A065331 -id:A065331 - cp's OEIS frontend

A365211 The sum of divisors d of n such that gcd(d, n/d) is a 5-rough number (A007310).

1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 36, 31, 42, 28, 40, 30, 72, 32, 33, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 68, 57, 93, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 65, 84, 144, 68

Offset: 1

Amiram Eldar, Aug 26 2023

First differs from A034448 at n = 25.

The number of these divisors is A365210(n).

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

Cf. A000203, A007310, A034448, A065330, A065331, A069184, A186099, A365210.

f[p_, e_] := If[p <= 3 , 1 + p^e, (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] <= 3, 1 + f[i,1]^f[i,2], (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1)));}

Multiplicative with a(p^e) = 1 + p^e for p = 2 and 3, and a(p^e) = (p^(e+1)-1)/(p-1) for a prime p >= 5.
a(n) <= A000203(n), with equality if and only if n is neither divisible by 4 nor by 9.
a(n) >= A034448(n), with equality if and only if n is not divisible by a square of a prime >= 5.
a(n) = A000203(A065330(n)) * A034448(A065331(n)).
Dirichlet g.f.: (1 - 1/2^(2*s-1)) * (1 - 1/3^(2*s-1)) * zeta(s)*zeta(s-1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 91*Pi^2/1296 = 0.69300463... .

A385007 The largest unitary divisor of n that is a biquadratefree number (A046100).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 3, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69

Offset: 1

Amiram Eldar, Jun 15 2025

First differs from A053165 at n = 32 = 2^5: a(32) = 1 while A053165(32) = 2.
First differs from A383764 at n = 32 = 2^5: a(32) = 1 while A383764(32) = 32.
Equivalently, a(n) is the least divisor d of n such that n/d is a 4-full number (A036967).

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

Cf. A013661, A036967, A046100, A053165, A058035, A077610, A383764.

The largest unitary divisor of n that is: A000265 (odd), A006519 (power of 2), A055231 (squarefree), A057521 (powerful), A065330 (5-rough), A065331 (3-smooth), A350388 (square), A350389 (exponentially odd), A360539 (cubefree), A360540 (cubefull), A366126 (cube), A367168 (exponentially 2^n), this sequence (biquadratefree).

f[p_, e_] := If[e < 4, p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] < 4, f[i, 1]^f[i, 2], 1)); }

a(n) = 1 if and only if n is a 4-full number (A036967).
a(n) = n if and only if n is a biquadratefree number (A046100).
Multiplicative with a(p^e) = p^e if e <= 3, and 1 otherwise.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(1-s) - p^(-s) + p^(2-2*s) - p^(1-2*s) - p^(2-3*s) + p^(3-3*s) - p^(3-4*s) + p^(-4*s)).
Sum_{k=1..n} a(k) ~ c * zeta(2) * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^6 + 1/p^8 - 1/p^9) = 0.56331392082909224894... .

A081320 Largest 3-smooth divisor of n-th Fibonacci number.

Original entry on oeis.org

1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 144, 1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 288, 1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 432, 1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 576, 1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 144, 1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 864, 1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 144, 1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 1152, 1

Offset: 1

Reinhard Zumkeller, May 20 2003

nonn

Conjecture: for n>12 and n>0 modulo 12: a(n)=a(n-12) and a(12*k)=A065331(k)*144.

The first part of the conjecture follows from the fact that the Fibonacci numbers are a strong divisibility sequence. - Charles R Greathouse IV, Sep 24 2012

			Fibonacci(36) = 14930352 = 2^4 * 3^3 * 17 * 19 * 107, therefore a(36) = 2^4 * 3^3 = 432.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A003586, A065331, A000045.

a[n_] := Times @@ ({2, 3}^IntegerExponent[Fibonacci[n], {2, 3}]);
Table[a[n], {n, 1, 1000}] (* Jean-François Alcover, Oct 15 2021 *)

fibord(n,p)=if(n==0, return(oo)); my(u=3,t); while((t=((Mod([1,1;1,0],p^u))^n)[1,2])==0, u*=2); valuation(t,p)
a(n)=if(n%12, return(gcd(fibonacci(n%12),24))); 3^fibord(n,3)<Charles R Greathouse IV, Nov 13 2015

a(n) = A065331(A000045(n)).

A179276 Largest 3-smooth number <= n.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 6, 8, 9, 9, 9, 12, 12, 12, 12, 16, 16, 18, 18, 18, 18, 18, 18, 24, 24, 24, 27, 27, 27, 27, 27, 32, 32, 32, 32, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 48, 48, 48, 48, 48, 48, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 64, 64, 64, 64, 64, 64, 64, 64, 72, 72

Offset: 1

Reinhard Zumkeller, Jul 07 2010

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A003586, A065331, A065333, A081328, A071521.

smooth3Q[n_] := FactorInteger[n][[-1, 1]] <= 3;
a[n_] := For[k = n, True, k--, If[smooth3Q[k], Return[k]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 14 2021 *)

A065333(a(n)) = 1;
a(A003586(n)) = A003586(n);
A065331(n) <= a(n).
a(n) = A003586(A071521(n)). - Ridouane Oudra, Aug 24 2025

Definition corrected by Georg Fischer, Aug 05 2021

A305720 Square array T(n, k) read by antidiagonals, n > 0 and k > 0; for any prime number p, the p-adic valuation of T(n, k) is the product of the p-adic valuations of n and of k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 16, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 8, 1, 4, 5, 4, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 9, 64, 1, 6, 1, 64, 9, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 4, 1, 8, 7, 8

Offset: 1

Rémy Sigrist, Jun 09 2018

The array T is completely multiplicative in both parameters.
For any n > 0 and prime number p, T(n, p) is the highest power of p dividing n.
For any function f associating a nonnegative value to any pair of nonnegative values and such that f(0, 0) = 0, we can build an analog of this sequence, say P_f, such that for any prime number p and any n and k > 0 with p-adic valuations i and j, the p-adic valuation of P_f(n, k) equals f(i, j):
   f(i, j)    P_f
   -------    ---
   i * j      T (this sequence)
   i + j      A003991 (product)
   abs(i-j)   A089913
   min(i, j)  A003989 (GCD)
   max(i, j)  A003990 (LCM)
   i AND j    A059895
   i OR j     A059896
   i XOR j    A059897
If log(N) denotes the set {log(n) : n is in N, the set of the positive integers}, one can define a binary operation on log(N): with prime factorizations n = Product p_i^e_i and k = Product p_i^f_i, set log(n) o log(k) = Sum_{i} (e_i*f_i) * log(p_i). o has the premises of a scalar product even if log(N) isn't a vector space. T(n, k) can be viewed as exp(log(n) o log(k)). - Luc Rousseau, Oct 11 2020

			Array T(n, k) begins:
  n\k|    1    2    3    4    5    6    7    8    9   10
  ---+--------------------------------------------------
    1|    1    1    1    1    1    1    1    1    1    1
    2|    1    2    1    4    1    2    1    8    1    2  -> A006519
    3|    1    1    3    1    1    3    1    1    9    1  -> A038500
    4|    1    4    1   16    1    4    1   64    1    4
    5|    1    1    1    1    5    1    1    1    1    5  -> A060904
    6|    1    2    3    4    1    6    1    8    9    2  -> A065331
    7|    1    1    1    1    1    1    7    1    1    1  -> A268354
    8|    1    8    1   64    1    8    1  512    1    8
    9|    1    1    9    1    1    9    1    1   81    1
   10|    1    2    1    4    5    2    1    8    1   10  -> A132741

Cf. A003989, A003990, A003991, A006519, A007947, A038500, A054496, A059895, A059896, A059897, A060904, A065331, A089913, A132741, A268354, A268357.

T[n_, k_] := With[{p = FactorInteger[GCD[n, k]][[All, 1]]}, If[p == {1}, 1, Times @@ (p^(IntegerExponent[n, p] * IntegerExponent[k, p]))]];
Table[T[n-k+1, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 11 2018 *)

T(n, k) = my (p=factor(gcd(n, k))[,1]); prod(i=1, #p, p[i]^(valuation(n, p[i]) * valuation(k, p[i])))

T(n, k) = T(k, n) (T is commutative).
T(m, T(n, k)) = T(T(m, n), k) (T is associative).
T(n, k) = 1 iff gcd(n, k) = 1.
T(n, n) = A054496(n).
T(n, A007947(n)) = n.
T(n, 1) = 1.
T(n, 2) = A006519(n).
T(n, 3) = A038500(n).
T(n, 4) = A006519(n)^2.
T(n, 5) = A060904(n).
T(n, 6) = A065331(n).
T(n, 7) = A268354(n).
T(n, 8) = A006519(n)^3.
T(n, 9) = A038500(n)^2.
T(n, 10) = A132741(n).
T(n, 11) = A268357(n).

A382490 The number of infinitary 3-smooth divisors of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 29 2025

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

Cf. A000120, A003586, A007310, A007814, A036537, A037445, A065331, A007949, A138302, A382488.

a[n_] := 2^(DigitCount[IntegerExponent[n, 2], 2, 1] + DigitCount[IntegerExponent[n, 3], 2, 1]); Array[a, 100]

a(n) = 1 << (hammingweight(valuation(n, 2)) + hammingweight(valuation(n, 3)));

Multiplicative with a(p^e) = 2^A000120(e) of p <= 3, and 1 otherwise.
a(n) = 2^(A000120(A007814(n)) + A000120(A007949(n))).
a(n) = A037445(A065331(n)).
a(n) = A037445(n) if and only if n is 3-smooth (A003586).
a(n) = A382488(n) if and only if n is an exponentially 2^n number (A138302).
a(n) = A072078(n) if and only if n is a product of a 5-rough number (A007310) and a 3-smooth number whose number of divisors is a power of 2 (i.e., in both A003586 and A036537).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/3) * Product_{k>=0} (1+1/2^(2^k-1))*(1+2/3^(2^k)) = 2.36739050930467832207... .

cp's OEIS Frontend

A365211 The sum of divisors d of n such that gcd(d, n/d) is a 5-rough number (A007310).

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A385007 The largest unitary divisor of n that is a biquadratefree number (A046100).

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A081320 Largest 3-smooth divisor of n-th Fibonacci number.

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A179276 Largest 3-smooth number <= n.

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A305720 Square array T(n, k) read by antidiagonals, n > 0 and k > 0; for any prime number p, the p-adic valuation of T(n, k) is the product of the p-adic valuations of n and of k.

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A382490 The number of infinitary 3-smooth divisors of n.

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