A065331 -id:A065331 - cp's OEIS frontend

A169611 Number of prime divisors of n that are not greater than 3, counted with multiplicity.

0, 1, 1, 2, 0, 2, 0, 3, 2, 1, 0, 3, 0, 1, 1, 4, 0, 3, 0, 2, 1, 1, 0, 4, 0, 1, 3, 2, 0, 2, 0, 5, 1, 1, 0, 4, 0, 1, 1, 3, 0, 2, 0, 2, 2, 1, 0, 5, 0, 1, 1, 2, 0, 4, 0, 3, 1, 1, 0, 3, 0, 1, 2, 6, 0, 2, 0, 2, 1, 1, 0, 5, 0, 1, 1, 2, 0, 2, 0, 4, 4, 1, 0, 3, 0, 1, 1, 3, 0, 3, 0, 2, 1, 1, 0, 6, 0, 1, 2, 2, 0, 2, 0, 3, 1

Offset: 1

Juri-Stepan Gerasimov, Dec 03 2009

nonn

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

Cf. A001222, A106799.

Cf. A007814, A007949, A065331.

a169611 = a001222 . a065331  -- Reinhard Zumkeller, Nov 19 2015

A169611 := proc(n) local f; a := 0 ; for f in ifactors(n)[2] do if op(1,f) <= 3 then a := a+op(2,f) ; end if; end do: return a; end proc: seq(A169611(n),n=1..100) ; # R. J. Mathar, Dec 04 2009

f[n_] := Plus @@ Last /@ Select[ FactorInteger@ n, 1 < #[[1]] < 4 &]; Array[f, 105] (* Robert G. Wilson v, Dec 19 2009 *)

A169611(n)=valuation(n,2)+valuation(n,3)  \\ M. F. Hasler, Aug 24 2012

a(n) = A001222(n) - A106799(n).
a(n) = A007814(n) + A007949(n). - R. J. Mathar, Dec 04 2009
a(n) = A001222(A065331(n)). - Reinhard Zumkeller, Nov 19 2015
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/2. - Amiram Eldar, Jan 16 2022

Definition corrected by M. F. Hasler, Aug 24 2012

A355582 a(n) is the largest 5-smooth divisor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 1, 8, 9, 10, 1, 12, 1, 2, 15, 16, 1, 18, 1, 20, 3, 2, 1, 24, 25, 2, 27, 4, 1, 30, 1, 32, 3, 2, 5, 36, 1, 2, 3, 40, 1, 6, 1, 4, 45, 2, 1, 48, 1, 50, 3, 4, 1, 54, 5, 8, 3, 2, 1, 60, 1, 2, 9, 64, 5, 6, 1, 4, 3, 10, 1, 72, 1, 2, 75, 4, 1, 6, 1, 80

Offset: 1

Amiram Eldar, Jul 08 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Cf. A006519, A007814, A007949, A038500, A051037, A060904, A065331, A112765, A132741, A165725, A355583, A355584.

Cf. A379005 (rgs-transform), A379006 (ordinal transform).

a[n_] := Times @@ ({2, 3, 5}^IntegerExponent[n, {2, 3, 5}]); Array[a, 100]

a(n) = 3^valuation(n, 3) * 5^valuation(n, 5) << valuation(n, 2);

from sympy import multiplicity as v
def a(n): return 2**v(2, n) * 3**v(3, n) * 5**v(5, n)
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Jul 08 2022

Multiplicative with a(p^e) = p^e if p <= 5 and 1 otherwise.
a(n) = A006519(n) * A038500(n) * A060904(n).
a(n) = 2^A007814(n) * 3^A007949(n) * 5^A112765(n).
a(n) = n / A165725(n).
Dirichlet g.f.: zeta(s)*(2^s-1)*(3^s-1)*(5^s-1)/((2^s-2)*(3^s-3)*(5^s-5)). - Amiram Eldar, Dec 25 2022
Sum_{k=1..n} a(k) ~ 2*n*log(n)^3 / (45*log(2)*log(3)*log(5)) + O(n*log(n)^2). - Vaclav Kotesovec, Apr 20 2025

A072079 Sum of 3-smooth divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 1, 12, 1, 15, 13, 3, 1, 28, 1, 3, 4, 31, 1, 39, 1, 7, 4, 3, 1, 60, 1, 3, 40, 7, 1, 12, 1, 63, 4, 3, 1, 91, 1, 3, 4, 15, 1, 12, 1, 7, 13, 3, 1, 124, 1, 3, 4, 7, 1, 120, 1, 15, 4, 3, 1, 28, 1, 3, 13, 127, 1, 12, 1, 7, 4, 3, 1, 195, 1, 3, 4, 7, 1, 12, 1, 31, 121

Offset: 1

Reinhard Zumkeller, Jun 13 2002

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

Cf. A000203, A003586, A065331, A072078.

Cf. A001620, A082633.

f[p_, e_] := If[p > 3, 1, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

a(n) = (2^(valuation(n, 2)+1)-1)*(3^(valuation(n, 3)+1)-1)/2; \\ Amiram Eldar, Dec 01 2022

a(n) = (2^(A007814(n)+1)-1)*(3^(A007949(n)+1)-1)/2.
a(n) = A000203(A065331(n)).
Multiplicative with a(2^e) = 2^(e+1)-1, a(3^e) = (3^(e+1)-1)/2, a(p^e) = 1, p>3. Christian G. Bower, May 20 2005
From Amiram Eldar, Dec 01 2022: (Start)
Dirichlet g.f.: zeta(s)*(2^s/(2^s-2))*(3^s/(3^s-3)).
Sum_{k=1..n} a(k) ~ c_1 * (n * log(n)^2 + c_2 * n * log(n) + c_3 * n), where c_1 = 1/(2*log(2)*log(3)) = 0.656598..., c_2 = (2*gamma - 2 + log(6)) = 0.9461907..., and c_3 = (log(6)^2 + log(2)*log(3))/6 - (log(6)-2)*(1-gamma) - 2*gamma_1 = 0.895656..., gamma is Euler's constant (A001620), and gamma_1 is the 1st Stieltjes constant (A082633). (End)

A171182 Period 6: repeat [0, 1, 1, 1, 0, 2].

Original entry on oeis.org

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

Offset: 1

Juri-Stepan Gerasimov, Dec 04 2009, Dec 07 2009

The number of divisors d of n of the form d=2 or 3. - Vladimir Shevelev, May 21 2010
a(n) = s(n+6), where s(k) is the number of partitions of k into distinct parts such that max(p) = 2 + min(p) for k >= 1, and (s(0)..s(6)) = (0,0,0,0,1,0,2). - Clark Kimberling, Apr 15 2014
Number of r X s integer-sided rectangles such that r < s, r + s = 2n, r | s and (s - r)/2 | s. - Wesley Ivan Hurt, Apr 24 2020
Number of positive integer solutions, (r,s,t), of the equation r^2 + t*s^2 = (n + 6)^2, where r + s = n + 6 and t < r <= s. For example, when n=6 we have the two solutions (4,8,2) and (6,6,3) since 4^2 + 2*8^2 = 12^2 and 6^2 + 3*6^2 = 12^2. - Wesley Ivan Hurt, Oct 04 2020

Vladimir Shevelev, A recursion for divisor function over divisors belonging to a prescribed finite sequence of positive integers and a solution of the Lahiri problem for divisor function sigma_x(n), arXiv:0903.1743 [math.NT], 2009. [From _Vladimir Shevelev_, May 21 2010]
Index entries for linear recurrences with constant coefficients, signature (-1,0,1,1).

Cf. A178142. - Vladimir Shevelev, May 21 2010
Cf. A115357.
Cf. A001221, A059841, A065331, A079978, A089128, A169611.
Number of distinct prime factors <= p: this sequence (p=3), A178146 (p=5), A210679 (p=7).

&cat[[0, 1, 1, 1, 0, 2]^^20]; // Wesley Ivan Hurt, Jun 19 2016

A171182:=n->(1+floor((n-3)^2/2)) mod 3: seq(A171182(n), n=1..100); # Wesley Ivan Hurt, Aug 27 2014

PadRight[{},110,{0,1,1,1,0,2}] (* Harvey P. Dale, Jan 28 2013 *)
LinearRecurrence[{-1, 0, 1, 1},{0, 1, 1, 1},105] (* Ray Chandler, Aug 26 2015 *)

a(n) = A115357(n-2) for n>1. - R. J. Mathar, Dec 09 2009
a(2) = 1, a(3) = 1, a(5) = 0, otherwise a(n) = a(n-2) + a(n-3) - a(n-5), where we put a(n) = 0, if n<0. - Vladimir Shevelev, May 21 2010
a(n) = floor(((n+1) mod 6)/3) + 2*floor(((n+5) mod 6)/5). - Gary Detlefs, Feb 15 2014
From Wesley Ivan Hurt, Aug 27 2014: (Start)
G.f.: (2+2*x+x^2)/(1+x-x^3-x^4).
a(n) + a(n-1) = a(n-3) + a(n-4) for n>4.
a(n) = (1 + floor((n-3)^2/2)) mod 3. (End)
a(n) = (5 + 3*cos(n*Pi) + 4*cos(2*n*Pi/3))/6. - Wesley Ivan Hurt, Jun 19 2016
From Amiram Eldar, Sep 16 2023: (Start)
Additive with a(p^e) = 1 if p <= 3, and 0 otherwise.
a(n) = A059841(n) + A079978(n).
a(n) = A001221(A089128(n)).
a(n) = A001221(A065331(n)). (End)

Edited by Charles R Greathouse IV, Mar 23 2010

A244417 Exponents of 6 in appearing in the 6-adic value of 1/n, n>=1 (A244416).

Original entry on oeis.org

0, 1, 1, 2, 0, 1, 0, 3, 2, 1, 0, 2, 0, 1, 1, 4, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 3, 2, 0, 1, 0, 5, 1, 1, 0, 2, 0, 1, 1, 3, 0, 1, 0, 2, 2, 1, 0, 4, 0, 1, 1, 2, 0, 3, 0, 3, 1, 1, 0, 2, 0, 1, 2, 6, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 2, 0, 1, 0, 4, 4, 1, 0, 2, 0, 1, 1, 3, 0, 2, 0, 2, 1, 1, 0, 5, 0, 1, 2, 2

Offset: 1

Wolfdieter Lang, Jul 02 2014

For the definition of 'g-dic value of 1/n' see a comment on A244416. In the Mahler reference, p. 7, the present exponent of 6 is there called f = f(1/n) for g = 6.

			See A244416.

Kurt Mahler, p-adic numbers and their functions, second ed., Cambridge University Press, 1981.

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A122841, A244416, A007814 (g=2), A007949 (g=3), A244415 (g=4), A112765 (g=5), A051903, A065331.

Cf. also A322026, A322316.

a[n_] := Max[IntegerExponent[n, {2, 3}]]; Array[a, 100] (* Amiram Eldar, Aug 19 2024 *)

A244417(n) = max(valuation(n,2), valuation(n,3)); \\ Antti Karttunen, Dec 04 2018

a(n) = 0 if n is congruent 1 or 5 (mod 6). a(n) = max(A007814(n), A007949(n)) if n == 0 (mod 6). a(n) = A007814(n) if n == 2 or 4 (mod 6) and a(n) = A007949(n) if n == 3 (mod 6).
a(n) = max(A007814(n), A007949(n)), in all cases. - Antti Karttunen, Dec 04 2018
From Amiram Eldar, Aug 19 2024: (Start)
a(n) = A051903(A065331(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 13/10. (End)

A248909 Completely multiplicative with a(p) = p if p = 6k+1 and a(p) = 1 otherwise.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 13, 7, 1, 1, 1, 1, 19, 1, 7, 1, 1, 1, 1, 13, 1, 7, 1, 1, 31, 1, 1, 1, 7, 1, 37, 19, 13, 1, 1, 7, 43, 1, 1, 1, 1, 1, 49, 1, 1, 13, 1, 1, 1, 7, 19, 1, 1, 1, 61, 31, 7, 1, 13, 1, 67, 1, 1, 7, 1, 1, 73, 37, 1, 19, 7, 13, 79, 1, 1

Offset: 1

Tom Edgar, Mar 06 2015

To compute a(n) replace primes not of the form 6k+1 in the prime factorization of n by 1.
The first place this sequence differs from A170824 is at n = 49.
For p prime, a(p) = p if p is a term in A002476 and a(p) = 1 if p = 2, p = 3 or p is a term in A007528.
a(n) is the largest term of A004611 that divides n. - Peter Munn, Mar 06 2021

			a(49) = 49 because 49 = 7^2 and 7 = 6*1 + 1.
a(15) = 1 because 15 = 3*5 and neither of these primes is of the form 6k+1.
a(62) = 31 because 62 = 31*2, 31 = 6*5 + 1, and 2 is not of the form 6k+1.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Index to divisibility sequences

Sequences used in a definition of this sequence: A002476, A004611, A007528, A020639, A028234, A032742.
Equivalent sequence for distinct prime factors: A170824.
Equivalent sequences for prime factors of other forms: A000265 (2k+1), A343430 (3k-1), A170818 (4k+1), A097706 (4k-1), A343431 (6k-1), A065330 (6k+/-1), A065331 (<= 3).

A248909 := proc(n)
    local a,pf;
    a := 1 ;
    for pf in ifactors(n)[2] do
        if modp(op(1,pf),6) = 1 then
            a := a*op(1,pf)^op(2,pf) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Mar 14 2015

f[p_, e_] := If[Mod[p, 6] == 1, p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

a(n) = {my(f = factor(n)); for (i=1, #f~, if ((f[i,1] - 1) % 6, f[i, 1] = 1);); factorback(f);} \\ Michel Marcus, Mar 11 2015

from sympy import factorint
def A248909(n):
    y = 1
    for p,e in factorint(n).items():
        y *= (1 if (p-1) % 6 else p)**e
    return y # Chai Wah Wu, Mar 15 2015

n=100; sixnplus1Primes=[x for x in primes_first_n(100) if (x-1)%6==0]
[prod([(x^(x in sixnplus1Primes))^y for x,y in factor(n)]) for n in [1..n]]

(define (A248909 n) (if (= 1 n) n (* (if (= 1 (modulo (A020639 n) 6)) (A020639 n) 1) (A248909 (A032742 n))))) ;; Antti Karttunen, Jul 09 2017

a(1) = 1; for n > 1, if A020639(n) = 1 (mod 6), a(n) = A020639(n) * a(A032742(n)), otherwise a(n) = a(A028234(n)). - Antti Karttunen, Jul 09 2017

a(n) = a(A065330(n)). - Peter Munn, Mar 06 2021

A099316 Greatest 3-smooth number dividing the n-th minimal number.

Original entry on oeis.org

1, 2, 4, 6, 12, 16, 24, 36, 48, 12, 64, 24, 144, 36, 192, 48, 72, 576, 144, 24, 36, 192, 1024, 36, 1296, 48, 72, 576, 3072, 144, 4096, 144, 5184, 36, 1296, 192, 216, 9216, 288, 12288, 576, 432, 3072, 576, 144, 5184, 72, 1296, 36864, 36, 1296, 9216, 46656, 288

Offset: 1

Reinhard Zumkeller, Oct 12 2004

nonn

A minimal number is the smallest number with a given number of divisors, see A007416.

David A. Corneth, Table of n, a(n) for n = 1..20000 (first 1000 terms from Amiram Eldar)

Cf. A007416, A065331, A099315, A003586, A099312, A099314.

a(n) = A065331(A007416(n)).

A265398 Perform one x^2 -> x+1 reduction for the polynomial with nonnegative integer coefficients that is encoded in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 4, 6, 6, 15, 8, 9, 12, 35, 12, 77, 30, 18, 16, 143, 18, 221, 24, 45, 70, 323, 24, 36, 154, 27, 60, 437, 36, 667, 32, 105, 286, 90, 36, 899, 442, 231, 48, 1147, 90, 1517, 140, 54, 646, 1763, 48, 225, 72, 429, 308, 2021, 54, 210, 120, 663, 874, 2491, 72, 3127, 1334, 135, 64, 462, 210, 3599, 572, 969, 180, 4087, 72

Offset: 1

Antti Karttunen, Dec 15 2015

Completely multiplicative with a(2) = 2, a(3) = 3, a(prime(k)) = prime(k-1) * prime(k-2) for k > 2. - Andrew Howroyd & Antti Karttunen, Aug 04 2018

Antti Karttunen, Table of n, a(n) for n = 1..10080

Cf. A064989, A065330, A065331.

Cf. also A192232, A206296, A265399.

a[n_] := a[n] = Module[{k, p, e}, Which[n<4, n, PrimeQ[n], k = PrimePi[n]; Prime[k-1] Prime[k-2], True, Product[{p, e} = pe; a[p]^e, {pe, FactorInteger[n]}]]];
a /@ Range[1, 72] (* Jean-François Alcover, Sep 20 2019 *)
f[p_, e_] := If[p < 5, p, NextPrime[p,-1]*NextPrime[p,-2]]^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 01 2022 *)

A065330(n) = { while(0 == (n%2), n = n/2); while(0 == (n%3), n = n/3); n; }
A065331 = n -> n/A065330(n);
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A265398(n) = { my(a); if(1 == n, n, a = A064989(A065330(n)); A064989(a)*a*A065331(n)); };

r(p) = {my(q = precprime(p-1)); q*precprime(q-1)};
a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i,1]<5, f[i,1], r(f[i,1]))^f[i,2])}; \\ Amiram Eldar, Dec 01 2022

(definec (A265398 n) (if (= 1 n) n (* (A065331 n) (A064989 (A065330 n)) (A064989 (A064989 (A065330 n))))))

a(1) = 1; for n > 1, a(n) = A064989(A064989(A065330(n))) * A064989(A065330(n)) * A065331(n).

Sum_{k=1..n} a(k) = c * n^3, where c = (1/3) * Product_{p prime} (p^3-p^2)/(p^3-a(p)) = 0.093529982... . - Amiram Eldar, Dec 01 2022

Keyword mult added by Antti Karttunen, Aug 04 2018

A265399 Repeatedly perform x^2 -> x+1 reduction for polynomial (with nonnegative integer coefficients) encoded in prime factorization of n, until the polynomial is at most degree 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 6, 18, 8, 9, 12, 108, 12, 1944, 36, 18, 16, 209952, 18, 408146688, 24, 54, 216, 85691213438976, 24, 36, 3888, 27, 72, 34974584955819144511488, 36, 2997014624388697307377363936018956288, 32, 324, 419904, 108, 36, 104819342594514896999066634490728502944926883876041385836544, 816293376, 5832, 48

Offset: 1

Antti Karttunen, Dec 15 2015

In terms of integers: apply A265398 as many times as necessary to n, until it gets 3-smooth, one of the terms of A003586.

Completely multiplicative with a(2) = 2, a(3) = 3, a(p) = a(A265398(p)) for p > 3. - Andrew Howroyd & Antti Karttunen, Aug 04 2018

Antti Karttunen, Table of n, a(n) for n = 1..60

Cf. A003586 (fixed points), A065331.

Cf. also A192232, A206296, A265398, A265752, A265753.

f[p_, e_] := If[p < 5, p, a[NextPrime[p, -1] * NextPrime[p, -2]]]^e; a[1] = 1; a[n_] := a[n] = Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Sep 07 2023 *)

\\ Needs also code from A265398.
A265399(n) = if(A065331(n) == n, n, A265399(A265398(n)));
for(n=1, 60, write("b265399.txt", n, " ", A265399(n)));

(definec (A265399 n) (if (= (A065331 n) n) n (A265399 (A265398 n))))

If A065331(n) = n [that is, when n is one of 3-smooth numbers, A003586] then a(n) = n, otherwise a(n) = a(A265398(n)).
Other identities. For all n >= 1:
a(n) = 2^A265752(n) * 3^A265753(n).

Keyword mult added by Antti Karttunen, Aug 04 2018

A382488 The number of unitary 3-smooth divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 2

Offset: 1

Amiram Eldar, Mar 29 2025

Period 6: repeat [1, 2, 2, 2, 1, 4].
Decimal expansion of 407380/333333.
Continued fraction expansion of 10/(6 + sqrt(66)) (with offset 0).

Amiram Eldar, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Cf. A005117, A003586, A034444, A065331, A072078, A181982, A382487.

The number of unitary prime(k)-smooth divisors of n: A134451 (k = 1), this sequence (k = 2), A382489 (k = 3).

Table[{1, 2, 2, 2, 1, 4}, {12}] // Flatten

a(n) = [1, 2, 2, 2, 1, 4][(n-1) % 6 + 1];

Multiplicative with a(p^e) = 2 if p <= 3, and 1 otherwise.
a(n) = A034444(A065331(n)).
a(n) = A034444(n) if and only if n is 3-smooth (A003586).
a(n) = A072078(n) if and only if n is squarefree (A005117).
a(n) = abs(A181982(n+9)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2.
G.f.: -(4*x^6 + x^5 + 2*x^4 + 2*x^3 +2*x^2 + x)/(x^6 - 1).
Dirichlet g.f.: (1 + 1/2^s) * (1 + 1/3^s) * zeta(s).

cp's OEIS Frontend

A169611 Number of prime divisors of n that are not greater than 3, counted with multiplicity.

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A355582 a(n) is the largest 5-smooth divisor of n.

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A072079 Sum of 3-smooth divisors of n.

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A171182 Period 6: repeat [0, 1, 1, 1, 0, 2].

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A244417 Exponents of 6 in appearing in the 6-adic value of 1/n, n>=1 (A244416).

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A248909 Completely multiplicative with a(p) = p if p = 6k+1 and a(p) = 1 otherwise.

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Maple

Mathematica

PARI

Python

Sage

Scheme

Formula

A099316 Greatest 3-smooth number dividing the n-th minimal number.

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