A067259 -id:A067259 - cp's OEIS frontend

A368712 The maximal exponent in the prime factorization of the cubefree numbers.

0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1

Offset: 1

Amiram Eldar, Jan 04 2024

The asymptotic density of occurrences of 1 is zeta(3)/zeta(2) = 0.730762... (A253905), and the asymptotic density of occurrences of 2 is 1 - zeta(3)/zeta(2) = 0.269237... .

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

Cf. A004709, A005117, A008966, A013929, A033150, A051903, A067259.
Cf. A002117, A013661, A253905.
Similar sequences: A368710, A368711, A368713.

s[n_] := If[n == 1, 0, Max @@ Last /@ FactorInteger[n]]; s /@ Select[Range[100], Max[FactorInteger[#][[;; , 2]]] < 3 &]
(* or *)
f[n_] := Module[{e = Max @@ FactorInteger[n][[;; , 2]]}, If[e < 3, e, Nothing]]; f[1] = 0; Array[f, 100]

lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = vecmax(factor(k)[,2]); if(e < 3, print1(e, ", ")));}

from sympy import mobius, integer_nthroot, factorint
def A368712(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return max(factorint(m).values(),default=0) # Chai Wah Wu, Aug 12 2024

a(n) = A051903(A004709(n)).
a(n) = 2 - A008966(A004709(n)) for n >= 2.
Except for n = 1, a(n) = 1 or 2.
a(n) = 1 if and only if A004709(n) is squarefree (A005117).
a(n) = 2 if and only if A004709(n) > 1 and is nonsquarefree (A013929), i.e., A004709(n) is in A067259.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 - zeta(3)/zeta(2) = 2 - A253905 = 1.269237030598... .

A384655 a(n) = Sum_{k=1..n} A051903(gcd(n,k)).

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 1, 7, 4, 6, 1, 11, 1, 8, 7, 15, 1, 14, 1, 17, 9, 12, 1, 25, 6, 14, 13, 23, 1, 22, 1, 31, 13, 18, 11, 36, 1, 20, 15, 39, 1, 30, 1, 35, 26, 24, 1, 53, 8, 32, 19, 41, 1, 44, 15, 53, 21, 30, 1, 59, 1, 32, 34, 63, 17, 46, 1, 53, 25, 46, 1, 81, 1, 38

Offset: 1

Amiram Eldar, Jun 06 2025

nonn

The terms of this sequence can be calculated efficiently using the 1st formula. The value the of function f(n, k) is equal to the number of integers i from 1 to n such that gcd(i, n) is 1 if k = 1, or k-free if k >= 2 (k-free numbers are numbers that are not divisible by a k-th power other than 1). E.g., f(n, 1) = A000010(n), f(n, 2) = A063659(n), and f(n, 3) = A254926(n).

			a(4) = A051903(gcd(4,1)) + A051903(gcd(4,2)) + A051903(gcd(4,3)) + A051903(gcd(4,4)) = A051903(1) + A051903(2) + A051903(1) + A051903(4) = 0 + 1 + 0 + 2 = 3.

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

Cf. A000010, A033150, A005117, A050873, A051903, A051953, A063659, A067259, A254926, A357310, A383156, A384656 (unitary analog).

e[n_] := If[n == 1, 0, Max[FactorInteger[n][[;;, 2]]]]; a[n_] := Sum[e[GCD[n, k]], {k, 1, n}]; Array[a, 100]
(* or *)
f[p_, e_, k_] := p^e - If[e < k, 0, p^(e - k)]; a[n_] := Module[{fct = FactorInteger[n], emax, s}, emax = Max[fct[[;; , 2]]]; s = emax * n; Do[s -= Times @@ (f[#1, #2, k] & @@@ fct), {k, 1, emax}]; s]; a[1] = 0; Array[a, 100]

e(n) = if(n == 1, 0, vecmax(factor(n)[,2]));
a(n) = sum(k = 1, n, e(gcd(n, k)));

a(n) = if(n == 1, 0, my(f = factor(n), p = f[,1], e = f[,2], emax = vecmax(e), s = emax*n); for(k = 1, emax, s -= prod(i = 1, #p, p[i]^e[i] - if(e[i] < k, 0, p[i]^(e[i]-k)))); s);

a(n) = Sum_{k=1..A051903(n)} (n - f(n, k)) = A051903(n) * n  - Sum_{k=1..A051903(n)} f(n, k), where f(n, k) is multiplicative for a given k, with f(p^e, k) = p^e - p^(e-k) if e >= k and f(p^e, k) = p^e if e < k.
a(n) = 1 if and only if n is prime.
a(n) >= 2 if and only if n is composite.
a(n) >= A051953(n) with equality if and only if n is squarefree.
a(n) >= 2*n - A000010(n) - A063659(n) with equality if and only if n is cubefree that is not squarefree (i.e., n in A067259, or equivalently, A051903(n) = 2).
a(p^e) = (p^e-1)/(p-1) for a prime p and e >= 1.
a(n) < c*n and lim sun_{n->oo} a(n)/n = c, where c is Niven's constant (A033150).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Sum{k>=1} (1-1/zeta(2*k)) = 0.49056393035179738598... .

A325240 Numbers whose minimum prime exponent is 2.

Original entry on oeis.org

4, 9, 25, 36, 49, 72, 100, 108, 121, 144, 169, 196, 200, 225, 288, 289, 324, 361, 392, 400, 441, 484, 500, 529, 576, 675, 676, 784, 800, 841, 900, 961, 968, 972, 1089, 1125, 1152, 1156, 1225, 1323, 1352, 1369, 1372, 1444, 1521, 1568, 1600, 1681, 1764, 1800

Offset: 1

Gus Wiseman, Apr 15 2019

nonn

Or barely powerful numbers, a subset of powerful numbers A001694.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose minimum multiplicity is 2 (counted by A244515).
Powerful numbers (A001694) that are not cubefull (A036966). - Amiram Eldar, Jan 30 2023

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    9: {2,2}
   25: {3,3}
   36: {1,1,2,2}
   49: {4,4}
   72: {1,1,1,2,2}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  121: {5,5}
  144: {1,1,1,1,2,2}
  169: {6,6}
  196: {1,1,4,4}
  200: {1,1,1,3,3}
  225: {2,2,3,3}
  288: {1,1,1,1,1,2,2}
  289: {7,7}
  324: {1,1,2,2,2,2}
  361: {8,8}
  392: {1,1,1,4,4}
  400: {1,1,1,1,3,3}

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

Positions of 2's in A051904.
Maximum instead of minimum gives A067259.
Cf. A001221, A001222, A001358, A001694, A007774, A036966, A051903, A052485, A118914, A244515, A325241.
Cf. A065483, A082695.

Select[Range[1000],Min@@FactorInteger[#][[All,2]]==2&]

is(n)={my(e=factor(n)[,2]); n>1 && vecmin(e) == 2; } \\ Amiram Eldar, Jan 30 2023

from math import isqrt, gcd
from sympy import integer_nthroot, factorint, mobius
def A325240(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
        for w in range(1,integer_nthroot(x,5)[0]+1):
            if all(d<=1 for d in factorint(w).values()):
                for y in range(1,integer_nthroot(z:=x//w**5,4)[0]+1):
                    if gcd(w,y)==1 and all(d<=1 for d in factorint(y).values()):
                        c += integer_nthroot(z//y**4,3)[0]
        return c
    return bisection(f,n,n**2) # Chai Wah Wu, Oct 02 2024

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Product_{p prime} (1 + 1/(p^2*(p-1))) = A082695 - A065483 = 0.6038122832... . - Amiram Eldar, Jan 30 2023

A328305 Numbers that are cubefree, but not squarefree and whose first arithmetic derivative is not squarefree, but some k-th (with k >= 2) derivative is.

Original entry on oeis.org

50, 99, 207, 306, 531, 549, 725, 747, 819, 931, 1083, 1175, 1611, 1775, 1899, 2057, 2075, 2299, 2331, 2367, 2499, 2525, 2842, 2853, 2891, 3425, 3577, 3610, 3771, 3789, 3843, 4059, 4149, 4311, 4475, 4575, 4626, 4693, 4775, 4998, 5239, 5274, 5341, 5547, 5634, 5706, 5715, 5746, 5819, 5949, 6147, 6223, 6275, 6381, 6413, 6475, 6575

Offset: 1

Antti Karttunen, Oct 13 2019

nonn

Numbers n for which A051903(n) = 2 and A328248(n) > 2.

			50 is not squarefree, as 50 = 2 * 5^2, and neither its arithmetic derivative A003415(50) = 45 = 3^2 * 5 is squarefree, but its second derivative A003415(45) = 39 = 3*13 is, thus 50 is included in this sequence.

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

Cf A003415, A051903, A328248, A328253.

Subsequence of A067259, A328303, A328304 and A328321.

A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A328248(n) = { my(k=1); while(n && !issquarefree(n), k++; n = A003415checked(n)); (!!n*k); };
isA067259(n) = (2==A051903(n));
isA328305(n) = (isA067259(n)&&(A328248(n)>2));

A375072 Biquadratefree numbers (A046100) that are not cubefree (A004709).

Original entry on oeis.org

8, 24, 27, 40, 54, 56, 72, 88, 104, 108, 120, 125, 135, 136, 152, 168, 184, 189, 200, 216, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 360, 375, 376, 378, 392, 408, 424, 440, 456, 459, 472, 488, 500, 504, 513, 520, 536, 540, 552, 568, 584, 594, 600

Offset: 1

Amiram Eldar, Jul 29 2024

Subsequence of A176297 and first differs from it at n = 41: A176297(41) = 432 = 2^4 * 3^3 is not a term of this sequence.
Numbers whose prime factorization has least one exponent that equals 3 and no higher exponent.
Numbers k such that A051903(k) = 3.
The asymptotic density of this sequence is 1/zeta(4) - 1/zeta(3) = A215267 - A088453 = 0.0920310303408826983406... .

Intersection of A046100 and A176297.
Cf. A004709, A051903, A067259.
Cf. A002117, A013662, A088453, A215267.

Select[Range[600], Max[FactorInteger[#][[;; , 2]]] == 3 &]

is(k) = k > 1 && vecmax(factor(k)[,2]) == 3;

from sympy import mobius, integer_nthroot
def A375072(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**4-x//k**3) for k in range(1, integer_nthroot(x,4)[0]+1))+sum(mobius(k)*(x//k**3) for k in range(integer_nthroot(x,4)[0]+1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 05 2024

A375142 Numbers whose powerful part (A057521) is a power of a squarefree number that is larger than 1 (A072777).

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164

Offset: 1

Amiram Eldar, Aug 01 2024

Subsequence of A013929 and first differs from it at n = 27: A013929(27) = 72 = 2^3 * 3^2 is not a term of this sequence.
Numbers whose prime factorization has one distinct exponent that does not equal 1.
Numbers that are a product of a squarefree number (A005117) and a power of a different squarefree number that is not squarefree.
The asymptotic density of this sequence is Sum_{k>=2} (d(k)-1)/zeta(2) = 0.36113984820338109927..., where d(k) = zeta(k) * Product_{p prime} (1 + Sum_{i=k+1..2*k-1} (-1)^i/p^i), if k is even, and d(k) = (zeta(2*k)/zeta(k)) * Product_{p prime} (1 + 2/p^k + Sum_{i=k+1..2*k-1} (-1)^(i+1)/p^i) if k is odd > 1.

			12 = 2^2 * 3 is a term because its powerful part, 4 = 2^2, is a power of a squarefree number, 2, that is larger than 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical notes. III. Certain equally distributed sets of integers, Pacific Journal of Mathematics, Vol. 12, No. 1 (1962), pp. 77-84.
Index entries for sequences computed from exponents in factorization of n.

Cf. A005117, A057521.
Subsequence of A013929.
Subsequences: A067259, A072777, A190641, A336591.

q[n_] := Count[Union[FactorInteger[n][[;; , 2]]], _?(# > 1 &)] == 1; Select[Range[200], q]

is(k) = {my(e = select(x -> (x > 1), Set(factor(k)[,2]))); #e == 1;}

A380692 Numbers k such that the least prime dividing k is larger than the maximum exponent in the prime factorization of k; a(1) = 1 by convention.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 99

Offset: 1

Amiram Eldar, Jan 30 2025

First differs from A368110 at n = 68: A368110(68) = 98 is not a term of this sequence.
First differs from its subsequence A380695 at n = 428: a(428) = 625 is not a term of A380695.
Numbers k such that A020639(k) > A051903(k).
All the squarefree numbers (A005117) are terms, and all the odd terms of A067259 are terms of this sequence.
Disjoint union of the sequences S_k, k >= 1, where S_k is the sequence of p-rough numbers (numbers whose prime factors are all greater than or equal to p), with p = nextprime(k) = A151800(k), whose maximum exponent in their prime factorization is k (i.e., numbers that are (k+1)-free but not k-free, where k-free numbers are numbers whose prime factorization exponents do not exceed k).
The asymptotic density of this sequence is Sum_{i>=1} d(i) = 0.68213349032332767778..., where d(i), the density of S_i, equals f(i+1) * Product_{primes p <= i} ((1-1/p)/(1-1/p^(i+1))) - f(i) * Product_{primes p <= i} ((1-1/p)/(1-1/p^i)), f(i) = 1/zeta(i) if i >= 2, and f(1) = 0.

			6 = 2^1 * 3^1 is a term since 2 > 1.
8 = 2^3 is not a term since 2 < 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Rough Number.
Wikipedia, Rough number.

Cf. A020639, A051903, A151800, A368110.
Subsequence of A380693.
Subsequences: A005117, A136327, the intersection of A005408 and A067259, A380694, A380695.

q[k_] := k == 1 || Module[{f = FactorInteger[k]}, f[[1, 1]] > Max[f[[;; , 2]]]]; Select[Range[100], q]

isok(k) = if(k == 1, 0, my(f = factor(k), e = f[, 2]); f[1, 1] > vecmax(e));

A328304 Numbers that are cubefree, but not squarefree and whose arithmetic derivative is not squarefree.

Original entry on oeis.org

4, 12, 20, 28, 36, 44, 50, 52, 60, 68, 76, 84, 92, 99, 100, 116, 124, 132, 140, 148, 156, 164, 172, 180, 188, 196, 204, 207, 212, 220, 225, 228, 236, 244, 252, 260, 268, 275, 276, 284, 292, 300, 306, 308, 316, 332, 340, 348, 356, 364, 372, 380, 388, 396, 404, 412, 420, 428, 436, 441, 444, 452, 460, 468, 476, 484, 492, 508, 516, 524, 525

Offset: 1

Antti Karttunen, Oct 13 2019

nonn

Numbers n for which A051903(n) = 2 and A051903(A003415(n)) > 1.

			4 = 2^2 is cubefree but not squarefree, and its arithmetic derivative A003415(4) = 4 is not squarefree, thus 4 is included in this sequence.
225 = 3^2 * 5^2 is cubefree but not squarefree, and its arithmetic derivative A003415(225) = 240 = 2^4 * 3 * 5 is not squarefree, thus 225 is included in this sequence.

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

Cf. A003415, A008966, A051903.
Intersection of A067259 and A328303. Intersection of A067259 and A328321.
Cf. A328305 (a subsequence).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
isA067259(n) = (2==A051903(n));
isA328303(n) = !issquarefree(A003415(n));
isA328304(n) = (isA067259(n)&&isA328303(n));

A382965 The number of non-unitary prime divisors of the n-th cubefree number that is not squarefree.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Apr 10 2025

The positive terms of A376366.

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

Cf. A002117, A046660, A056170, A067259, A013661, A369427, A376366, A382966, A382968.

f[k_] := If[k == 1, Nothing, Module[{e = FactorInteger[k][[;; , 2]]}, If[Max[e] == 2, Count[e, 2], Nothing]]]; Array[f, 150]

list(lim) = {my(e); for(k = 2, lim, e = factor(k)[, 2]; if(vecmax(e) == 2, print1(#select(x -> x == 2, e), ", ")));}

a(n) = A056170(A067259(n)).
a(n) = A046660(A067259(n)).
a(n) = A369427(A067259(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2)/(zeta(2)-zeta(3))) * Sum_{p prime} ((p-1)/(p^3-1)) = 1.10872412837037270926... .

A072358 Number of cubefree numbers <= n which are not squarefree.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 11, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 18, 18, 18, 18, 18, 18, 18, 18, 19

Offset: 1

Reinhard Zumkeller, Jul 18 2002

nonn

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

Cf. A060431, A013928.
Cf. A008966, A212793.
Cf. A067259, A004709, A005117, A059956, A088453.

a072358 n = a072358_list !! (n-1)
a072358_list = scanl1 (+) $
   zipWith (*) a212793_list $ map (1 -) a008966_list
-- Reinhard Zumkeller, May 27 2012

Accumulate @ Table[Boole[Max @ FactorInteger[n][[;; , 2]] == 2], {n, 1, 100}] (* Amiram Eldar, Feb 16 2021 *)

from math import isqrt
from sympy import mobius, integer_nthroot
def A072358(n): return sum(mobius(k)*(n//k**3-n//k**2) for k in range(1, integer_nthroot(n,3)[0]+1))-sum(mobius(k)*(n//k**2) for k in range(integer_nthroot(n,3)[0]+1,isqrt(n)+1)) # Chai Wah Wu, Aug 12 2024

a(n) = A060431(n) - A013928(n+1).
a(n) = Sum_{k=1..n} (A212793(k) * (1 - A008966(k))). - Reinhard Zumkeller, May 27 2012
a(n) ~ c * n, where c = 1/zeta(3) - 1/zeta(2) = A088453 - A059956 = 0.22398... - Amiram Eldar, Feb 16 2021

cp's OEIS Frontend

A368712 The maximal exponent in the prime factorization of the cubefree numbers.

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A384655 a(n) = Sum_{k=1..n} A051903(gcd(n,k)).

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A325240 Numbers whose minimum prime exponent is 2.

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A328305 Numbers that are cubefree, but not squarefree and whose first arithmetic derivative is not squarefree, but some k-th (with k >= 2) derivative is.

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A375072 Biquadratefree numbers (A046100) that are not cubefree (A004709).

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A375142 Numbers whose powerful part (A057521) is a power of a squarefree number that is larger than 1 (A072777).

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A380692 Numbers k such that the least prime dividing k is larger than the maximum exponent in the prime factorization of k; a(1) = 1 by convention.

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