A361430 -id:A361430 - 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

A360908 Multiplicative with a(p^e) = 2*e - 1.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 5, 3, 1, 1, 3, 1, 1, 1, 7, 1, 3, 1, 3, 1, 1, 1, 5, 3, 1, 5, 3, 1, 1, 1, 9, 1, 1, 1, 9, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 7, 3, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 3, 1, 1, 3, 11, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 3, 3, 1, 1, 1, 7, 7, 1, 1, 3, 1, 1

Offset: 1

Vaclav Kotesovec, Feb 25 2023

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A048691, A322327, A361430, A367822.

a[n_] := Times @@ ((2*Last[#] - 1) & /@ FactorInteger[n]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Feb 25 2023 *)

for(n=1, 100, print1(direuler(p=2, n, (1+(1+1/X)/(1-1/X)^2))[n], ", "))

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1]=2*f[k,2]-1; f[k,2]=1); factorback(f); \\ Michel Marcus, Feb 25 2023

Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 2/(p^s*(p^s-1))).
Sum_{k=1..n} a(k) ~ c*n, where c = A367822 = Product_{p prime} (1 + 2/(p*(p-1))) = 3.279577150984783607372919498914633983999130708105267540952619534539808381...
a(n) = A361430(n^2). - Amiram Eldar, Feb 11 2024

A298826 a(n) = A298825(n)/n.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 2, -1, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -2, 0, 0, 0, 0, 4, 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, 5, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -2

Offset: 1

Mats Granvik, Jan 27 2018

sign

From Mats Granvik, Apr 06 2019: (Start)
Positions of nonzero entries appear to be given by A001694.
The limit lim_{k->oo} Sum_{n=1..k} a(n)/n appears to converge to some number.
Sum_{n=1..30000} a(n)/n = 1.31897... This appears to be in agreement with:
(1/30000)*Sum_{n<=30000} log(A014963(n))*log(A014963(n+2)) = 1.30351...
If the limit can be proven to converge to a number greater than 1, then it is true that Sum_{n<=X} log(A014963(n))*log(A014963(n+2)) > X, where ">" means "greater than" as in usual inequality notation.
The twin prime conjecture, according to Terence Tao on his blog, is that Sum_{n<=X} log(A014963(n))*log(A014963(n+2)) >> X where ">>" means "asymptotically greater than". There he also says that the first Hardy Littlewood conjecture states that Sum_{n<=X} log(A014963(n))*log(A014963(n+h)) = S(h)*X + o(X), where "S(h)" is the singular series.
Compare this to the prime number theorem which is lim_{X->oo} (Sum_{n<=X} log(A014963(n)))/X = 1.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..2500
Mats Granvik, Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture.
Terence Tao, Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges. See second formula.

Cf. A298674, A298824, A298825, A001694, A361430.

nn = 90;
A = Table[Boole[Mod[n, k] == 0], {n, nn}, {k, nn}];
B = Table[If[Mod[k, n] == 0, MoebiusMu[n]*n, 0], {n, nn}, {k, nn}];
T = (A.B);
TwinMangoldt = Table[a = T[[All, kk]];
    F1 = Table[If[Mod[n, k] == 0, a[[n/k]], 0], {n, nn}, {k, nn}];
    b = T[[All, kk + 2]];
    F2 = Table[ If[Mod[n, k] == 0, b[[n/k]], 0], {n, nn}, {k, nn}];
    (F1.F2)[[All, 1]], {kk, nn - 2}];
TT = Transpose[TwinMangoldt];
Table[Sum[TT[[n, k]], {k, n}]/n, {n, nn - 2}]
(* This faster alternate conjectured program agrees with Antti Karttunen's precomputed list of numbers. *)
nn = 108; b = 4*Select[Range[1, nn, 2], SquareFreeQ]; bb = Table[DivisorSigma[0, n]*(MoebiusMu[n] + Sum[If[b[[j]] == n, LiouvilleLambda[n]*2/3, 0], {j, 1, Length[b]}]), {n, 1, nn}];
cc = Table[Sum[If[Mod[n, k] == 0, bb[[n/k]]*DivisorSigma[0, k], 0], {k, 1, n}], {n, 1, nn}] (* Mats Granvik, Mar 17 2019 *)
(* Dirichlet generating function *) s=7; nn=2500; N[Zeta[s]^2*Product[(1 - 2 Prime[j]^(-s)), {j, 1, nn}]*(1 + Sum[1/2/2^(n*(s - 1)), {n, 2, nn}]), 40] (* Mats Granvik, Apr 06 2019 *)

up_to = 256;
DirConv(ma,h) = { my(u = matsize(ma)[1], md = matrix(u,u)); for(n=1,u-h,for(k=1,u,md[n,k] = sumdiv(k,d,ma[n,d]*ma[n+h,k/d]))); (md); };
A298826list(up_to) = { my(h=2, matA = matrix(up_to+h,up_to+h,n,k,!(n%k)), matB = matrix(up_to+h,up_to+h,n,k,(!(k%n))*moebius(n)*n), matT = matA*matB, matD = DirConv(matT,2)); vector(up_to,i,(1/i)*sum(j=1,i,matD[j,i])); };
v298826 = A298826list(up_to);
A298826(n) = v298826[n]; \\ Antti Karttunen, Sep 30 2018

a(n) = Sum_{k=1..n} (A298674(k, n))/n = A298825(n)/n.
Conjecture: a(n) = (-1)^(n+1) * Sum_{d|n} A076479(d). - Daniel Suteu, Apr 04 2019
From Mats Granvik, Apr 06 2019: (Start)
The Dirichlet generating function, after Daniel Suteu above and Álvar Ibeas in A076479, appears to be: Sum_{n>=1} a(n)/n^s = zeta(s)^2*(Product_{j>=1} (1 - 2*prime(j)^(-s)))*(1 + Sum_{n>=2} ((1/2)/2^(n*(s - 1)))).
Conjectured formula: Let b(n) = 4*A056911(n) and c(n) = A000005(n)*A008683(n) + Sum_{j=1..length(b(1..N))} [b(j)=n]*A008836(n)*2/3) then a(n) = Sum_{k=1..n}[k|n] c(n/k)*A000005(k). (End)
Conjecture: a(n) = (-1)^(n+1) * Sum_{d|n} mu(d)*tau(d)*tau(n/d). - Ridouane Oudra, Nov 19 2019
The conjectured Dirichlet generating function simplifies to: Sum_{n>=1} a(n)/n^s = zeta(s)^2*(Product_{j>=1} (1 - 2*prime(j)^(-s)))*(1 + 2^(1 - s)/(2^s - 2)). - Steven Foster Clark, Sep 12 2022
Conjecture: abs(a(n)) = A361430(n). - Vaclav Kotesovec, Mar 12 2023
Conjecture: a(n) is multiplicative with a(p^e) = (-1)^p * (e-1) for prime p and e > 0. That conforms to the conjectured Dirichlet generating function (compare Steven Foster Clark, Sep 12 2022). - Werner Schulte, Jun 09 2025

More terms from Antti Karttunen, Sep 30 2018

A370329 a(n) is the number of coreful divisors of the n-th powerful number that are also powerful numbers.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 4, 1, 1, 5, 2, 3, 1, 2, 1, 2, 6, 3, 1, 1, 2, 4, 1, 4, 7, 4, 1, 3, 2, 1, 2, 3, 6, 1, 1, 2, 8, 1, 5, 3, 6, 2, 1, 5, 3, 4, 1, 8, 1, 1, 2, 4, 4, 9, 1, 2, 6, 1, 1, 9, 2, 2, 2, 1, 2, 1, 1, 4, 5, 1, 10, 1, 2, 1, 3, 8, 6, 3, 10, 1, 6, 2, 1, 7, 2, 3

Offset: 1

Amiram Eldar, Feb 15 2024

A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n (see A307958).

The positive terms of A361430.

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

Cf. A001694, A002117, A062503, A078434, A307958, A360908 (analogous with squares), A361430, A370328.

f[p_, e_] := e - 1; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{max = 10^4}, s /@ Union@ Flatten@ Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]]

lista(kmax) = {my(e); for(k = 1, kmax, e = factor(k)[,2]; if(k == 1 || vecmin(e) > 1, print1(prod(i = 1, #e, e[i]-1), ", ")));}

a(n) = A361430(A001694(n)).
a(n) = 1 if and only if n is the square of a squarefree number (A062503).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3/2) * zeta(3) * Product_{p prime} (1 + 2/p^2 + 2/p^(5/2) - 1/p^3 - 2/p^(7/2) - 2/p^4) = 6.91748056612108993003... . (The infinite product of primes is the value of f(1/2) in A361430).

A364988 a(n) is the sum of coreful divisors d of n such that n/d is also a coreful divisor.

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 0, 6, 3, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 12, 0, 0, 0, 0, 30, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 62, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 39, 0, 0, 0, 0

Offset: 1

Amiram Eldar, Aug 15 2023

A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n (see A307958).

The number of these divisors is A361430(n).

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

Cf. A001694, A307958, A361430.

Similar sequences: A000203, A057723 (sum of coreful divisors).

f[p_, e_] := (p^e - 1)/(p-1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^f[i,2] - 1)/(f[i,1] - 1) - 1);}

Multiplicative with a(p^e) = (p^e - 1)/(p-1) - 1.
Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 + (2*p - p^s*(p+1))/p^(2*s)).
a(n) > 0 if and only if n is powerful (A001694).
a(n) <= n with equality only when n = 1.
a(p^2) = p for a prime p.

A335850 Cubefull highly composite numbers: numbers with a record number of cubefull divisors (A190867).

Original entry on oeis.org

1, 8, 16, 32, 64, 128, 256, 512, 1024, 1728, 2592, 5184, 7776, 10368, 15552, 20736, 31104, 46656, 62208, 93312, 124416, 186624, 248832, 373248, 559872, 746496, 1119744, 1492992, 2239488, 2985984, 3359232, 4478976, 6718464, 8957952, 13436928, 17915904, 26873856

Offset: 1

Amiram Eldar, Jun 26 2020

nonn

The analogous sequence of squarefull highly composite numbers is the sequence of highly powerful numbers (A005934).
The corresponding record values are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, ... (see the link for more values).
Also, indices of records in A361430, i.e., numbers k with a record number of coreful divisors d such that k/d is also a coreful divisor of k (a coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, see A307958). - Amiram Eldar, Aug 15 2023

Amiram Eldar, Table of n, a(n) for n = 1..832
Amiram Eldar, Table of n, a(n), A190867(a(n)) for n = 1..832

Subsequence of A025487.

Cf. A002182, A005934, A190867, A307958, A361430.

f[p_, e_] := Max[1, e-1] ; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]); s = {}; dm = 0; Do[d1 = d[n]; If[d1 > dm, dm = d1; AppendTo[s, n]], {n, 1, 10^5}]; s

d(n) = vecprod(apply(x->max(1, x-1), factor(n)[, 2]));
lista(kmax) = {my(dm = 0, d1); for(k = 1, kmax, d1 = d(k); if(d1 > dm, dm = d1; print1(k, ", ")));} \\ Amiram Eldar, Aug 15 2023

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]]

A380691 Number of divisors d | k, d < k/d, such that (d, k/d) are neither unitary nor both coreful, where k is neither squarefree nor prime power (in A126706).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 2, 1, 1, 3, 1, 1, 3, 2, 2, 2, 1, 4, 1, 1, 2, 2, 3, 3, 1, 1, 4, 1, 2, 2, 2, 2, 2, 4, 1, 1, 2, 2, 1, 2, 4, 3, 1, 4, 1, 1, 1, 3, 5, 2, 1, 2, 5, 2, 2, 3, 2, 1, 3, 1, 4, 2, 4, 2, 2, 2, 2, 1, 6, 1, 1, 1, 2, 2, 5, 2, 1, 4, 1

Offset: 1

Michael De Vlieger, Feb 09 2025

A divisor d | k is said to be coreful if rad(d) = rad(k), where rad = A007947.
In other words, half the number of divisors d | k such that both gcd(d, k/d) > 1 and rad(d) != rad(k/d).
Divisors d and k/d have at least 1 prime factor in common and at least one prime factor q divides one but not the other divisor. Thus, the reference domain S is the intersection of nonsquarefree numbers (k in A013929) and numbers that are not prime powers (k in A024619).
Let S = { prime p : p | d } and let T = { prime p : p | k/d }. Then this sequence counts divisor pairs (d, k/d), d < k/d, such that the symmetric difference of S and T is not empty. For instance, for k = 24 = 2*12 = 4*6, where, in both cases, the product P of the symmetric difference is 3. For k = 180 = 2*90 = 3*60 = 6*30 = 10*18 = 12*15, the products of symmetric differences are 15, 10, 5, 15, and 10, respectively. In the case of 10*18, it is evident that neither rad(10) = rad(180) nor rad(18) = rad(30).

			Table of n, a(n) listing divisors d and S(n)/d for select values of n:
    n  S(n) a(n)  d*S(n)/d
  ---------------------------------------------------------------------
    1    12   1   2*6
    2    18   1   3*6
    3    20   1   2*10
    4    24   2   2*12, 4*6
    5    28   1   2*14
    6    36   2   2*18, 3*12
    7    40   2   2*20, 4*10
   10    48   3   2*24, 4*12, 6*8
   26    96   4   2*48, 4*24, 6*16, 8*12
   57   180   5   2*90, 3*60, 6*30, 10*18, 12*15
   77   240   6   2*120, 4*60, 6*40, 8*30, 10*24, 12*20
  123   360   8   2*180, 3*120, 4*90, 6*60, 10*36, 12*30, 15*24, 18*20

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

Cf. A000005, A001221, A007947, A013929, A024619, A034444, A126706, A361430.

Table[1/2*(DivisorSigma[0, k] - 2^PrimeNu[k] - Apply[Times, FactorInteger[k][[All, -1]] - 1]), {k, Select[Range[12, 240], Nor[PrimePowerQ[#], SquareFreeQ[#]] &] }]

Let tau = A000005, omega = A001221, rad = A007947, and S = A126706.
a(n) = card({ d | k : d < k/d, gcd(d, k/d) > 1, rad(d) != rad(k/d) }), k = S(n).
For k in S(n), a(n) = 1/2 * tau(k) - 2^omega(k) - Product_{p|k} m-1, where p^m | k but p^(m-1) does not divide k.
For k = S(n), a(n) = 1/2 * (A000005(k) - A034444(k) - A361430(k)).

cp's OEIS Frontend

A376936 Powerful numbers divisible by cubes of 2 distinct primes.

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A360908 Multiplicative with a(p^e) = 2*e - 1.

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A298826 a(n) = A298825(n)/n.

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A370329 a(n) is the number of coreful divisors of the n-th powerful number that are also powerful numbers.

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A364988 a(n) is the sum of coreful divisors d of n such that n/d is also a coreful divisor.

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A335850 Cubefull highly composite numbers: numbers with a record number of cubefull divisors (A190867).

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A379593 Numbers that set records in A379592.

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A380691 Number of divisors d | k, d < k/d, such that (d, k/d) are neither unitary nor both coreful, where k is neither squarefree nor prime power (in A126706).

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