43200, 54000, 55296, 62208, 64800, 73728, 86400, 88200, 93312, 108000, 115200, 124416, 162000, 165888, 172800, 194400, 221184, 259200

Offset: 1

Michael De Vlieger, Nov 16 2024

Products of primorials that are powerful but not perfect powers.

			Prime power decomposition of the first 12 terms:
   a(1) =   72 = 2^3 * 3^2
   a(2) =  288 = 2^5 * 3^2
   a(3) =  432 = 2^4 * 3^3
   a(4) =  864 = 2^5 * 3^3
   a(5) = 1152 = 2^7 * 3^2
   a(6) = 1800 = 2^3 * 3^2 * 5^2
   a(7) = 2592 = 2^5 * 3^4
   a(8) = 3456 = 2^7 * 3^3
   a(9) = 4608 = 2^9 * 3^2
  a(10) = 5400 = 2^3 * 3^3 * 5^2
  a(11) = 6912 = 2^8 * 3^3
  a(12) = 7200 = 2^5 * 3^2 * 5^2

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

Cf. A001597, A001694, A002110, A007947, A052486, A025487, A286708, A364930, A377854.

(* First load function f in A025487, then: *)
Select[Rest@ Union@ Flatten@ f[14],
 And[Divisible[#, Apply[Times, #2[[All, 1]] ]^2],
   GCD @@ #2[[All, -1]] == 1] & @@ {#, FactorInteger[#]} &]

Intersection of A286708 \ A001597 and A025487.
Intersection of A052486 and A025487.
Proper subset of A364930, in turn a proper subset of A369374.
Proper subset of A377854.

A203662 Achilles number whose largest proper divisor is also an Achilles number.

Original entry on oeis.org

864, 1944, 3888, 4000, 5400, 6912, 9000, 10584, 10800, 10976, 17496, 18000, 21168, 21600, 24696, 25000, 26136, 30375, 31104, 32000, 34992, 36000, 36504, 42336, 42592, 43200, 48600, 49000, 49392, 50000, 52272, 55296, 62208, 62424, 68600, 69984

Offset: 1

Antonio Roldán, Jan 04 2012

nonn

Exponent of smallest prime divisor of n is greater than or equal to 3.

Both N and the largest proper divisor of N share the same prime factors with different exponents.

			17496 is in the sequence because 17496=2^3*3^7 (Achilles number) and the largest proper divisor 8748=2^2*3^7 is also an Achilles number.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Donovan Johnson)

Cf. A052486, A143610.

(* First run the program for A052486 to define achillesQ *) Select[Range[50000], achillesQ[#] && achillesQ[Divisors[#][[-2]]] &] (* Alonso del Arte, Jan 05 2012 *)

# uses program in A052486
from itertools import count, islice
from math import gcd
from sympy import factorint
def A203662_gen(): # generator of terms
    def g(x):
        (f:=factorint(x))[min(f)]-=1
        return (x,f.values())
    return map(lambda x:x[0],filter(lambda x:all(d>1 for d in x[1]) and gcd(*x[1])==1,map(g,(A052486(i) for i in count(1)))))
A203662_list = list(islice(A204662_gen(),20)) # Chai Wah Wu, Sep 10 2024

A247246 Differences of consecutive Achilles numbers.

Original entry on oeis.org

36, 92, 88, 104, 40, 68, 148, 27, 125, 64, 104, 4, 153, 27, 171, 29, 20, 196, 232, 144, 56, 312, 280, 108, 188, 199, 113, 67, 189, 72, 344, 16, 112, 232, 268, 63, 45, 392, 292, 32, 76, 8, 80, 587, 50, 147, 456, 184, 288, 488, 115, 772, 137, 36, 40, 212, 248

Offset: 1

Eric Chen, Nov 28 2014

29 is the first prime in this sequence, and it equals 1352 - 1323. Clearly, if the difference is prime, then these two Achilles numbers must be relatively prime, so primes appear in this sequence rarely. However, are there infinitely many n such that a(n) is prime?

The number 1 can also appear in this sequence, because it equals 5425069448 - 5425069447 = (2^3 * 26041^2) - (7^3 * 41^2 * 97^2). Does every natural number appear in this sequence? If so, do they appear infinitely often?

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carlos Rivera, Problem 53, The Prime Puzzles and Problems Connection.
Eric Weisstein's World of Mathematics, Achilles number.
Wikipedia, Achilles number.

Cf. A052486, A076446, A053289.

f:= proc(n) local E; E:= map(t->t[2], ifactors(n)[2]); min(E)>1 and igcd(op(E))=1 end proc:
Achilles:= select(f, [$1..10^5]):
seq(Achilles[i+1]-Achilles[i],i=1..nops(Achilles)-1); # Robert Israel, Dec 13 2014

achillesQ[n_] := With[{ee = FactorInteger[n][[All, 2]]}, Min[ee] > 1 && GCD @@ ee == 1];
Select[Range[10^4], achillesQ] // Differences (* Jean-François Alcover, Sep 26 2020 *)

isA052486(n) = { n>1 & vecmin(factor(n)[, 2])>1 & !ispower(n); }
lista(nn) = {v = select(n->isA052486(n), vector(nn, i, i)); vector(#v-1, n, v[n+1] - v[n]);} \\ Michel Marcus, Nov 29 2014

from math import isqrt
from sympy import mobius, integer_nthroot
def A247246(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 kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x+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)
        c -= squarefreepi(integer_nthroot(x,3)[0])-l+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length()))
        return c
    return -(a:=bisection(f,n,n))+bisection(lambda x:f(x)+1,a,a) # Chai Wah Wu, Sep 10 2024

a(n) = A052486(n+1) - A052486(n).

A304339 Fixed point of f starting with n, where f(x) = x/(largest perfect power divisor of x).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 1, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 3, 1, 26, 1, 7, 29, 30, 31, 1, 33, 34, 35, 1, 37, 38, 39, 5, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 2, 55, 7, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 2

Offset: 1

Gus Wiseman, May 11 2018

nonn

All terms are squarefree numbers. First differs from A304328 at a(500) = 1, A304328(500) = 4.

			f maps 500 -> 4 -> 1 -> 1, so a(500) = 1.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000961, A001597, A001694, A005117, A007916, A052485, A052486, A059404, A091050, A160400, A203025, A294068, A303707, A304327, A304328.

radQ[n_]:=And[n>1,GCD@@FactorInteger[n][[All,2]]===1];
op[n_]:=n/Last[Select[Divisors[n],!radQ[#]&]];
Table[FixedPoint[op,n],{n,200}]

a(n)={while(1, my(m=1); fordiv(n, d, if(ispower(d), m=max(m,d))); if(m==1, return(n)); n/=m)} \\ Andrew Howroyd, Aug 26 2018

A366854 Powers k^m such that k is neither squarefree nor prime powers, and m > 1.

Original entry on oeis.org

144, 324, 400, 576, 784, 1296, 1600, 1728, 1936, 2025, 2304, 2500, 2704, 2916, 3136, 3600, 3969, 4624, 5184, 5625, 5776, 5832, 6400, 7056, 7744, 8000, 8100, 8464, 9216, 9604, 9801, 10000, 10816, 11664, 12544, 13456, 13689, 13824, 14400, 15376, 15876, 17424, 18225

Offset: 1

Michael De Vlieger, Jan 01 2024

Analogous to A303606 = { k^m : Omega(k) = omega(k) > 1, m > 1 }, i.e., squarefree composite k (in A120944) raised to m > 1. Proper subset of A131605, itself a proper subset of A286708, which is in turn a proper subset of A126706. This sequence does not intersect Achilles numbers A052486.

			Let b(n) = A126706(n).
a(1) = b(1)^2 = 12^2 = 144. Since 144 = 2^4*3^2, it is both powerful and a perfect power.
a(2) = b(2)^2 = 18^2 = 324.
a(3) = b(3)^2 = 20^2 = 400.
a(8) = b(1)^3 = 12^3 = 1728, etc.

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

Cf. A001597, A001694, A052486, A120944, A126706, A131605, A246547, A286708, A303606.

nn = 20000; i = 1; k = 2;
MapIndexed[Set[S[First[#2]], #1] &,
  Select[Range@ Sqrt[nn], Nor[SquareFreeQ[#], PrimePowerQ[#]] &] ];
Union@ Reap[
  While[j = 2;
    While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 2,
    k++; i++] ][[-1, 1]]

This sequence is A126706(i)^m, m > 1.
A131605 = union of {1}, A303606, and {a(n)}.
A286708 = union of A303606, {a(n)}, and A052486.
A001597 = union of {1}, A246547, A303606, and {a(n)}.
A001694 = union of A246547, A303606, {a(n)}, and A052486.

A369118 k is a term if and only if k is a composite number where the bases and the exponents of its factors in the prime decomposition are all odd primes.

Original entry on oeis.org

27, 125, 243, 343, 1331, 2187, 2197, 3125, 3375, 4913, 6859, 9261, 12167, 16807, 24389, 29791, 30375, 35937, 42875, 50653, 59319, 68921, 78125, 79507, 83349, 84375, 103823, 132651, 148877, 161051, 166375, 177147, 185193, 205379, 226981, 273375, 274625

Offset: 1

Peter Luschny, Jan 17 2024

nonn

Every term is divisible by a cube.
If n and k are terms then n*k is a term if and only if gcd(n, k) = 1.
From Michael De Vlieger, Jan 19 2024: (Start)
Prime exponents of prime power factors p^m | k imply that k is a powerful number. Hence this sequence is a proper subset of A001694, and k is of the form a^2 * b^3.
Prime exponents m imply either perfect powers k in A001597 such that all the m are the same, or an Achilles number k (in A052486) if the exponents differ. This is because prime p divides itself but is coprime to primes q != p. Therefore this sequence is not a subsequence of A001597.
The sequence consists of composite prime powers (A246547) and powerful numbers that are not prime powers (A286708), both of which are numbers that are not squarefree (A013929). (End)

			25015118625 = 3^5 * 5^3 * 7^7 is a term.
3125 = 5^5 and 3375 = 3^3 * 5^3 are terms but 3125*3375 is not a term.

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

Cf. A002808 (superset), A001694 (superset).
A051674 is a subsequence for n>1.
Intersects A001597, A013929, A052486, A246547, A286708.
Subsequence of A381825.

A369118Q[n_] := OddQ[n] && AllTrue[FactorInteger[n], OddQ[#] && PrimeQ[#]&, 2];
Select[Range[500000], A369118Q] (* Paolo Xausa, Jan 19 2024 *)

isok(k) = k > 1 && (k % 2 && #select(x -> (x <= 2) || !isprime(x), factor(k)[, 2]) == 0); \\ Amiram Eldar, Mar 08 2025

def isA369118(n):
    return (n > 1 and is_odd(n) and all(is_odd(f[1]) and is_prime(f[1])
           for f in factor(n)))
print([n for n in range(1, 300000) if isA369118(n)])

Sum_{n>=1} 1/a(n) = -1 + Product_{prime >= 3} (1 + Sum_{prime q >= 3} 1/p^q) = 0.05534030537711484966... . - Amiram Eldar, Mar 08 2025

A378859 Achilles numbers that are abundant.

Original entry on oeis.org

72, 108, 200, 288, 392, 432, 500, 648, 800, 864, 968, 972, 1152, 1352, 1372, 1568, 1800, 1944, 2000, 2592, 2700, 3200, 3456, 3528, 3872, 3888, 4000, 4500, 4608, 5000, 5292, 5400, 5408, 5488, 6272, 6912, 7200, 8712, 8748, 9000, 9248, 9800, 10368, 10584, 10800, 10976, 11552, 12168, 12348, 12500, 12800, 13068, 13500, 14112, 15488

Offset: 1

Massimo Kofler, Dec 09 2024

nonn

33075 is the smallest odd term.

The set of distinct prime factors of a term can be any set P of primes such that Product_{p in P} p/(p-1) > 2. - Robert Israel, Jan 29 2025

			72=2^3*3^2 is a term because it is an Achilles number (powerful but imperfect, see A052486) and it is smaller than the sum of its proper divisors (1+2+3+4+6+8+9+12+18+24+36=123).
108=2^2*3^3 is a term because it is an Achilles number (powerful but imperfect, see A052486) and it is smaller than the sum of its proper divisors (1+2+3+4+6+9+12+18+27+36+54=172).

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

Intersection of A005101 and A052486.

filter:= proc(n) local F, E, t; F:= ifactors(n)[2]; E:= F[..,2]; min(E)>1 and igcd(op(E))=1 and mul((t[1]^(1+t[2])-1)/(t[1]-1), t = F) > 2*n end proc:
select(filter, [$1..10^5]); # Robert Israel, Jan 28 2025

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[16000], q] (* Amiram Eldar, Dec 09 2024 *)

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A072413 Numbers k such that the LCM of exponents in the prime factorization of k does not equal the product of the exponents.

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A304328 a(n) = n/(largest perfect power divisor of n).

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A072412 Numbers k such that the LCM of exponents in the prime factorization of k does not equal the largest exponent.

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A378002 Achilles numbers that are products of primorials.

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A203662 Achilles number whose largest proper divisor is also an Achilles number.

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A247246 Differences of consecutive Achilles numbers.

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A304339 Fixed point of f starting with n, where f(x) = x/(largest perfect power divisor of x).

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