A320966 -id:A320966 - cp's OEIS frontend

A379592 Number of coreful divisor pairs (d, k/d), d | k, d < k/d, such that only one divisor divides the other, where k is in A320966.

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

Offset: 1

Michael De Vlieger, Dec 28 2024

nonn

Number of ways to write k = A320966(n) as a product of numbers i and j, i < j, such that rad(i) = rad(j) = rad(k), and either i | j or j | i, where rad = A007947 is the squarefree kernel.

Analogous to A370329, where the reference domain is A001694 instead of A320966.

			Let s(n) = A320966(n).
a(1) = 1 since s(1) = 8 = 2*4.
a(2) = 1 since s(2) = 16 = 2*8.
a(3) = 1 since s(3) = 27 = 3*9.
a(4) = 2 since s(4) = 32 = 2*16 = 4*8.
a(10) = 3 since s(10) = 128 = 2*64 = 4*32 = 8*16.
a(23) = 4 since s(23) = 512 = 2*256 = 4*128 = 8*64 = 16*32.
a(181) = 7 since s(181) = 20736 = 6*3456 = 12*1728 = 18*1152 = 24*864 = 36*576 = 48*432 = 72*288, etc.

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

Cf. A320966, A370329, A379552.

nn = 5400; rad[x_] := Times @@ FactorInteger[x][[All, 1]];
s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] > 0 &];
Table[k = s[[n]];
  Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
    _?(And[rad[#1] == rad[#2],
       Xor[Divisible[#2, #1],
           Divisible[#1, #2]]] & @@ # &)], {n, Length[s]}]

A379523 Sum of coreful divisors d | k such that gcd(d, k/d) > 1 and rad(d) = rad(k/d), with d | k/d and d < d/k, where k is in A320966 and rad = A007947.

Original entry on oeis.org

6, 10, 12, 30, 54, 18, 30, 24, 30, 126, 30, 30, 42, 120, 238, 90, 60, 56, 42, 50, 126, 60, 510, 162, 130, 168, 60, 336, 70, 150, 234, 66, 240, 110, 990, 90, 378, 432, 84, 132, 78, 112, 210, 270, 546, 90, 110, 456, 330, 150, 2046, 1092, 182, 714, 102, 350, 260

Offset: 1

Michael De Vlieger, Jan 15 2025

nonn

Alternatively, sum of divisors d | k such that omega(d) = omega(k/d) = omega(k), d | k/d, and d < k/d, where omega = A001221.

			a(1) = 6 since s(1) = 8 = 2*4; 2 | 4 but 4 > 2; 2+4 = 6. a(1) = (2^3-1)/(2-1)-1 = 6.
a(2) = 10 since s(2) = 16 = 2*8; 2 | 8 but 8 > 2; 2+8 = 10. a(2) = (2^4-1)/(2-1)-(2^2)-1 = 10.
a(3) = 12 since s(3) = 27 = 3*9; 3 | 9 but 9 > 3; 3+9 = 12. a(3) = (3^3-1)/(3-1)-1 = 12.
a(4) = 30 since s(4) = 32 = (2^5-1)/(2-1)-1 = 30.
a(5) = 54 since s(5) = 64 = (2^6-1)/(2-1)-(2^3)-1 = 54.
a(6) = 18 since s(6) = 72 = 6*12; 6 | 12 but 12 > 6; 6+12 = 18.
a(8) = 24 since s(8) = 108 = 6*18; 6 | 24 but 24 > 6; 6+18 = 24, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n) n = 1..10^5.

Cf. A000079, A001221, A007947, A246549, A320966, A364988.

nn = 2500;
s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] > 0 &];
Map[Function[n,
  DivisorSum[n, # &,
    And[PrimeNu[#1] == PrimeNu[#2] == #3,
        Xor[Divisible[#2, #1], Divisible[#1, #2]]] & @@
    {#, n/#, PrimeNu[n]} &]], s]

Let s(n) = A320966(n).
a(n) <= A364988(s(n)).
Using Iverson brackets:
For s(n) = 2^e, e > 2, (i.e., in A000079):
a(n) = 2^e - [e mod 2 = 0]*sqrt(2^e)-2
     = 2^e - 2*(e-1) for even e or 2^e-2 for odd e.
     = A364988(s(n)) - [e mod 2 = 0]*2^(e/2) for s(n) = 2^e, e > 2.
For s(n) = p^e, e > 2, (i.e., in A246549):
a(n) = A364988(s(n)) - [e mod 2 = 0]*p^(e/2)
     = (p^e - 1)/(p-1) - [e mod 2 = 0]*sqrt(p^e) - 1.

A001694 Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers).

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000

Offset: 1

N. J. A. Sloane

Numbers of the form a^2*b^3, a >= 1, b >= 1.
In other words, if the prime factorization of n is Product_k p_k^e_k then all e_k are greater than 1.
Numbers n such that Sum_{d|n} phi(d)*phi(n/d)*mu(d) > 0; places of nonzero A300717. - Benoit Cloitre, Nov 30 2002
This sequence is closed under multiplication. The primitive elements are A168363. - Franklin T. Adams-Watters, May 30 2011
Complement of A052485. - Reinhard Zumkeller, Sep 16 2011
The number of terms less than or equal to 10^k beginning with k = 0: 1, 4, 14, 54, 185, 619, 2027, 6553, 21044, ...: A118896. - Robert G. Wilson v, Aug 11 2014
a(10^n): 1, 49, 3136, 253472, 23002083, 2200079025, 215523459072, 21348015504200, 2125390162618116, ... . - Robert G. Wilson v, Aug 15 2014
a(m) mod prime(n) > 0 for m < A258599(n); a(A258599(n)) = A001248(n) = prime(n)^2. - Reinhard Zumkeller, Jun 06 2015
From Des MacHale, Mar 07 2021: (Start)
A number m is powerful if and only if |R/Z(R)| = m, for some finite non-commutative ring R.
A number m is powerful if and only if |G/Z(G)| = m, for some finite nilpotent class two group G (Reference Aine Nishe). (End)
Numbers n such that Sum_{k=1..n} phi(gcd(n,k))*mu(gcd(n,k)) > 0. - Richard L. Ollerton, May 09 2021

			1 is a term because for every prime p that divides 1, p^2 also divides 1.
2 is not a term since 2 divides 2 but 2^2 does not.
4 is a term because 2 is the only prime that divides 4 and 2^2 does divide 4. - _N. J. A. Sloane_, Jan 16 2022

G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277-307.
Aleksandar Ivić, The Riemann Zeta-Function, Wiley, NY, 1985, see p. 407.
Richard A. Mollin, Quadratics, CRC Press, 1996, Section 1.6.
Aine NiShe, Commutativity and Generalisations in Finite Groups, Ph.D. Thesis, University College Cork, 2000.
Paulo Ribenboim, Meine Zahlen, meine Freunde, 2009, Springer, 9.1 Potente Zahlen, pp. 241-247.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 54, exercise 10 (in the third edition 2015, p. 63, exercise 70).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe, terms 1001..5000 from G. C. Greubel)
Paul T. Bateman and Emil Grosswald, On a theorem of Erdős and Szekeres, Illinois J. Math. 2:1 (1958), pp. 88-98.
Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, A further investigation on covering systems with odd moduli, arXiv:2507.16135 [math.NT], 2025. See p. 3.
Valentin Blomer, Binary quadratic forms with large discriminants and sums of two squareful numbers II, Journal of the London Mathematical Society 71:1 (2005), pp. 69-84.
Chris K. Caldwell, Powerful Numbers.
Tsz Ho Chan, Arithmetic Progressions Among Powerful Numbers, J. Int. Seq., Vol. 26 (2023), Article 23.1.1.
Tsz Ho Chan, A note on three consecutive powerful numbers, arXiv:2503.21485 [math.NT], 2025.
Jean-Marie De Koninck, Nicolas Doyon, and Florian Luca, Powerful Values of Quadratic Polynomials, J. Int. Seq. 14 (2011), Article 11.3.3.
Paul Erdős and George Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), 7 (1935), 95-102. [Zahlen i-ter Art, p. 101]
Solomon W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77 (1970), 848-852.
K. Schneider, PlanetMath.org, Squarefull Number.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395.
D. Suryanarayana and R. Sita Rama Chandra Rao, The distribution of square-full integers, Ark. Mat., Volume 11, Number 1-2 (1973), 195-201.
Eric Weisstein's World of Mathematics, Powerful Number.
Eric Weisstein's World of Mathematics, Squareful.
Wikipedia, Powerful number.
Index entries for sequences related to powerful numbers.

Disjoint union of A062503 and A320966.
Cf. A007532 (Powerful numbers, definition (2)), A005934, A005188, A003321, A014576, A023052 (Powerful numbers, definition (3)), A046074, A013929, A076871, A258599, A001248, A112526, A168363, A224866, A261883, A300717.
Cf. A052485 (complement), A076446 (first differences), A376361, A376362.

a001694 n = a001694_list !! (n-1)
a001694_list = filter ((== 1) . a112526) [1..]
-- Reinhard Zumkeller, Nov 30 2012

isA001694 := proc(n) for p in ifactors(n)[2] do if op(2,p) = 1 then return false; end if; end do; return true; end proc:
A001694 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if isA001694(a) then return a; end if; end do; end if; end proc:
seq(A001694(n),n=1..20) ; # R. J. Mathar, Jun 07 2011

Join[{1}, Select[ Range@ 1250, Min@ FactorInteger[#][[All, 2]] > 1 &]]
(* Harvey P. Dale, Sep 18 2011; modified by Robert G. Wilson v, Aug 11 2014 *)
max = 10^3; Union@ Flatten@ Table[a^2*b^3, {b, max^(1/3)}, {a, Sqrt[max/b^3]}] (* Robert G. Wilson v, Aug 11 2014 *)
nextPowerfulNumber[n_] := Block[{r = Range[ Floor[1 + n^(1/3)]]^3}, Min@ Select[ Sort[ r*Floor[1 + Sqrt[n/r]]^2], # > n &]]; NestList[ nextPowerfulNumber, 1, 55] (* Robert G. Wilson v, Aug 16 2014 *)

isA001694(n)=n=factor(n)[,2];for(i=1,#n,if(n[i]==1,return(0)));1 \\ Charles R Greathouse IV, Feb 11 2011

list(lim,mn=2)=my(v=List(),t); for(m=1,sqrtnint(lim\1,3), t=m^3; for(n=1,sqrtint(lim\t), listput(v,t*n^2))); Set(v) \\ Charles R Greathouse IV, Jul 31 2011; edited Sep 22 2015

is=ispowerful \\ Charles R Greathouse IV, Nov 13 2012

from sympy import factorint
A001694 = [1]+[n for n in range(2,10**6) if min(factorint(n).values()) > 1]
# Chai Wah Wu, Aug 14 2014

from math import isqrt
from sympy import mobius, integer_nthroot
def A001694(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, 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
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 09 2024

sloane.A001694.list(54) # Peter Luschny, Feb 08 2015

A112526(a(n)) = 1. - Reinhard Zumkeller, Sep 16 2011
Bateman & Grosswald prove that there are zeta(3/2)/zeta(3) x^{1/2} + zeta(2/3)/zeta(2) x^{1/3} + O(x^{1/6}) terms up to x; see section 5 for a more precise error term. - Charles R Greathouse IV, Nov 19 2012
a(n) = A224866(n) - 1. - Reinhard Zumkeller, Jul 23 2013
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6). - Ivan Neretin, Aug 30 2015
Sum_{n>=1} 1/a(n)^s = zeta(2*s)*zeta(3*s)/zeta(6*s), s > 1/2 (Golomb, 1970). - Amiram Eldar, Oct 02 2022

More terms from Henry Bottomley, Mar 16 2000

Definition expanded by Jonathan Sondow, Jan 03 2016

A376936 Powerful numbers divisible by cubes of 2 distinct primes.

Original entry on oeis.org

216, 432, 648, 864, 1000, 1296, 1728, 1944, 2000, 2592, 2744, 3375, 3456, 3888, 4000, 5000, 5184, 5400, 5488, 5832, 6912, 7776, 8000, 9000, 9261, 10000, 10125, 10368, 10584, 10648, 10800, 10976, 11664, 13500, 13824, 15552, 16000, 16200, 16875, 17496, 17576, 18000

Offset: 1

Michael De Vlieger, Oct 16 2024

Numbers m with coreful divisors d, m/d such that neither d | m/d nor m/d | d, i.e., numbers m such that there exists a divisor pair (d, m/d) such that rad(d) = rad(m/d) but gcd(d, m/d) > 1 is neither d nor m/d, where rad = A007947. Divisors in each pair must be dissimilar and each in A126706.
Proper subset of A320966.
Contains A372695, A177493, and A162142. Does not contain A085986.

			216 is in the sequence since rad(12) | rad(18), but 12 does not divide 18 and 18 does not divide 12.
432 is a term since rad(18) | rad(24), but 18 does not divide 24 and 24 does not divide 18.
Table of coreful divisors d, a(n)/d such that neither d | a(n)/d nor a(n)/d | d for select a(n)
   n |   a(n)   divisor pairs d X a(n)/d
  ---------------------------------------------------------------------------
   1 |   216:   12 X 18;
   2 |   432:   18 X 24;
   3 |   648:   12 X 54;
   4 |   864:   24 X 36, 18 X 48;
   5 |  1000:   20 X 50;
   6 |  1296:   24 X 54;
   7 |  1728:   18 X 96, 36 X 48;
   8 |  1944:   12 X 162, 36 X 54;
   9 |  2000:   40 X 50;
  10 |  2592:   24 X 108, 48 X 54;
  11 |  2744:   28 X 98;
  12 |  3375:   45 X 75;
  13 |  3456:   18 X 192, 36 X 96, 48 X 72;
  22 |  7776:   24 X 324, 48 X 162, 54 X 144, 72 X 108;
  58 | 31104:   48 X 648, 54 X 576, 96 X 324, 108 X 288, 144 X 216, 162 X 192

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Notes on this sequence

Cf. A007947, A030078, A085986, A126706, A162142, A177493, A286708, A307958, A320966, A361430, A370329, A372695, A376514.

Cf. A082020, A082695.

Union@ Select[
  Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[20000],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &]

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - (15/Pi^2) * (1 + Sum_{prime} 1/((p-1)*(p^2+1))) = 0.021194288968234037106579437374641326044... . - Amiram Eldar, Nov 08 2024

A320965 Squares divisible by a single cube > 1.

Original entry on oeis.org

16, 81, 144, 324, 400, 625, 784, 1936, 2025, 2401, 2500, 2704, 3600, 3969, 4624, 5625, 5776, 7056, 8100, 8464, 9604, 9801, 13456, 13689, 14641, 15376, 15876, 17424, 19600, 21609, 21904, 22500, 23409, 24336, 26896, 28561, 29241, 29584, 30625, 35344, 39204, 41616, 42849

Offset: 1

Hugo Pfoertner, Oct 25 2018

nonn

Numbers whose prime factorization has a single exponent that is equal to 4 and all the rest, if they exist, are equal to 2. - Amiram Eldar, Jun 25 2022

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A000290, A000578, A046099, A060687, A062320, A216427, A320966, A321070.

sdscQ[n_]:=Count[Rest[Divisors[n]],?(IntegerQ[Surd[#,3]]&)]==1; Select[ Range[ 250]^2,sdscQ] (* _Harvey P. Dale, Aug 18 2020 *)
q[n_] := (e = Sort[FactorInteger[n][[;;, 2]], Greater])[[1]] == 4 && AllTrue[Rest[e], # == 2 &]; Select[Range[200]^2, q] (* Amiram Eldar, Jun 25 2022 *)

From Amiram Eldar, Jun 25 2022: (Start)
a(n) = A060687(n)^2.
Sum_{n>=1} 1/a(n) = (15/Pi^2) * Sum_{k>=2} (-1)^k * P(2*k) = 0.09603403868516162554..., where P(s) is the prime zeta function. (End)

A386428 Odd numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, k > 0, r > 1, and gcd(p,r) = 1.

Original entry on oeis.org

28125, 153125, 253125, 378125, 528125, 903125, 1128125, 1378125, 1653125, 2278125, 2628125, 3003125, 3341637, 3403125, 4278125, 4753125, 5253125, 5778125, 6903125, 7503125, 8128125, 8778125, 9282325, 10153125, 10878125, 11628125, 12403125, 12778713, 14028125, 14878125, 15753125, 16653125, 17578125, 18193357, 18528125

Offset: 1

Antti Karttunen, Aug 17 2025

nonn

Powerful numbers (A001694) that satisfy Euler's criterion for odd perfect numbers (A228058). That is, terms of A228058 whose special factor is not a prime, but a prime power.
If N = q^k*n^2 (i.e., a number of the form A228058) is an odd perfect number with special prime q, then the assertion that k must be 1 is known as the Descartes-Frenicle-Sorli conjecture on odd perfect numbers. In other words, the conjecture stipulates that certainly this subsequence of A228058 does not contain any odd perfect numbers.
a(1) = A228058(520); a(1..12) = 5^5 * A028375(2..13).

Antti Karttunen, Table of n, a(n) for n = 1..811
Jose A. B. Dris, Conditions Equivalent to the Descartes-Frenicle-Sorli Conjecture on Odd Perfect Numbers, Notes on Number Theory and Discrete Mathematics, Vol. 23, No. 2 (2017), pp. 12-20, arXiv preprint, arXiv:1610.01868 [math.NT], 2016.

Intersection of A001694 (or A320966) and A228058.

Cf. A028375.

isA386428(n) = if(!(n%2)||(omega(n)<2), 0, my(f=factor(n), y=0); for(i=1, #f~, if(1==(f[i, 2]%4), if((1==y)||1==f[i,2]||(1!=(f[i, 1]%4)), return(0), y=1), if(f[i, 2]%2, return(0)))); (y));

isA386428(n) = (ispowerful(n) && isA228058(n)); \\ See A228058.

A378900 Squares of numbers divisible by the squares of two distinct primes.

Original entry on oeis.org

1296, 5184, 10000, 11664, 20736, 32400, 38416, 40000, 46656, 50625, 63504, 82944, 90000, 104976, 129600, 153664, 156816, 160000, 186624, 194481, 202500, 219024, 234256, 250000, 254016, 291600, 331776, 345744, 360000, 374544, 419904, 455625, 456976, 467856, 490000

Offset: 1

Michael De Vlieger, Dec 12 2024

Also, the squares in A376936.
Proper subset of A378767, in turn a proper subset of A286708, the intersection of A001694 and A024619.
Numbers that have 3 kinds of coreful divisor pairs (d, k/d), d | k, i.e., rad(d) = rad(k/d) = rad(k) where rad = A007947. These kinds are described as follows:
Type A: d = k/d, which pertain to square k (in A000290).
Type B: d | k/d, d < k/d, which pertain to k in A320966, powerful numbers divisible by a cube.
Type C: neither d | k/d nor k/d | d, which pertain to k in A376936.
Since divisors d, k/d may either divide or not divide the other, there are no other cases.
In addition the following kinds of divisor pairs are also seen:
Type D: (d, k/d) such that d | k/d but there exists a factor Q | k/d that does not divide d. Then omega(d) < omega(k/d) = omega(k).
Type E: Nontrivial unitary divisor pairs (d, k/d) such that gcd(d, k/d) = 1, d > 1, k/d > 1. Let prime power factor p^m | k be such that m is maximized. Then set d = p^m and it is clear that for any k in A024619, there exists at least 1 nontrivial unitary divisor pair.

			Let b = A036785.
Table of the first 12 terms of this sequence, showing examples of types A, B, and C of coreful pairs of divisors.
   n    a(n)   Factors of a(n)    b(n)   Type B       Type C
  -------------------------------------------------------------
   1    1296   2^4  * 3^4          36    6 * 216      24 * 54
   2    5184   2^6  * 3^4          72    6 * 864      48 * 108
   3   10000   2^4  * 5^4         100   10 * 1000     40 * 250
   4   11664   2^4  * 3^6         108    6 * 1944     24 * 486
   5   20736   2^8  * 3^4         144    6 * 3456     54 * 384
   6   32400   2^4  * 3^4 * 5^2   180   30 * 1080    120 * 270
   7   38416   2^4  * 7^4         196   14 * 2744     56 * 686
   8   40000   2^6  * 5^4         200   10 * 4000     80 * 500
   9   46656   2^6  * 3^6         216    6 * 7776     48 * 972
  10   50625   3^4  * 5^4         225   15 * 3375    135 * 375
  11   63504   2^4  * 3^4 * 7^2   252   42 * 1512    168 * 378
  12   82944   2^10 * 3^4         288    6 * 13824    54 * 1536

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Listing of select divisor pairs of a(n), n = 1..12, showing divisor pairs of type A in light gray, type B in orange and purple, and type C in black.

Cf. A000290, A001694, A007947, A024619, A036785, A126706, A286708, A320966, A376936, A378768.

Cf. A013661, A082020.

s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}],   IntegerQ@ Sqrt[#] &] &[500000];
Union@ Select[s, Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &]

a(n) = A036785(n)^2.

Sum_{n>=1} 1/a(n) = Pi^2/6 - (15/Pi^2) * (1 + Sum_{p prime} 1/(p^4-1)) = 0.0015294876575980711757... . - Amiram Eldar, Dec 21 2024

A378984 Squares in A378769.

Original entry on oeis.org

32400, 63504, 90000, 129600, 156816, 202500, 219024, 254016, 291600, 345744, 360000, 374544, 467856, 490000, 518400, 571536, 627264, 685584, 777924, 810000, 876096, 960400, 1016064, 1089936, 1166400, 1210000, 1245456, 1382976, 1411344, 1440000, 1498176, 1587600

Offset: 1

Michael De Vlieger, Dec 15 2024

Let omega = A001221, bigomega = A001222, and rad = A007947.
Numbers k that have all types of divisor pairs (d, k/d), d | k, that are listed in both A378769 and A378900. These are listed below:
Type A*: (Nontrivial) unitary divisor pairs, i.e., d coprime to k/d. The rest of the types are in cototient.
Type B*: gcd(d, k/d) > 1, rad(d) !| k/d, rad(k/d) !| d. These exist for k in A375055.
Type C: d < k/d, d | k/d but rad(k/d) !| d. Implies rad(k/d) = rad(k) and omega(d) < omega(k/d). These exist for k in A126706.
Type D: Either rad(d) | k/d, rad(k/d) !| d or vice versa. These exist for k in A378767.
Type E*: d = k/d = sqrt(k).
Type F: rad(d) = rad(k/d) = rad(k), d < k/d, d | k/d. These exist for k in A320966.
Type G*: rad(d) = rad(k/d) = rad(k), neither d | k/d nor k/d | d. These exist for k in A376936.
Asterisks denote symmetric types.
Since numbers d and k/d are either coprime or not, and if not, the squarefree kernel of one either divides the other or not, and if so, d divides k/d or not, and if so, d = k/d or not, there are no other types.
Smallest odd term is a(45) = 2480625.
Square roots not A350372: sqrt(810000) = 900 is not in A350372.

			a(1) = 32400 = 2^4 * 3^4 * 5^2 has the following divisor pair types:
  Type A: 16 * 2025, Type B: 48 * 675, Type C: 2 * 16200, Type D: 8 * 4050
  Type E: 180 * 180, Type F: 30 * 1080, Type G: 120 * 270.
a(2) = 63504 = 2^4 * 3^4 * 7^2 has the following divisor pair types:
  Type A: 16 * 3969, Type B: 48 * 1323, Type C: 2 * 31752, Type D: 8 * 7938
  Type E: 252 * 252, Type F: 42 * 1512, Type G: 168 * 378.
a(3) = 90000 = 2^4 * 3^2 * 5^4 has the following divisor pair types:
  Type A: 9 * 10000, Type B: 18 * 5000, Type C: 2 * 45000, Type D: 8 * 11250
  Type E: 300 * 300, Type F: 30 * 3000, Type G: 120 * 750, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Diagram listing all divisor pairs for a(n), n = 1..8, showing type A in white, type B in light gray, type C in green or red, type D in blue or gold, type E in dark gray, type F in orange or purple, and type G in black.
Michael De Vlieger, Diagram listing divisor pairs (d, k/d) for k = a(n), n = 1..60, showing only those with the smallest d and using the same color scheme as above, for each type and its reversal if the type is nonsymmetric.

Cf. A000290, A001221, A001222, A007947, A126706, A320966, A350372, A375055, A376936, A378767, A378769, A378900.

Cf. A013661, A082020.

s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[2^21],  IntegerQ@ Sqrt[#] &];
t = Select[s, Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
Select[t, PrimeOmega[#] > PrimeNu[#] > 2 &]

This sequence is { k = s^2 : rad(k)^2 | k,
  bigomega(k) > omega(k) > 2, p^3 | k and q^3 | k for distinct primes p, q }.
Intersection of A378769 and A378900.
Intersection of A000290, A375055, and A376936.
Sum_{n>=1} = Pi^2/6 - (15/Pi^2) * (1 + Sum_{p prime} (1/(p^4-1))) - ((Sum_{p prime} (1/(p^2*(p^2-1))))^2 - Sum_{p prime} (1/(p^4*(p^2-1)^2)))/2 = 0.00015490158528995570146... . - Amiram Eldar, Dec 21 2024

A377847 Powerful numbers that are divisible by the cube of a single prime.

Original entry on oeis.org

8, 16, 27, 32, 64, 72, 81, 108, 125, 128, 144, 200, 243, 256, 288, 324, 343, 392, 400, 500, 512, 576, 625, 675, 729, 784, 800, 968, 972, 1024, 1125, 1152, 1323, 1331, 1352, 1372, 1568, 1600, 1800, 1936, 2025, 2048, 2187, 2197, 2304, 2312, 2401, 2500, 2700, 2704, 2888, 2916

Offset: 1

Amiram Eldar, Nov 09 2024

Numbers whose prime factorization contains one exponent that equals 3, and all the others, if they exist, are equal to 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Subsequence of A001694, A320966 and A377846.
A030078 is a subsequence.
Cf. A082020.

q[n_] := Module[{e = Sort[FactorInteger[n][[;; , 2]]]}, e[[-1]] > 2 && (Length[e] == 1 || e[[-2]] == 2)]; With[{max = 3000}, Select[Union@ Flatten@Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], q]]

is(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2])); e[1] > 1 && e[#e] > 2 && (#e == 1 || e[#e - 1] == 2));

Sum_{n>=1} 1/a(n) = (15/Pi^2) * Sum_{p prime} 1/((p-1)*(p^2+1)) = 0.40258439321745859629... .

A379593 Numbers that set records in A379592.

Original entry on oeis.org

8, 32, 128, 512, 2048, 8192, 20736, 41472, 82944, 165888, 186624, 373248, 746496, 1492992, 2985984, 5971968, 6718464, 11943936, 23887872, 26873856, 53747712, 107495424, 214990848, 241864704, 429981696, 859963392, 967458816, 1719926784, 3439853568, 3869835264, 7464960000

Offset: 1

Michael De Vlieger, Dec 30 2024

nonn

Proper subset of the intersection of A025487 and A320966.
Let k be a powerful number (in A001694) and let coreful d | k be such that k/d is also coreful, i.e., rad(d) = rad(d/k) = rad(k), where rad = A007947 is the squarefree kernel. Suppose d < d/k. Then coreful d may either divide k/d or not (indeed, if d/k < d, k/d may either divide d or not).
Then we have either d | k/d (the cardinality of such divisors is A379592(n) for k = A320966(n)) or d does not divide k/d (the cardinality of such divisors is A379552(n) for k = A376936(n)). (The case d = k/d, both certainly coreful, of course pertains to perfect squares k in A000290.)
Coreful divisors are counted by A361430 across natural numbers, and A370329 across powerful numbers A001694. Numbers that set records in A361430 (and A370329) are in A005934 (highly powerful numbers), with records in A036965.

			Let b(n) = A379592(n).
Table showing exponents of prime power factors of a(n) for n = 1..12. Example: a(7) = 20736 = 2^8*3^4, so "8.4" appears in the "exp." column.
   n      a(n)  exp. b(a(n))
  --------------------------
   1        8    3       1   2*4
   2       32    5       2   2*16 = 4*8
   3      128    7       3   2*64 = 4*32 = 8*16
   4      512    9       4   2*256 = 4*128 = 8*64 = 16*32
   5     2048   11       5   2*1024 = 4*512 = 8*256 = 16*128 = 32=64
   6     8192   13       6   2*4096 = 4*2048 = 8*1024 = 16*512 = 32*256 = 64*128
   7    20736    8.4     7
   8    41472    9.4     8
   9    82944   10.4     9
  10   165888   11.4    10
  11   186624    8.6    11
  12   373248    9.6    12

Michael De Vlieger, Table of n, a(n) for n = 1..119
Michael De Vlieger, Prime power decomposition of a(n), n = 1..119.

Cf. A005934, A025487, A320966, A361430, A370329, A379592, A379594.

(* Load function f at A025487 *)
r = 0; s = Union@ Flatten@ f[10]; nn = Length[s];
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
  Transpose@ Reap[Monitor[
    Do[k = s[[i]];
      If[# > r, r = #; Sow[k]] &@
        Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
          _?(And[rad[#1] == rad[#2],
            Xor[Divisible[#2, #1],
                Divisible[#1, #2]]] & @@ # &)], {i, nn}], {i, nn}] ][[-1, 1]]

cp's OEIS Frontend

A379592 Number of coreful divisor pairs (d, k/d), d | k, d < k/d, such that only one divisor divides the other, where k is in A320966.

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A379523 Sum of coreful divisors d | k such that gcd(d, k/d) > 1 and rad(d) = rad(k/d), with d | k/d and d < d/k, where k is in A320966 and rad = A007947.

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A001694 Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers).

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A376936 Powerful numbers divisible by cubes of 2 distinct primes.

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A320965 Squares divisible by a single cube > 1.

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A386428 Odd numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, k > 0, r > 1, and gcd(p,r) = 1.

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A378900 Squares of numbers divisible by the squares of two distinct primes.

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