A120944 -id:A120944 - cp's OEIS frontend

A358971 a(1) = 1. Thereafter a(n) is least novel k != n such that rad(k) = rad(n), where rad is A007947.

1, 4, 9, 2, 25, 12, 49, 16, 3, 20, 121, 6, 169, 28, 45, 8, 289, 24, 361, 10, 63, 44, 529, 18, 5, 52, 81, 14, 841, 60, 961, 64, 99, 68, 175, 48, 1369, 76, 117, 50, 1681, 84, 1849, 22, 15, 92, 2209, 36, 7, 40, 153, 26, 2809, 72, 275, 98, 171, 116, 3481, 30, 3721

Offset: 1

David James Sycamore, Dec 07 2022

nonn

In other words, for n > 1, a(n) is the least novel k other than n which has not occurred earlier whose squarefree kernel is equal to the squarefree kernel of n.
Conjectured to be a permutation of the positive integers with primes appearing in natural order. Primes are minima, 1 and primes squared are records.
From Michael De Vlieger, Dec 07 2022: (Start)
Some consequences of definition:
There are no fixed points outside of a(1) = 1.
Prime power p^e implies a(p^e) = p^(e+1) for odd e, else p^(e-1). Hence a(p) = p^2 comprise maxima, while a(p^2) = p comprise minima.
Let lpf(m) = least prime factor of m. Squarefree m implies a(m) = lpf(m)*m and a(lpf(m)*m) = m, as seen in scatterplot in rays with slope p and 1/p, respectively. Therefore squarefree numbers are sequestered along or below a(n/2) = n/2.
Let K = rad(n); a(n) and n (such that a(n) != n) belong to the same sequence K*R_K, where R_K is the list of K-regular numbers, 1 and those whose prime divisors are restricted to p | K. For example, if K = 6, then a(n) and n belong to 6*A003586, and if K = 10, then a(n) and n belong to 10*A003592.
Observation: For m in A286708, abs(a(m) - m) is relatively small. (End)
This sequence is a self-inverse permutation of the positive integers: for any squarefree number s > 1, let v_s be the list of numbers with radical s, then for any k > 0, a(v_s(2*k)) = v_s(2*k-1) and a(v_s(2*k-1)) = v_s(2*k). - Rémy Sigrist, Dec 08 2022

			a(2) = 4 because 4 is the least number (not equal to 2) which has the same squarefree kernel as 2.
a(4) = 2 because 2 is the least unused number (not equal to 4) having the same squarefree kernel as 4

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, highlighting primes in red, composite prime powers (in A246547) in gold, composite squarefree numbers (A120944) in green, numbers neither squarefree nor prime power (in A126706) in blue, with numbers in A286706 in large light blue. Gold and light blue numbers are in A001694. Maxima are a(p) = p^2, minima are a(p^2) = p.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^10, using the same color coding as immediately above, labeled and showing quasi-rays with slopes p and 1/p.
Michael De Vlieger, Log log scatterplot of a(n) n = 1..2^16, showing n | a(n) in green, a(n) | n in red, and other terms in blue.
Index entries for sequences that are permutations of the natural numbers

Cf. A000040, A001248, A005117, A007947, A253288, A358916.

nn = 61; c[] = False; q[] = 1; f[n_] := f[n] = Times @@ FactorInteger[n][[All, 1]]; a[1] = 1; c[1] = True; u = 2; Do[Which[PrimeQ[n], k = n^2, PrimeQ@ Sqrt[n], k = Sqrt[n], True, k = f[n]; m = q[k]; While[Nand[! c[k m], k m != n, Divisible[k, f[m]]], m++]; While[Nor[c[q[k] k], Divisible[k, f[q[k]]]], q[k]++]; k *= m]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Dec 07 2022 *)

For squarefree n, a(a(n)) = n; a(p) = p^2 for p prime, and a(p^2) = p.

More terms from Michael De Vlieger, Dec 07 2022

A369417 Powerful numbers k with multiple distinct prime factors such that rad(k) is not a primorial, where rad(k) = A007947(k).

Original entry on oeis.org

100, 196, 200, 225, 392, 400, 441, 484, 500, 675, 676, 784, 800, 968, 1000, 1089, 1125, 1156, 1225, 1323, 1352, 1372, 1444, 1521, 1568, 1600, 1764, 1936, 2000, 2025, 2116, 2312, 2500, 2601, 2704, 2744, 2888, 3025, 3087, 3136, 3200, 3249, 3267, 3364, 3375, 3528

Offset: 1

Michael De Vlieger, Jan 22 2024

nonn

Numbers k such that Omega(k) > omega(k) > 1, where all prime power factors p^m have exponents m > 1, such that squarefree kernel rad(k) not in A002110, where Omega = A001222 and omega = A001221.

			Let S = A366413 = {A120944 \ A002110}.
This sequence is the union of the following infinite sets:
S(1)^2 * A003592 = 10^2 * A003592 = {100, 200, 400, 500, 800, 1000, ...}
                 = { m*S(1)^2 : rad(m) | S(1) }.
S(2)^2 * A003591 = 14^2 * A003591 = {196, 392, 784, 1372, 1568, ...}
                 = { m*S(2)^2 : rad(m) | S(2) }.
S(3)^2 * A003593 = 15^2 * A003593 = {225, 675, 1125, 2025, 3375, ...}
                 = { m*S(3)^2 : rad(m) | S(3) }, etc.

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

Cf. A001221, A001222, A002110, A055932, A120944, A126706, A286708, A366413, A369374.

With[{nn = 2^14},
  Select[
    Select[
      Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
    Not@*PrimePowerQ],
  Nand[EvenQ[#],
    Union@ Differences@ PrimePi[FactorInteger[#][[All, 1]]] == {1}] &] ]

{a(n)} = { A286708 \ A055932 }.
{a(n)} = { m*s^2 : Omega(s) = omega(s) > 1, s not in A002110, rad(m) | s }.
A286708 is the union of A369374 and this sequence.

A377713 Squarefree composite k such that floor(log n/log lpf(k)) <= omega(k), where lpf = A020639 and omega = A001221.

Original entry on oeis.org

6, 15, 21, 35, 55, 65, 77, 85, 91, 95, 115, 119, 133, 143, 161, 187, 203, 209, 217, 221, 247, 253, 259, 287, 299, 301, 319, 323, 329, 341, 377, 385, 391, 403, 407, 437, 451, 455, 473, 481, 493, 517, 527, 533, 551, 559, 583, 589, 595, 611, 629, 649, 667, 671, 689

Offset: 1

Michael De Vlieger, Nov 04 2024

Also squarefree composite k such that there exist no numbers m such that rad(m) | k and omega(m) > omega(k).
The only even term is 6.
Let P(i) = A002110(i). Numbers k = prime(i) * P(i+j)/P(i) < prime(i)^(i+j) with j ≥ 1 implies k such that omega(k) = j+1 is in the sequence.
The number k = p*m is a solution where squarefree m with lpf(m) > p is such that m < p^omega(m). For example, k = 5*7 is in the sequence since 7 < 5^2.
The number of a(n) such that lpf(a(n)) = p is finite. For example, the only terms divisible by 3 are {6, 15, 21}.

			6 is in the sequence since floor(log_2 6) = 1+floor(log_2 3) = omega(6) = 2.
10 is not in the sequence since floor(log_2 5) = 2 and omega(10) = 2, thus 1+floor(log_2 5) > omega(10). Seen another way, 2^3 < 10, but omega(8) > omega(10).
15 is in the sequence since floor(log_3 15) = 1+floor(log_3 5) = omega(15) = 2.
21 is in the sequence because 1+floor(log_3 7) = omega(21) = 2.
33 = 3*11 is not in the sequence because 11 > 3^2.
115 = 5*23 is in the sequence because 23 < 5^2.
145 = 5*29 is not in the sequence since 29 > 5^2, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Numbers k for which floor(log k / log lpf(k)) <= bigomega(k), 2024.

Cf. A000961, A001221, A002110, A007947, A020639, A120944, A376846, A377590, A377591.

s = Select[Range[1000], And[SquarefreeQ[#], CompositeQ[#]] &];
Select[s, Floor@ Log[FactorInteger[#][[1, 1]], #] <= PrimeOmega[#] &]

A029561 Quasi-Carmichael numbers to base 2: squarefree composites n such that prime p|n ==> p-2|n-2.

Original entry on oeis.org

1595, 6785, 53867, 67727, 102377, 296003, 740027, 961877, 998867, 1048817, 1270055, 1365377, 4086227, 7715567, 12266267, 15017921, 24960245, 33759467, 34918697, 36265385, 38342397, 41048777, 44535647, 48056087, 56264987, 58515347

Offset: 1

David W. Wilson

nonn

These are called 2-Korselt numbers by Beouallegue et al.

Giovanni Resta, Table of n, a(n) for n = 1..393 (terms < 10^12)
Index entries for sequences related to Carmichael numbers.
K. Bouallegue, O. Echi, R. G. E. Pinch, Korselt numbers and sets, Int. J. Numb. Theory 6 (2) (2010) 257-269

Cf. A120944, A002997.

qcm[n_, d_] := Block[{p, e}, {p, e} = Transpose@FactorInteger@n; Length[p] > 1 && Max[e] == 1 && ! MemberQ[p, d] && Max@ Mod[n-d, p-d] == 0]; Select[Range[10^5],  qcm[#, 2] &] (* Giovanni Resta, May 21 2013 *)

A069158 a(n) = Product{d|n} mu(d), product over positive divisors, d, of n, where mu(d) = Moebius function (A008683).

Original entry on oeis.org

1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, -1, 0, 1, 1, -1, 0, 0, 1, 0, 0, -1, 1, -1, 0, 1, 1, 1, 0, -1, 1, 1, 0, -1, 1, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, 1, 1, -1, 0, -1, 1, 0, 0, 1, 1, -1, 0, 1, 1, -1, 0, -1, 1, 0, 0, 1, 1, -1, 0, 0, 1, -1, 0, 1, 1, 1, 0, -1, 0, 1, 0, 1, 1, 1, 0, -1, 0, 0, 0, -1, 1, -1, 0, 1, 1

Offset: 1

Leroy Quet, Apr 08 2002

sign

Absolute value of a(n) = absolute value of mu(n).
Differs from A080323 at n=2, 105, 165, 195, 231, ..., 15015,..., 19635,.. (cf. A046389, A046391, ...) - R. J. Mathar, Dec 15 2008
Not multiplicative: For example a(2)*a(15) <> a(30). - R. J. Mathar, Mar 31 2012
Row products of table A225817. - Reinhard Zumkeller, Jul 30 2013

			a(6) = mu(1)*mu(2)*mu(3)*mu(6) = 1*(-1)*(-1)*1 = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A008683, A008966, A010051, A013929, A046389, A046391, A080323, A225817.

a069158 = product . a225817_row  -- Reinhard Zumkeller, Jul 30 2013

f := function(n); t1 := &*[MoebiusMu(d) : d in Divisors(n) ]; return t1; end function;

A069158 := proc(n)
    mul(numtheory[mobius](d),d=numtheory[divisors](n)) ;
end proc: # R. J. Mathar, May 28 2016

a[n_] := Product[MoebiusMu[d], {d, Divisors[n]}]; Array[a, 106] (* Jean-François Alcover, Feb 22 2018 *)

a(n) = vecprod(apply(moebius, divisors(n))); \\ Amiram Eldar, Feb 10 2025

a(n) = 0 if mu(n) = 0 (A013929); a(n) = -1 if n = prime; a(n) = 1 if n = squarefree composite (A120944) or 1.

a(n) = A008966(n) - 2*A010051(n). - Amiram Eldar, Feb 10 2025

A128555 a(n) = the smallest positive multiple of d(n) that does not occur earlier in the sequence, where d(n) is the number of positive divisors of n.

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 10, 12, 9, 16, 14, 18, 20, 24, 28, 5, 22, 30, 26, 36, 32, 40, 34, 48, 15, 44, 52, 42, 38, 56, 46, 54, 60, 64, 68, 27, 50, 72, 76, 80, 58, 88, 62, 66, 78, 84, 70, 90, 21, 96, 92, 102, 74, 104, 100, 112, 108, 116, 82, 120, 86, 124, 114, 7, 128, 136, 94, 126

Offset: 1

Leroy Quet, Mar 10 2007

nonn

This sequence is a permutation of the positive integers.

a(2^(p+1)) = p, where p is prime. - Michael De Vlieger, Dec 07 2022

			8 has 4 positive divisors. So a(8) is the smallest positive multiple of 4 that has yet to appear in the sequence. 4 and 8 occur among the first 7 terms of the sequence, but 12 does not. So a(8) = 12.

Ivan Neretin, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, composite prime powers (in A246547) in gold, squarefree composites (A120944) in green, numbers neither prime power nor squarefree (A126706) in blue, with numbers in A286708 in large light blue. Highlighted in light green are squarefree composites divisible by 6.

Cf. A000005, A128556, A358820 (inverse).

A128555 := proc(nmin) local a,n,d,k ; a := [1,2] ; while nops(a) < nmin do n := nops(a)+1 ; d := numtheory[tau](n) ; k := 1; while k*d in a do k := k+1 ; od; a := [op(a),k*d] ; od: RETURN(a) ; end: A128555(80) ; # R. J. Mathar, Oct 09 2007

a = {1}; Do[AppendTo[a, Min[Complement[Range[Max[a] + 1]*DivisorSigma[0,n], a]]], {n, 2, 68}]; a (* Ivan Neretin, May 03 2015 *)
nn = 120; c[] = False; q[] = 1; Do[d = DivisorSigma[0, n]; m = q[d]; While[c[m d], m++]; If[m == q[d], While[c[m d], m++]; q[d] = m]; Set[{a[n], c[m d]}, {m d, True}], {n, nn}]; Array[a, nn] (* Michael De Vlieger, Dec 07 2022 *)

from itertools import count, islice
from sympy import divisor_count as d
def agen():
    seen = set()
    for n in count(1):
        dn = d(n)
        m = dn
        while m in seen: m += dn
        yield m
        seen.add(m)
print(list(islice(agen(), 68))) # Michael S. Branicky, Dec 08 2022

More terms from R. J. Mathar, Oct 09 2007

A157488 a(1) = 1; for n > 1, a(n) = product of exponential divisors of n.

Original entry on oeis.org

1, 2, 3, 8, 5, 6, 7, 16, 27, 10, 11, 72, 13, 14, 15, 128, 17, 108, 19, 200, 21, 22, 23, 144, 125, 26, 81, 392, 29, 30, 31, 64, 33, 34, 35, 46656, 37, 38, 39, 400, 41, 42, 43, 968, 675, 46, 47, 3456, 343, 500, 51, 1352, 53, 324, 55, 784, 57, 58, 59, 1800, 61, 62, 1323, 4096

Offset: 1

Jaroslav Krizek, Mar 01 2009

nonn

The exponential divisors of a number n = Product p(i)^e(i) are all numbers of the form Product p(i)^s(i) where s(i) divides e(i) for all i.

Not multiplicative: a(3)=3 (e-divisor 3^1), a(4)=8 (e-divisors 2^1 and 2^2), but a(12)=72 (e-divisors 3*2 and 3*2^2) <> a(3)*a(4). - R. J. Mathar, Apr 14 2011

			For n = 16 = 2^4 = the a(16) = 2^(A000203(4)) = 2^7 = 128. e-divisors of number 16 is 2, 4, 16, their product is 128.

Amiram Eldar, Table of n, a(n) for n = 1..10000
József Sándor, A note on exponential divisors and related arithmetic functions, Scientia Magna, Vol. 1, No. 1 (2005), pp. 97-101.

Cf. A000005, A000027, A000040, A000203, A006881, A000961, A049419, A051377, A120944, A322791.

Similar sequences: A007955, A061537, A274029.

[ &*[ d: d in Divisors(n) | forall(t) { p: p in P | v gt 0 and e mod v eq 0 where v is Valuation(d, p) where e is Valuation(n, p) } where P is PrimeDivisors(n) ]: n in [2..64] ]; // Klaus Brockhaus, May 26 2009

f[p_, e_] := p^(DivisorSigma[1, e]/DivisorSigma[0, e]); a[n_] :=(Times @@ (f @@@ (fct = FactorInteger[n])))^(Times @@ DivisorSigma[0, Last /@ fct]); Array[a, 100] (* Amiram Eldar, Jun 03 2020 *)

a(1) = 1, a(p) = p, a(p*q) = p*q, a(p*q...*z) = pq...z, a(p^k) = p^(A000203(k)), for p, q, ..., z distinct primes and k > 1 an integer.
From Amiram Eldar, Jun 03 2020: (Start)
If n = Product_{i} p_i^e_i then a(n) = Product_{i} p_i^(sigma(e_i) * d_exp(n) / d(e_i)), where d_exp(n) = Product_{i} d(e_i) is the number of exponential divisors of n (A049419), d(e) and sigma(e) are the number of divisors (A000005) of e and their sum (A000203).
a(n) <= A007955(n) with equality if and only if n is noncomposite. (End)

a(1) = 1 from N. J. A. Sloane, Mar 02 2009

a(60) corrected by Klaus Brockhaus, May 26 2009

A158522 Dirichlet inverse of number of unitary divisors of n (A034444).

Original entry on oeis.org

1, -2, -2, 2, -2, 4, -2, -2, 2, 4, -2, -4, -2, 4, 4, 2, -2, -4, -2, -4, 4, 4, -2, 4, 2, 4, -2, -4, -2, -8, -2, -2, 4, 4, 4, 4, -2, 4, 4, 4, -2, -8, -2, -4, -4, 4, -2, -4, 2, -4, 4, -4, -2, 4, 4, 4, 4, 4, -2, 8, -2, 4, -4, 2, 4, -8, -2, -4, 4, -8, -2, -4, -2, 4, -4, -4, 4, -8, -2, -4, 2, 4, -2

Offset: 1

Jaroslav Krizek, Mar 20 2009

Abs{a(n)} = A034444(n). Examples of Dirichlet convolutions with function a(n), i.e., b(n) = Sum_{d|n} a(d)*c(n/d): a(n) * A034444(n) = A063524(n), a(n) * A000005(n) = A010052(n), a(n) * A000027(n) = A074722(n), a(n) * A000012(n) = A008836(n).

Möbius transform of Liouville's lambda function (A008836). - Wesley Ivan Hurt, Jun 22 2024

			a(60) = a(2^2*3*5) = [(-1)^2*2]*[(-1)^1*2]*[(-1)^1*2] = 2*(-2)*(-2) = 8.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A034444, A063524, A000005, A010052, A000027, A074722, A001222, A001221, A000040, A006881, A120944, A000961.

Table[LiouvilleLambda[n] 2^PrimeNu[n], {n, 1, 50}] (* Geoffrey Critzer, Mar 07 2015 *)

for(n=1,20, print1((-1)^bigomega(n)* 2^omega(n), ", ")) \\ G. C. Greubel, May 21 2017

a(n) = (-1)^A001222(n)*A034444(n) = (-1)^A001222(n)*2^A001221(n), for n >= 2.
Multiplicative with a(p^e) = 2*(-1)^e, p prime, e>0. a(p^0) = 1.
Dirichlet g.f.: zeta(2s)/(zeta(s))^2. - R. J. Mathar, Apr 02 2011
a(n) = Sum_{d|n} (-1)^Omega(d) * mu(n/d). - Wesley Ivan Hurt, Jun 22 2024

A160097 Number of non-exponential divisors of n.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, May 01 2009

nonn

The non-exponential divisors d|n of a number n = Product_i p(i)^e(i) are divisors d not of the form Product_i p(i)^s(i), s(i)|e(i) for all i.

			a(8) = 2 because 1 and 2^2 are non-exponential divisors of 8 = 2^3. 2^2 is a non-exponential divisor because 2^2 = 4 divides 8, but the exponent 2 = s(1) does not divide the exponent 3 = e(1).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A049419, A000040, A006881, A120944, A000961, A053810.

Cf. A001620, A327837.

f1[p_, e_] := e + 1; f2[p_, e_] := DivisorSigma[0, e]; a[1] = 1; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; Array[a, 100] (* Amiram Eldar, Oct 26 2021 *)

A049419(n) = { my(f = factor(n), m = 1); for(k=1, #f~, m *= numdiv(f[k, 2])); m; } \\ After Jovovic's formula for A049419.
A160097(n) = if(1==n,n,(numdiv(n) - A049419(n))); \\ Antti Karttunen, May 25 2017

a(n) = A000005(n) - A049419(n) for n >= 2.
a(1) = 1, a(p) = 1, a(p*q) = 3, a(p*q*...*z) = 2^k - 1, where the indices are p=primes (A000040), p*q = product of two distinct primes (A006881), and generally p*q*...*z = product of k (k > 0) distinct primes (A120944).
a(p^k) = k + 1 - A000005(k), where p are primes (A000040), p^k are prime powers A000961 (n>1), k = natural numbers (A000027).
a(p^q) = q - 1, where p and q are primes (A000040), and p^q = prime powers of primes (A053810).
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*A001620 - A327837 - 1). - Amiram Eldar, Feb 03 2025

Edited by R. J. Mathar, May 08 2009

A202387 Squarefree Smith numbers, cf. A006753.

Original entry on oeis.org

22, 58, 85, 94, 166, 202, 265, 274, 319, 346, 355, 382, 391, 438, 454, 483, 517, 526, 535, 562, 627, 634, 645, 654, 663, 690, 706, 762, 778, 861, 895, 913, 915, 922, 958, 985, 1086, 1111, 1165, 1219, 1255, 1282, 1507, 1581, 1626, 1633, 1642, 1678, 1795, 1822

Offset: 1

Reinhard Zumkeller, Dec 19 2011

Intersection of A006753 and A005117;
also squarefree hoax numbers: intersection of A019506 and A005117;
squarefree composite numbers m such that sum of digits of m = sum of digits of all prime factors of m.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Smith Number
Eric Weisstein's World of Mathematics, Hoax Number
Wikipedia, Smith number

Cf. A007953, A027746, A120944.

a202387 n = a202387_list !! (n-1)
a202387_list = [x | x <- a120944_list,
                    a007953 x == sum (map a007953 (a027746_row x))]

cp's OEIS Frontend

A358971 a(1) = 1. Thereafter a(n) is least novel k != n such that rad(k) = rad(n), where rad is A007947.

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A369417 Powerful numbers k with multiple distinct prime factors such that rad(k) is not a primorial, where rad(k) = A007947(k).

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A377713 Squarefree composite k such that floor(log n/log lpf(k)) <= omega(k), where lpf = A020639 and omega = A001221.

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A029561 Quasi-Carmichael numbers to base 2: squarefree composites n such that prime p|n ==> p-2|n-2.

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A069158 a(n) = Product{d|n} mu(d), product over positive divisors, d, of n, where mu(d) = Moebius function (A008683).

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A128555 a(n) = the smallest positive multiple of d(n) that does not occur earlier in the sequence, where d(n) is the number of positive divisors of n.

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A157488 a(1) = 1; for n > 1, a(n) = product of exponential divisors of n.

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