A065330 -id:A065330 - cp's OEIS frontend

A165725 Largest divisor of n coprime to 30. I.e., a(n) = max { k | gcd(n, k) = k and gcd(k, 30) = 1 }.

1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 11, 1, 13, 7, 1, 1, 17, 1, 19, 1, 7, 11, 23, 1, 1, 13, 1, 7, 29, 1, 31, 1, 11, 17, 7, 1, 37, 19, 13, 1, 41, 7, 43, 11, 1, 23, 47, 1, 49, 1, 17, 13, 53, 1, 11, 7, 19, 29, 59, 1, 61, 31, 7, 1, 13, 11, 67, 17, 23, 7, 71, 1, 73, 37, 1, 19, 77, 13, 79, 1, 1, 41, 83, 7

Offset: 1

Barry Wells (wells.barry(AT)gmail.com), Sep 25 2009

This is the sequence of the largest divisor of n which is coprime to 30. The product of the first 3 prime numbers is 2*3*5=30. This sequence gives the largest factor of n which does not include 2, 3 or 5 in its prime factorization.

			The largest factor of 1, 2, 3, 4, 5 and 6 not including the primes 2, 3 and 5 is 1. 7 is prime and therefore its sequence value is 7. For p > 5, p prime, gives a(p) = p. As 14 = 2*7, a(14)= 7. As 98 = 2*7*7, a(98)= 49.

Barry Wells, Table of n, a(n) for n = 1..1024 [a(392) and a(704) corrected by Sean A. Irvine]

A051037 gives the smooth five numbers, numbers whose prime divisor only include 2, 3 and 5. A132740 gives the largest divisor of n coprime to 10. A065330 gives a(n) = max { k | gcd(n, k) = k and gcd(k, 6) = 1 }.
Largest divisor of n coprime to a prime factor of 30: A000265 (2), A038502 (3), A132739 (5).
Cf. A355582.

a[n_] := n / Times @@ ({2, 3, 5}^IntegerExponent[n, {2, 3, 5}]); Array[a, 100] (* Amiram Eldar, Jul 10 2022 *)

a(n)=n>>valuation(n,2)/3^valuation(n,3)/5^valuation(n,5) \\ Charles R Greathouse IV, Jul 16 2017

From Amiram Eldar, Jul 10 2022: (Start)
Multiplicative with a(p^e) = p^e if p >= 7 and 1 otherwise.
a(n) = n/A355582(n). (End)
Sum_{k=1..n} a(k) ~ (5/24) * n^2. - Amiram Eldar, Nov 28 2022
Dirichlet g.f.: zeta(s-1)*(2^s-2)*(3^s-3)*(5^s-5)/((2^s-1)*(3^s-1)*(5^s-1)). - Amiram Eldar, Jan 04 2023

A225857 Numbers of the form 2^i3^j(12k+1) or 2^i3^j(12k+5), i, j, k >= 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 16, 17, 18, 20, 24, 25, 26, 27, 29, 30, 32, 34, 36, 37, 39, 40, 41, 45, 48, 49, 50, 51, 52, 53, 54, 58, 60, 61, 64, 65, 68, 72, 73, 74, 75, 77, 78, 80, 81, 82, 85, 87, 89, 90, 96, 97, 98, 100, 101, 102, 104, 106, 108, 109, 111

Offset: 1

Ralf Stephan, May 18 2013

From Peter Munn, Nov 11 2023: (Start)
Numbers k whose 5-rough part, A065330(k), is congruent to 1 modulo 4.
Contains all nonzero squares.
Positive integers in the multiplicative subgroup of rationals generated by 2, 3, 5 and integers congruent to 1 modulo 12. Thus, the sequence is closed under multiplication and, provided the result is an integer, under division.
This subgroup has index 2 and does not include -1, so is the complement of its negation. In respect of the sequence, the index 2 property implies we can take any absent positive integer m, and divide by m all terms that are multiples of m to get the complementary sequence, A225858.
Likewise, the sequence forms a subgroup of index 2 of the positive integers under the operation A059897(.,.).
(End)
The asymptotic density of this sequence is 1/2. - Amiram Eldar, Nov 14 2023

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

Complement of A225858.

Cf. A059897, A065330, A225837.

[n: n in [1..200] | d mod 4 eq 1 where d is n div (2^Valuation(n,2)*3^Valuation(n,3))]; // Bruno Berselli, May 16 2013

Select[Range[120], Mod[#/Times @@ ({2, 3}^IntegerExponent[#, {2, 3}]), 4] == 1 &] (* Amiram Eldar, Nov 14 2023 *)

for(n=1,200,t=n/(2^valuation(n,2)*3^valuation(n,3));if((t%4==1),print1(n,",")))

from itertools import count
from sympy import integer_log
def A225857(n):
    def f(x):
        c = n
        for i in range(integer_log(x,3)[0]+1):
            i2 = 3**i
            for j in count(0):
                k = i2<x:
                    break
                m = x//k
                c += (m-7)//12+(m-11)//12+2
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 24 2025

Name clarified by Peter Munn, Nov 10 2023

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).

A385044 The number of unitary divisors of n that are 5-rough numbers (A007310).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 16 2025

The sum of these divisors is A385045(n), and the largest of them is A065330(n).

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

The unitary analog of A035218.
Cf. A007310, A034444, A065330, A385045.
Cf. A001620, A013661, A306016.
The number of unitary divisors of n that are: A000034 (power of 2), A055076 (exponentially odd), A056624 (square), A056671 (squarefree), A068068 (odd), A323308 (powerful), A365498 (cubefree), A365499 (biquadratefree), A368248 (cubefull), A380395 (cube), A382488 (3-smooth), A385042 (exponentially 2^n), this sequence (5-rough).

f[p_, e_] := If[p <= 3, 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x <= 3, 1, 2), factor(n)[, 1]));

Multiplicative with a(p^e) = 1 if p <= 3, and 2 if p >= 5.
a(n) = A034444(n)/A382488(n).
a(n) <= A034444(n), with equality if and only if n is 5-rough.
a(n) <= A035218(n).
Dirichlet g.f.: (zeta(s)^2/zeta(2*s)) * (1/((1+1/2^s)*(1+1/3^s))).
Sum_{k=1..n} a(k) ~ (n / (2 * zeta(2))) *(log(n) + 2*gamma - 1 + log(2)/3 + log(3)/4 - 2*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).

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... .

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

Original entry on oeis.org

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... .

A375670 The maximum exponent in the prime factorization of the largest 5-rough divisor of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Aug 23 2024

First differs from A106799 at n = 35.

The largest exponent among the exponents of the primes that are larger than 3 in the prime factorization of n.

Cf. A003586, A007310 (5-rough numbers), A051903, A065330, A106799, A375669.

a[n_] := Module[{m = n / Times@@({2,3}^IntegerExponent[n,{2,3}])}, If[m == 1, 0, Max[FactorInteger[m][[;; , 2]]]]]; Array[a, 100]

a(n) = {my(m = n >> valuation(n, 2)/3^valuation(n, 3)); if(m == 1, 0,vecmax(factor(m)[,2]));}

a(n) = A051903(A065330(n)).
a(n) = 0 if and only if n is a 3-smooth number (A003586).
a(n) = 1 if and only if n is a product of a squarefree 5-rough number larger than 1 and a 3-smooth number.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k * d(k) = 1.1034178389191320571029... , where d(k) is the asymptotic density of the occurrences of k in this sequence: d(1) = 3/(2*zeta(2)), and d(k) = (1/zeta(k+1)) / ((1-1/2^(k+1))*(1-1/3^(k+1))) - (1/zeta(k)) / ((1-1/2^k)*(1-1/3^k)) for k >= 2.
In general, the asymptotic mean of the maximum exponent in the prime factorization of the largest p-rough divisor of n is Sum_{k>=1} k * d(k), where d(1) = 1/(zeta(2) * f(p, 2)), d(k) = 1/(zeta(k+1) * f(p, k+1)) - 1/(zeta(k) * f(p, k)) for k >= 2, and f(p, m) = Product_{q prime < p} (1-1/q^m).

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... .

A257913 Pandigital numbers reordered so that the numbers A050278(n)/(2^k*3^m), where 2^k||A050278(n) and 3^m||A050278(n), appear in nondecreasing order.

Original entry on oeis.org

2845310976, 1379524608, 1745960832, 6398410752, 3076521984, 5892341760, 2305179648, 3718250496, 1578369024, 9145036728, 5392687104, 1356709824, 1607952384, 3215904768, 1485029376, 5638470912, 5619843072, 6185973240, 5234098176, 7246198035, 1072963584

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 12 2015

If two such numbers A050278(n_1)/(2^k_1*3^m_1) and A050278(n_2)/(2^k_2*3^m_2) are equal, then A050278(n_1) appears earlier than A050278(n_2) iff A050278(n_1)<A050278(n_2). For example, a(2)/(2^13*3^7)=a(3)/(2^7*3^11)= 77. There are 210189 such pairs.

Note that, a(1) = 2845310976 means that min(A050278(n)/(2^k*3^m)) = 2845310976/(2^19*3^4) = 67.

Cf. A050278, A257893, A257899, A257901, A257914, A065330.

cp's OEIS Frontend

A165725 Largest divisor of n coprime to 30. I.e., a(n) = max { k | gcd(n, k) = k and gcd(k, 30) = 1 }.

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A225857 Numbers of the form 2^i*3^j*(12k+1) or 2^i*3^j*(12k+5), i, j, k >= 0.

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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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A385044 The number of unitary divisors of n that are 5-rough numbers (A007310).

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

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

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

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A225857 Numbers of the form 2^i3^j(12k+1) or 2^i3^j(12k+5), i, j, k >= 0.