A072079 -id:A072079 - cp's OEIS frontend

A355579 Numbers k such that A072079(k)/k sets a new record.

1, 2, 4, 6, 12, 24, 36, 48, 72, 144, 288, 432, 864, 1728, 2592, 3456, 5184, 10368, 20736, 31104, 41472, 62208, 124416, 248832, 373248, 746496, 1492992, 2239488, 2985984, 4478976, 8957952, 17915904, 26873856, 53747712, 107495424, 161243136, 214990848, 322486272

Offset: 1

Amiram Eldar, Jul 08 2022

nonn

Numbers m such that A072079(m)/m > A072079(k)/k for all k < m.
All the terms are 3-smooth numbers (A003586).
Equivalently, 3-smooth numbers k such that A000203(k)/k sets a new record.
Analogous to superabundant numbers (A004394) with 3-smooth numbers only.

			The numbers of 3-smooth divisors of the first 6 positive integers are 1, 3, 4, 7, 1 and 12. The corresponding values of A072079(k)/k are 1, 3/2, 4/3, 7/4, 1/5 and 2. The record values, 1, 3/2, 7/4 and 2, occur at 1, 2, 4 and 6, the first 4 terms of this sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..4370

Subsequence of A003586 and A355578.

Cf. A000203, A004394, A072079.

s[n_] := Module[{e = IntegerExponent[n, {2, 3}], p}, p = {2, 3}^e; If[Times @@ p == n, (2^(e[[1]] + 1) - 1)*(3^(e[[2]] + 1) - 1)/(2*n), 0]]; sm = 0; seq = {}; Do[sn = s[n]; If[sn > sm, sm = sn; AppendTo[seq, n]], {n, 1, 10^6}]; seq

lista(nmax) = {my(list = List(), rmax = 0, e2, e3, r); for(n = 1, nmax, e2 = valuation(n, 2); e3 = valuation(n, 3); r = if(2^e2 * 3^e3 == n, (2^(e2 + 1) - 1)*(3^(e3 + 1) - 1)/(2*n), 0); if(r > rmax, rmax = r;  listput(list, n))); Vec(list)};

from fractions import Fraction
from sympy import multiplicity as v
from itertools import count, takewhile
def f(n): return Fraction((2**(v(2, n)+1)-1) * (3**(v(3, n)+1)-1)//2, n)
def smooth3(lim):
    pows2 = list(takewhile(lambda x: x record: record = v; records.append(argv)
    return records
print(aupto(10**9)) # Michael S. Branicky, Jul 08 2022

Limit_{n->oo} A072079(a(n))/a(n) = lim_{n->oo} A000203(a(n))/a(n) = 3.

A072078 Number of 3-smooth divisors of n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jun 13 2002

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

Cf. A000005, A003586, A007814, A007949, A072079, A122841, A322026.

[(Valuation(n,2)+1)*(Valuation(n,3)+1): n in [1..120]]; // Vincenzo Librandi, Mar 24 2015

a[n_] := DivisorSum[n, MoebiusMu[6*#]*DivisorSigma[0, n/#] &]; Array[a, 100] (* or *) a[n_] := ((1+IntegerExponent[n, 2])*(1+IntegerExponent[n, 3])); Array[a, 100] (* Amiram Eldar, Dec 03 2018 from the pari codes *)

a(n)=sumdiv(n,d,moebius(6*d)*numdiv(n/d)) \\ Benoit Cloitre, Jun 21 2007

A072078(n) = ((1+valuation(n,2))*(1+valuation(n,3))); \\ Antti Karttunen, Dec 03 2018

a(n) = A000005(A065331(n)).
a(n) = (A007814(n) + 1)*(A007949(n) + 1).
1/Product_{k>0} (1 - x^k + x^(2*k))^a(k) is g.f. for A000041(). - Vladeta Jovovic, Jun 07 2004
From Christian G. Bower, May 20 2005: (Start)
Multiplicative with a(2^e) = a(3^e) = e+1, a(p^e) = 1, p>3.
Dirichlet g.f.: 1/((1-1/2^s)*(1-1/3^s))^2 * Product{p prime > 3}(1/(1-1/p^s)). [corrected by Vaclav Kotesovec, Nov 20 2021] (End)
a(n) = Sum_{d divides n} mu(6d)*tau(n/d). - Benoit Cloitre, Jun 21 2007
Dirichlet g.f.: zeta(s)/((1-1/2^s)*(1-1/3^s)). - Ralf Stephan, Mar 24 2015; corrected by Vaclav Kotesovec, Nov 20 2021
Sum_{k=1..n} a(k) ~ 3*n. - Vaclav Kotesovec, Nov 20 2021

More terms from Benoit Cloitre, Jun 21 2007

A385006 The sum of the biquadratefree divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 15, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 15, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 60, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 15, 84, 144

Offset: 1

Amiram Eldar, Jun 15 2025

First differs from A365682 and A366992 at n = 32.

The number of these divisors is A252505(n), and the largest of them is A058035(n).

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

Cf. A046100, A048250, A058035, A073185, A252505, A365682, A366992.

The sum of divisors d of n such that d is: A000593 (odd), A033634 (exponentially odd), A035316 (square), A038712 (power of 2), A048250 (squarefree), A072079 (3-smooth), A073185 (cubefree), A113061 (cube), A162296 (nonsquarefree), A183097 (powerful), A186099 (5-rough), A353900 (exponentially 2^n), A385005 (cubefull), this sequence (biquadratefree).

f[p_, e_] := (p^Min[e+1, 4] - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; (p^min(e+1, 4) - 1)/(p - 1));}

Multiplicative with a(p^e) = (p^min(e+1, 4) - 1)/(p - 1).
In general, the sum of the k-free (numbers that are not divisible by a k-th power larger than 1) divisors of n is multiplicative with a(p^e) = (p^min(e+1, k) - 1)/(p - 1).
Dirichlet g.f.: zeta(s) * zeta(s-1) /zeta(4*s-4).
In general, the sum of the k-free divisors of n has Dirichlet g.f.: zeta(s)*zeta(s-1)/zeta(k*s-k).
Sum_{k=1..n} a(k) ~ (15/(2*Pi^2)) * n^2.
In general, the sum of the k-free divisors of n has an average order (Pi^2/(12*zeta(k))) * n^2.

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

Original entry on oeis.org

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).

A385137 The sum of divisors d of n such that n/d is a 3-smooth number (A003586).

Original entry on oeis.org

1, 3, 4, 7, 5, 12, 7, 15, 13, 15, 11, 28, 13, 21, 20, 31, 17, 39, 19, 35, 28, 33, 23, 60, 25, 39, 40, 49, 29, 60, 31, 63, 44, 51, 35, 91, 37, 57, 52, 75, 41, 84, 43, 77, 65, 69, 47, 124, 49, 75, 68, 91, 53, 120, 55, 105, 76, 87, 59, 140, 61, 93, 91, 127, 65, 132

Offset: 1

Amiram Eldar, Jun 19 2025

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

Cf. A003586, A064987, A072044, A072045, A072079.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), this sequence (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).

f[p_, e_] := If[p < 5, (p^(e+1) - 1)/(p - 1), p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(p < 5, (p^(e + 1) - 1)/(p - 1), p^e));}

a(n) = A064987(n)/A385138(n).
Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p <= 3, and p^e if p >= 5.
In general, the sum of divisors d of n such that n/d is q-smooth (not divisible by a prime larger than q) is multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p <= q, and p^e if p > q.
Dirichlet g.f.: zeta(s-1) / ((1 - 1/2^s) * (1 - 1/3^s)).
In general, the sum of divisors d of n such that n/d is q-smooth has Dirichlet g.f.: zeta(s-1) / Product_{p prime <= q} (1 - 1/q^s).
Sum_{k=1..n} a(k) ~ (3/4)*n^2.
In general, the sum of divisors d of n such that n/d is prime(k)-smooth has an average order c * n^2 / 2, where c = A072044(k-1)/A072045(k-1) for k >= 2.

A355584 a(n) is the sum of the 5-smooth divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 1, 15, 13, 18, 1, 28, 1, 3, 24, 31, 1, 39, 1, 42, 4, 3, 1, 60, 31, 3, 40, 7, 1, 72, 1, 63, 4, 3, 6, 91, 1, 3, 4, 90, 1, 12, 1, 7, 78, 3, 1, 124, 1, 93, 4, 7, 1, 120, 6, 15, 4, 3, 1, 168, 1, 3, 13, 127, 6, 12, 1, 7, 4, 18, 1, 195, 1, 3, 124, 7

Offset: 1

Amiram Eldar, Jul 08 2022

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

Sum of the p-smooth divisors of n: A038712 (2), A072079 (3), this sequence (5).

Cf. A000203, A007814, A007949, A051037, A112765, A355582, A355583.

a[n_] := (Times @@ ({2, 3, 5}^(IntegerExponent[n, {2, 3, 5}] + 1) - 1))/8; Array[a, 100]

a(n) = (2^(valuation(n, 2) + 1) - 1) * (3^(valuation(n, 3) + 1) - 1) * (5^(valuation(n, 5) + 1) - 1) / 8;

from sympy import multiplicity as v
def a(n): return (2**(v(2, n)+1)-1) * (3**(v(3, n)+1)-1) * (5**(v(5, n)+1)-1) // 8
print([a(n) for n in range(1, 77)]) # Michael S. Branicky, Jul 08 2022

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p <= 5, and 1 otherwise.
a(n) = (2^(A007814(n)+1)-1)*(3^(A007949(n)+1)-1)*(5^(A112765(n)+1)-1)/8.
a(n) = A000203(A355582(n)).
a(n) <= A000203(n), with equality if and only if n is in A051037.
Dirichlet g.f.: zeta(s)*(2^s/(2^s-2))*(3^s/(3^s-3))*(5^s/(5^s-5)). - Amiram Eldar, Dec 25 2022

A385005 The sum of the cubefull divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 28, 1, 1, 1, 1, 57, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 28, 1, 9, 1, 1, 1, 1, 1, 1, 1, 121, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 109, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 15 2025

The sum of the terms in A036966 that divide n.

The number of these divisors is A190867(n), and the largest of them is A360540(n).

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

Cf. A036966, A190867, A183097, A360540.

The sum of divisors d of n such that d is: A000593 (odd), A033634 (exponentially odd), A035316 (square), A038712 (power of 2), A048250 (squarefree), A072079 (3-smooth), A073185 (cubefree), A113061 (cube), A162296 (nonsquarefree), A183097 (powerful), A186099 (5-rough), A353900 (exponentially 2^n), this sequence (cubefull), A385006 (biquadratefree).

f[p_, e_] := (p^(e+1)-1)/(p-1) - p - If[e == 1, 0, p^2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; (p^(e+1)-1)/(p-1) - p - if(e == 1, 0, p^2));}

Multiplicative with a(p^e) = 1 if e <= 2, and a(p^e) = ((p^(e+1)-1) / (p-1)) - p - p^2 if e >= 3.

Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - p^(s-1) + 1/p^(3*s-3)).

A355578 Numbers whose sum of 3-smooth divisors sets a new record.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 108, 144, 192, 216, 288, 324, 384, 432, 576, 648, 768, 864, 972, 1152, 1296, 1536, 1728, 1944, 2304, 2592, 2916, 3072, 3456, 3888, 4608, 5184, 5832, 6912, 7776, 8748, 9216, 10368, 11664, 13824, 15552, 17496

Offset: 1

Amiram Eldar, Jul 08 2022

nonn

Numbers m such that A072079(m) > A072079(k) for all k < m.
All the terms are 3-smooth numbers (A003586).
Equivalently, 3-smooth numbers k such that A000203(k) sets a new record.
Analogous to highly abundant numbers (A002093) with 3-smooth numbers only.

			The numbers of 3-smooth divisors of the first 6 positive integers are 1, 3, 4, 7, 1 and 12. The record values, 1, 3, 4 and 12, occur at 1, 2, 3, 4 and 6, the first 5 terms of this sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..11233 (all < 10^60)

Subsequence of A003586.
A355579 is a subsequence.
Cf. A000203, A002093, A072079, A309015.

s[n_] := Module[{e = IntegerExponent[n, {2, 3}], p}, p = {2, 3}^e; If[Times @@ p == n, (2^(e[[1]] + 1) - 1)*(3^(e[[2]] + 1) - 1)/2, 0]]; sm = 0; seq = {}; Do[sn = s[n]; If[sn > sm, sm = sn; AppendTo[seq, n]], {n, 1, 18000}]; seq

lista(nmax) = {my(list = List(), smax = 0, e2, e3, s); for(n = 1, nmax, e2 = valuation(n, 2); e3 = valuation(n, 3); s = if(2^e2 * 3^e3 == n, (2^(e2 + 1) - 1)*(3^(e3 + 1) - 1)/2, 0); if(s > smax, smax = s;  listput(list, n))); Vec(list)};

from sympy import multiplicity as v
from itertools import count, takewhile
def f(n): return (2**(v(2, n)+1)-1) * (3**(v(3, n)+1)-1)//2
def smooth3(lim):
    pows2 = list(takewhile(lambda x: x record: record = v; records.append(argv)
    return records
print(aupto(10**5)) # Michael S. Branicky, Jul 08 2022

A382314 G.f. satisfies A(x) = 1/(1-x) + 2xA(x^2) + 3x^2A(x^3).

Original entry on oeis.org

1, 3, 4, 7, 1, 18, 1, 15, 13, 3, 1, 58, 1, 3, 4, 31, 1, 81, 1, 7, 4, 3, 1, 162, 1, 3, 40, 7, 1, 18, 1, 63, 4, 3, 1, 337, 1, 3, 4, 15, 1, 18, 1, 7, 13, 3, 1, 418, 1, 3, 4, 7, 1, 324, 1, 15, 4, 3, 1, 58, 1, 3, 13, 127, 1, 18, 1, 7, 4, 3, 1, 1161, 1, 3, 4, 7, 1, 18, 1, 31, 121, 3, 1, 58, 1, 3, 4, 15, 1, 81, 1, 7, 4, 3, 1, 1026

Offset: 0

Paul D. Hanna, Apr 14 2025

nonn

Logarithmic derivative of A382126.

			G.f.: A(x) = 1 + 3*x + 4*x^2 + 7*x^3 + x^4 + 18*x^5 + x^6 + 15*x^7 + 13*x^8 + 3*x^9 + x^10 + 58*x^11 + x^12 + 3*x^13 + 4*x^14 + 31*x^15 + ...
where A(x) = 1/(1-x) + 2*x*A(x^2) + 3*x^2*A(x^3).
RELATED SERIES.
The logarithm of the g.f. B(x) for A382126 yields the series
log(B(x)) = x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + x^5/5 + 18*x^6/6 + x^7/7 + 15*x^8/8 + 13*x^9/9 + 3*x^10/10 + x^11/11 + 58*x^12/12 + ... + a(n-1)*x^n/n + ...
where B(x) = B(x^2)*B(x^3)/(1-x) begins
B(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 11*x^6 + 13*x^7 + 20*x^8 + 26*x^9 + 36*x^10 + 44*x^11 + 66*x^12 + ... + A382126(n)*x^n + ...
so that A(x) = B'(x)/B(x).

Paul D. Hanna, Table of n, a(n) for n = 0..1030

Cf. A382126, A072079, A003586, A007814, A007949.

{a(n) = my(A=1+x +x*O(x^n)); for(i=1,#binary(n), A = 1/(1-x) + 2*x*subst(A,x,x^2) + 3*x^2*subst(A,x,x^3) + x*O(x^n)  ); polcoef(A,n)}
for(n=0,100,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1/(1-x) + 2*x*A(x^2) + 3*x^2*A(x^3).
(2) A(x) = (1-x)^2*(1+2*x)*(1+2*x+3*x^2)/((1-x)*(1-x^2)*(1-x^3)) + 4*x^3*A(x^4) + 12*x^5*A(x^6) + 9*x^8*A(x^9).
(3) A(x) = (1/x)*Sum_{n>=0} Sum_{k=0..n} binomial(n,k) * 2^(n-k)*3^k * x^(2^(n-k)*3^k) / (1 - x^(2^(n-k)*3^k)).
(4) A(x) = (1/x)*Sum_{n>=1} B(n) * A003586(n) * x^A003586(n)/(1 - x^A003586(n)) where B(n) = binomial(F2(n)+F3(n),F3(n)), with F2(n) = A007814(A003586(n)) and F3(n) = A007949(A003586(n)).
(5) A(x) = B'(x)/B(x) where B(x) = B(x^2)*B(x^3)/(1-x) is the g.f. of A382126.

cp's OEIS Frontend

A355579 Numbers k such that A072079(k)/k sets a new record.

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A072078 Number of 3-smooth divisors of n.

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A385006 The sum of the biquadratefree divisors of n.

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

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A385137 The sum of divisors d of n such that n/d is a 3-smooth number (A003586).

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A355584 a(n) is the sum of the 5-smooth divisors of n.

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A385005 The sum of the cubefull divisors of n.

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A382314 G.f. satisfies A(x) = 1/(1-x) + 2xA(x^2) + 3x^2A(x^3).