A065331 -id:A065331 - cp's OEIS frontend

A385046 The sum of the unitary divisors of n that are 3-smooth numbers (A003586).

1, 3, 4, 5, 1, 12, 1, 9, 10, 3, 1, 20, 1, 3, 4, 17, 1, 30, 1, 5, 4, 3, 1, 36, 1, 3, 28, 5, 1, 12, 1, 33, 4, 3, 1, 50, 1, 3, 4, 9, 1, 12, 1, 5, 10, 3, 1, 68, 1, 3, 4, 5, 1, 84, 1, 9, 4, 3, 1, 20, 1, 3, 10, 65, 1, 12, 1, 5, 4, 3, 1, 90, 1, 3, 4, 5, 1, 12, 1, 17

Offset: 1

Amiram Eldar, Jun 16 2025

The number of these divisors is A382488(n), and the largest of them is A065331(n).

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

The unitary analog of A072079.
Cf. A001620, A003586, A034448, A065331, A082633, A382488.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), A385045 (5-rough), this sequence (3-smooth), A385047 (power of 2), A385048 (cubefull), A385049 (biquadratefree).

f[n_, p_] := If[Divisible[n, p], p^IntegerExponent[n, p] + 1, 1]; a[n_] := f[n, 2]*f[n, 3]; Array[a, 100]

a(n) = if(n%2, 1, 2^valuation(n, 2)+1) * if(!(n%3), 3^valuation(n, 3)+1, 1);

Multiplicative with a(p^e) = p^e + 1 if p <= 3, and 1 if p >= 5.
a(n) = A034448(n)/A385045(n).
a(n) <= A034448(n), with equality if and only if n 3-smooth.
a(n) <= A072079(n).
Dirichlet g.f.: zeta(s) * ((1-1/2^(2*s-1))/(1-1/2^(s-1))) * ((1-1/3^(2*s-1))/(1-1/3^(s-1))).
Sum_{k=1..n} a(k) ~ (n/(6*log(2)*log(3))) * (log(n)^2 + c1*log(n) + c2), where c1 = 2*gamma - 2 + 7*log(2) + 5*log(3) - 2*log(6) = 5.916004..., c2 = 2 - 5*log(2) - 11*log(2)^2/6 - 3*log(3) - 5*log(3)^2/6 + 15*log(2)*log(3)/2 + (5*log(2) + 3*log(3) - 2)*gamma - 2*gamma_1 = 1.957142..., gamma is Euler's constant (A001620), and gamma_1 is the 1st Stieltjes constant (A082633).

A343431 Part of n composed of prime factors of the form 6k-1.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 11, 1, 1, 1, 5, 1, 17, 1, 1, 5, 1, 11, 23, 1, 25, 1, 1, 1, 29, 5, 1, 1, 11, 17, 5, 1, 1, 1, 1, 5, 41, 1, 1, 11, 5, 23, 47, 1, 1, 25, 17, 1, 53, 1, 55, 1, 1, 29, 59, 5, 1, 1, 1, 1, 5, 11, 1, 17, 23, 5, 71, 1, 1, 1, 25, 1, 11, 1, 1, 5, 1, 41, 83, 1, 85, 1, 29, 11, 89, 5

Offset: 1

Peter Munn, Apr 15 2021

Completely multiplicative with a(p) = p if p is of the form 6k-1 and a(p) = 1 otherwise.

Largest term of A259548 that divides n.

Equivalent sequence for distinct prime factors: A170825.
Equivalent sequences for prime factors of other forms: A000265 (2k+1), A343430 (3k-1), A170818 (4k+1), A097706 (4k-1), A248909 (6k+1), A065330 (6k+/-1), A065331 (<= 3), A355582 (<= 5).
Range of terms: A259548.

f[p_, e_] := If[Mod[p, 6] == 5, p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* after Amiram Eldar at A248909 *)

a(n) = {my(f = factor(n)); for (i=1, #f~, if ((f[i, 1] + 1) % 6, f[i, 1] = 1); ); factorback(f); } \\ after Michel Marcus at A248909

from math import prod
from sympy import factorint
def A343431(n): return prod(p**e for p, e in factorint(n).items() if not (p+1)%6) # Chai Wah Wu, Dec 26 2022

a(n) = n / A065331(n) / A248909(n) = A065330(n) / A248909(n).

A384042 The number of integers k from 1 to n such that gcd(n,k) is a 5-rough number (A007310).

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 7, 4, 6, 5, 11, 4, 13, 7, 10, 8, 17, 6, 19, 10, 14, 11, 23, 8, 25, 13, 18, 14, 29, 10, 31, 16, 22, 17, 35, 12, 37, 19, 26, 20, 41, 14, 43, 22, 30, 23, 47, 16, 49, 25, 34, 26, 53, 18, 55, 28, 38, 29, 59, 20, 61, 31, 42, 32, 65, 22, 67, 34, 46

Offset: 1

Amiram Eldar, May 18 2025

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

Cf. A000010, A003586, A007310, A065330, A065331, A372671.

The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), A384039 (powerful), A384040 (cubefull), A384041 (exponentially odd), this sequence (5-rough).

f[p_, e_] := If[p < 5, (p-1)*p^(e-1), p^e]; 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] < 5, (f[i,1]-1)*f[i,1]^(f[i,2]-1), f[i,1]^f[i,2]));}

Multiplicative with a(p^e) = (p-1)*p^(e-1) if p <= 3 and p^e if p >= 5.
a(n) >= A000010(n), with equality if and only if n is 3-smooth (A003586).
a(n) = A000010(A065331(n)) * A065330(n).
a(n) = 2 * n * phi(n)/phi(6*n) = n * A000010(n) / A372671(n).
Dirichlet g.f.: zeta(s-1) * (1-1/2^s) * (1-1/3^s).
Sum_{k=1..n} a(k) ~ n^2 / 3.

A065332 3-smooth numbers in their natural position, gaps filled with 0.

Original entry on oeis.org

1, 2, 3, 4, 0, 6, 0, 8, 9, 0, 0, 12, 0, 0, 0, 16, 0, 18, 0, 0, 0, 0, 0, 24, 0, 0, 27, 0, 0, 0, 0, 32, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 72, 0, 0, 0, 0, 0, 0, 0, 0, 81, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 96, 0, 0

Offset: 1

Reinhard Zumkeller, Oct 29 2001

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

Cf. A003586, A065330, A065331, A065333, A065334, A065335.

smooth3Q[n_] := n == 2^IntegerExponent[n, 2]*3^IntegerExponent[n, 3];
a[n_] := n Boole[smooth3Q[n]];
Array[a, 100] (* Jean-François Alcover, Oct 17 2021 *)

a(n) = if(n >> valuation(n, 2) == 3^valuation(n, 3), n, 0); \\ Amiram Eldar, Sep 16 2023

a(n) = if A065330(n) = 1 then n else 0.
a(n) = A065333(n) * n.
If a(k) > 0 then a(k) = (2^A065334(k)) * (3^A065335(k)).
From Amiram Eldar, Sep 16 2023: (Start)
Multiplicative with a(p^e) = p^e if p <= 3, and 0 otherwise.
Dirichlet g.f.: 6^s / ((2^s-2)*(3^s-3)).
Sum_{k=1..n} a(k) ~ (n/(log(2)*log(3))) * (log(n) + log(6)/2 - 1). (End)

A099315 Greatest 3-smooth number dividing the smallest number having exactly n divisors.

Original entry on oeis.org

1, 2, 4, 6, 16, 12, 64, 24, 36, 48, 1024, 12, 4096, 192, 144, 24, 65536, 36, 262144, 48, 576, 3072, 4194304, 72, 1296, 12288, 36, 192, 268435456, 144, 1073741824, 24, 9216, 196608, 5184, 36

Offset: 1

Reinhard Zumkeller, Oct 12 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..2000 (calculated from the b-file at A005179)

Cf. A003586, A005179, A065331, A099316, A099311, A099313.

a(n) = A065331(A005179(n)).

A146892 For definition see comments lines.

Original entry on oeis.org

1, 6, 6, 72, 72, 72, 6, 72, 72, 5184, 6, 5184, 72, 5184, 31104, 5184, 5184, 5184, 2592, 5184, 432, 373248, 36, 373248, 31104, 26873856, 26873856, 26873856, 373248, 31104, 36, 31104, 2239488, 2239488, 1934917632, 26873856, 31104, 2239488

Offset: 0

Yasutoshi Kohmoto, Apr 17 2009

Let USigma denote the unitary sigma function, A034448.
As in A146891, let PF_p(n) denote the largest power of the prime p dividing n. PF_2 is A006519, and PF_3 is A038500. Furthermore define PF_1(n)=1.
Extension to multi-prime-indices is done by multiplying the corresponding functions: PF_{p,q,..}(n) = PF_p(n)*PF_q(n)*... An example of this is PF_{2,3} = A065331.
[How to compute c(m)]
Case of Base Primes = {2}{3}
c(0)=2^m, b(0)=2^m
c(n)=c(n-1)/PF_2[USigma[b(n-1)]]*PF_3[USigma[b(n-1)]]
b(n)=USigma[b(n-1)]/ PF_2,3[USigma[b(n-1)]]
IF b(k)=1 THEN END
a(m)=c(k)
Sequence gives a(m)
Factorization of term becomes 2^r*3^s.

Cf. A146891.

A146892 := proc(n) local b,a,k ;
   b := [2^n] ;
   while op(-1,b) <> 1 do
       b := [op(b), A065330(A034448(op(-1,b))) ] ;
   od:
   a := 2^n ;
   for k from 2 to nops(b) do
       a := a/ A006519(A034448(op(k-1,b))) *A038500(A034448(op(k-1,b))) ;
   od:
   a ;
end: # R. J. Mathar, Jun 24 2009

More terms from R. J. Mathar, Jun 24 2009

A365210 The number of divisors d of n such that gcd(d, n/d) is a 5-rough number (A007310).

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, 3, 4, 2, 4, 2, 8, 2, 2, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 3, 6, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 2, 4, 8, 2, 4, 4, 8, 2, 4, 2, 4, 6, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Aug 26 2023

First differs from A034444 at n = 25.

The sum of these divisors is A365211(n).

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

Cf. A000005, A007310, A035218, A065330, A065331, A069733, A365211.

Cf. A001620, A013661, A073002.

f[p_, e_] := If[p <= 3 , 2, 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,1] <= 3, 2, f[i,2]+1));}

Multiplicative with a(p^e) = 2 for p = 2 and 3, and a(p^e) = e+1 for a prime p >= 5.
a(n) <= A000005(n), with equality if and only if n is neither divisible by 4 nor by 9.
a(n) >= A034444(n), with equality if and only if n is not divisible by a square of a prime >= 5.
a(n) = A000005(A065330(n)) * A034444(A065331(n)).
Dirichlet g.f.: (1-1/4^s) * (1-1/9^s) * zeta(s)^2.
Sum_{k=1..n} a(k) ~ (2*n/3) * (log(n) + 2*gamma - 1 + 2*log(2)/3 + log(3)/4), where gamma is Euler's constant (A001620).

A066262 a(n) = gcd(n, A066260(n)).

Original entry on oeis.org

1, 2, 3, 4, 1, 6, 1, 8, 9, 2, 1, 12, 1, 2, 3, 16, 1, 18, 1, 4, 3, 2, 1, 24, 1, 2, 27, 4, 1, 6, 1, 32, 3, 2, 1, 36, 1, 2, 3, 8, 1, 6, 1, 4, 9, 2, 1, 48, 1, 2, 3, 4, 1, 54, 5, 8, 3, 2, 1, 12, 1, 2, 9, 64, 1, 6, 1, 4, 3, 2, 1, 72, 1, 2, 3, 4, 1, 6, 1, 16, 81, 2, 1, 12, 1, 2, 3, 8, 1, 18, 1, 4, 3, 2, 5, 96

Offset: 1

Reinhard Zumkeller, Dec 10 2001

nonn

Same as A065331 through a(54); A065331(55)=1, but a(55)=5. - Jon E. Schoenfield, Jan 28 2022

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A066260.

Different from A065331.

Composite(n) = { my(k=n + primepi(n) + 1); while (k != n + primepi(k) + 1, k = n + primepi(k) + 1); return(k) }
{ for (n=1, 100, my(f=factor(n), a=1); for (i=1, matsize(f)[1], a*=Composite(primepi(f[i, 1]))^f[i, 2]); a=gcd(n, a); print1(a, ", ") ) } \\ Harry J. Smith, Feb 07 2010

A365208 The number of divisors d of n such that gcd(d, n/d) is a 3-smooth number (A003586).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 2, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 2, 4, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 4, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4

Offset: 1

Amiram Eldar, Aug 26 2023

First differs from A000005 at n = 25.

The sum of these divisors is A365209(n).

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

Cf. A000005, A003586, A034444, A065330, A065331, A365209.

Cf. A001620, A013661, A073002.

f[p_, e_] := If[p <= 3, e + 1, 2]; 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, f[i,2]+1, 2));}

Multiplicative with a(p^e) = e+1 if p = 2 or 3, and a(p^e) = 2 for a prime p >= 5.
a(n) <= A000005(n), with equality if and only if n is not divisible by a square of a prime >= 5.
a(n) >= A034444(n), with equality if and only if n is neither divisible by 4 nor by 9.
a(n) = A000005(A065331(n)) * A034444(A065330(n)).
Dirichlet g.f.: (4^s/(4^s-1)) * (9^s/(9^s-1)) * zeta(s)^2/zeta(2*s).
Sum_{k==1..n} a(k) ~ (9/Pi^2)*n*(log(n) + 2*gamma - 2*log(2)/3 - log(3)/4 - 2*zeta'(2)/zeta(2) - 1), where gamma is Euler's constant (A001620).

A365209 The sum of divisors d of n such that gcd(d, n/d) is a 3-smooth number (A003586).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 26, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 50, 78, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144

Offset: 1

Amiram Eldar, Aug 26 2023

First differs from A000005 at n = 25.

The number of these divisors is A365208(n).

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

Cf. A000203, A003586, A034448, A065330, A065331, A365208.

Cf. A002117, A013661, A306633.

f[p_, e_] := If[p <= 3, (p^(e+1)-1)/(p-1), 1 + p^e]; 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, (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1), 1 + f[i,1]^f[i,2]));}

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) for p = 2 or 3, and a(p^e) = 1 + p^e for a prime p >= 5.
a(n) <= A000203(n), with equality if and only if n is not divisible by a square of a prime >= 5.
a(n) >= A034448(n), with equality if and only if n is neither divisible by 4 nor by 9.
a(n) = A000203(A065331(n)) * A034448(A065330(n)).
Dirichlet g.f.: (4^s/(4^s-2)) * (9^s/(9^s-3)) * zeta(s)*zeta(s-1)/zeta(2*s-1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (54/91) * zeta(2)/zeta(3) = (54/91) * A306633 = 0.812037... .

cp's OEIS Frontend

A385046 The sum of the unitary divisors of n that are 3-smooth numbers (A003586).

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A343431 Part of n composed of prime factors of the form 6k-1.

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A384042 The number of integers k from 1 to n such that gcd(n,k) is a 5-rough number (A007310).

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A065332 3-smooth numbers in their natural position, gaps filled with 0.

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A099315 Greatest 3-smooth number dividing the smallest number having exactly n divisors.

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A146892 For definition see comments lines.

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A365210 The number of divisors d of n such that gcd(d, n/d) is a 5-rough number (A007310).

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A066262 a(n) = gcd(n, A066260(n)).

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A365208 The number of divisors d of n such that gcd(d, n/d) is a 3-smooth number (A003586).