A372695 -id:A372695 - cp's OEIS frontend

A376936 Powerful numbers divisible by cubes of 2 distinct primes.

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

A378767 Numbers k that are not prime powers which are divisible by a cube greater than 1.

Original entry on oeis.org

24, 40, 48, 54, 56, 72, 80, 88, 96, 104, 108, 112, 120, 135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 248, 250, 264, 270, 272, 280, 288, 296, 297, 304, 312, 320, 324, 328, 336, 344, 351, 352, 360, 368, 375, 376, 378, 384

Offset: 1

Michael De Vlieger, Dec 06 2024

Products m = j*k such that omega(k) = omega(m) > omega(j), where rad(j) | k but j does not divide k, with rad = A007947 and omega = A001221.
Proper subset of A126706.
This sequence is distinct from A362148, since this sequence also contains 216, 432, etc.

			Prime decomposition of select a(n) = m, showing m = j*k:
a(1) = 24 = 2^3 * 3 = 4 * 6.
a(2) = 40 = 2^3 * 5 = 4 * 10.
a(3) = 48 = 2^4 * 3 = 8 * 6.
a(4) = 54 = 2 * 3^3 = 9 * 6.
a(5) = 56 = 2^3 * 7 = 4 * 14.
a(6) = 72 = 2^3 * 3^2 = 4 * 18.
a(9) = 96 = 2^5 * 3 = 8 * 12 = 16 * 6.
a(130) = 864 = 2^5 * 3^2 = 8 * 108 = 9 * 96 = 16 * 54, etc.

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

Cf. A024619, A046099, A126706, A362148, A372695.

Select[Select[Range[2^10], AnyTrue[FactorInteger[#][[All, -1]], # > 2 &] &], Not@*PrimePowerQ]

{a(n)} = { k : omega(k) > 1, there exists p | k such that p^3 | k }.
Intersection of A046099 and A024619.
Union of A362148 and A372695.

A378768 Squares of powerful numbers that are not prime powers.

Original entry on oeis.org

1296, 5184, 10000, 11664, 20736, 38416, 40000, 46656, 50625, 82944, 104976, 153664, 160000, 186624, 194481, 234256, 250000, 331776, 419904, 455625, 456976, 614656, 640000, 746496, 810000, 937024, 944784, 1000000, 1185921, 1265625, 1327104, 1336336, 1500625, 1679616

Offset: 1

Michael De Vlieger, Dec 06 2024

Contained in A286708, which is a proper subset of A126706.

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

Cf. A000290, A001694, A126706, A286708, A372695.

Cf. A013662, A013664, A013670.

With[{nn = 2000}, Select[Rest@ Union[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}] ], Not@*PrimePowerQ]^2]

from math import isqrt
from sympy import integer_nthroot, primepi, mobius
def A378768(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+1+sum(primepi(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length()))
    return bisection(f, n, n)**2 # Chai Wah Wu, Dec 08 2024

a(n) = A286708(n)^2.
Intersection of A000290 and A286708.
Intersection of A000290 and A372695.
Sum_{n>=1} 1/a(n) = zeta(4)*zeta(6)/zeta(12) - Sum_{p prime} (1/(p^4-p^2)) - 1 = 0.0013772572536044025109... . - Amiram Eldar, Dec 10 2024

A372841 4-full numbers that are not prime powers.

Original entry on oeis.org

1296, 2592, 3888, 5184, 7776, 10000, 10368, 11664, 15552, 20000, 20736, 23328, 31104, 34992, 38416, 40000, 41472, 46656, 50000, 50625, 62208, 69984, 76832, 80000, 82944, 93312, 100000, 104976, 124416, 139968, 151875, 153664, 160000, 165888, 186624, 194481, 200000

Offset: 1

Michael De Vlieger, May 14 2024

nonn

Numbers k such that rad(k)^4 | k and omega(k) > 1. In other words, numbers with at least 2 distinct prime factors whose prime power factors have exponents that exceed 3.
Proper subset of the following sequences: A001694, A036966, A036967, A126706, A286708, A372695.
Smallest term k with omega(k) = m is k = A002110(m)^4.

			Table of smallest 12 terms:
   n      a(n)
  -----------------------
   1     1296 = 2^4 * 3^4
   2     2592 = 2^5 * 3^4
   3     3888 = 2^4 * 3^5
   4     5184 = 2^6 * 3^4
   5     7776 = 2^5 * 3^5
   6    10000 = 2^4 * 5^4
   7    10368 = 2^7 * 3^4
   8    11664 = 2^4 * 3^6
   9    15552 = 2^6 * 3^5
  10    20000 = 2^5 * 5^4
  11    20736 = 2^8 * 3^4
  12    23328 = 2^5 * 3^6

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

Cf. A001694, A007947, A024619, A036966, A036967, A126706, A286708, A372695.

With[{nn = 200000}, Rest@ Select[Union@ Flatten@ Table[a^7 * b^6 * c^5 * d^4, {d, Surd[nn, 4]}, {c, Surd[nn/(d^4), 5]}, {b, Surd[nn/(c^5 * d^4), 6]}, {a, Surd[nn/(b^6 * c^5 * d^4), 7]}], Not@*PrimePowerQ]]

from math import gcd
from sympy import primepi, integer_nthroot, factorint
def A372841(n):
    def f(x):
        c = n+x+1+sum(primepi(integer_nthroot(x, k)[0]) for k in range(4, x.bit_length()))
        for u in range(1,integer_nthroot(x,7)[0]+1):
            if all(d<=1 for d in factorint(u).values()):
                for w in range(1,integer_nthroot(a:=x//u**7,6)[0]+1):
                    if gcd(w,u)==1 and all(d<=1 for d in factorint(w).values()):
                        for y in range(1,integer_nthroot(z:=a//w**6,5)[0]+1):
                            if gcd(w,y)==1 and gcd(u,y)==1 and all(d<=1 for d in factorint(y).values()):
                                c -= integer_nthroot(z//y**5,4)[0]
        return c
    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
    return bisection(f,n,n) # Chai Wah Wu, Sep 10 2024

Intersection of A036967 and A024619.

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^3*(p-1))) - Sum_{p prime} 1/(p^3*(p-1)) - 1 = 0.0026996042121456100761... . - Amiram Eldar, May 17 2024

A379773 Numbers that set records in in A379772.

Original entry on oeis.org

24, 96, 384, 1080, 2160, 4320, 8640, 12960, 17280, 34560, 38880, 69120, 77760, 108000, 155520, 311040, 432000, 622080, 756000, 1512000, 2268000, 3024000, 4536000, 5292000, 6804000, 9072000, 12096000, 13608000, 21168000, 27216000, 47628000, 54432000, 74088000, 81648000

Offset: 1

Michael De Vlieger, Jan 04 2025

nonn

Proper subset of the intersection A025487 and A378767.

Conjecture: a(n) is powerful (i.e., in A286708) for n >= 68. Additionally, for some n much larger than 68, a(n) may be cubefull (i.e., in A372695).

			Let b(n) = A379772(n).
Table showing exponents of prime power factors of a(n) for n = 1..20.
Example: a(5) = 2160 = 2^4 * 3^3 * 5, hence we write "4.3.1".
   n     a(n)  Exp.   b(a(n))
  ----------------------------------
   1      24   3.1      1   4*6
   2      96   5.1      2   6*16 = 8*12
   3     384   7.1      3   6*64 = 12*32 = 16*24
   4    1080   3.3.1    5   4*270 = 9*120 = 12*90 = 18*60 = 30*36
   5    2160   4.3.1    6   8*270 = 9*240 = 18*120 = 24*90 = 30*72 = 36*60
   6    4320   5.3.1    9
   7    8640   6.3.1   10
   8   12960   5.4.1   11
   9   17280   7.3.1   13
  10   34560   8.3.1   14
  11   38880   5.5.1   16
  12   69120   9.3.1   17

Michael De Vlieger, Table of n, a(n) for n = 1..171
Michael De Vlieger, Prime power decomposition of a(n), n = 1..171.
Michael De Vlieger, List of (d, k/d), d < k/d, k = a(n), n = 1..24, such that gcd(d, k/d) > 1, and shown in blue, rad(d) | k/d though d does not divide k/d, but rad(k/d) does not divide d, while in gold, rad(d) does not divide k/d but rad(k/d) | d though k/d does not divide d.

Cf. A025487, A378767, A379772, A379774.

(* Load function f at A025487 *)
r = 0; s = Select[Union@ Flatten@ f[8][[3 ;; -1]], Not @* SquareFreeQ];
rad[x_] := Times @@ FactorInteger[x][[All, 1]]; nn = Length[s];
Reap[Do[k = s[[i]];
  If[# > r, r = #; Sow[k] ] &@
    Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
      _?((m = GCD @@ {##};
      And[! MemberQ[{1, #1, #2}, m],
        And[PrimeNu[#1] < PrimeNu[#2], Divisible[#2, rad[#1]]] & @@
        SortBy[{##}, PrimeNu]]) & @@ # &)], {i, nn}] ][[-1, 1]]

cp's OEIS Frontend

A376936 Powerful numbers divisible by cubes of 2 distinct primes.

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A378767 Numbers k that are not prime powers which are divisible by a cube greater than 1.

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A378768 Squares of powerful numbers that are not prime powers.

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A372841 4-full numbers that are not prime powers.

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A379773 Numbers that set records in in A379772.

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