A179746 -id:A179746 - cp's OEIS frontend

A217584 Numbers k such that d(k^2)/d(k) is an integer, where d(k) is the number of divisors of k.

1, 144, 324, 400, 784, 1936, 2025, 2500, 2704, 3600, 3969, 4624, 5625, 5776, 7056, 8100, 8464, 9604, 9801, 13456, 13689, 15376, 15876, 17424, 19600, 21609, 21904, 22500, 23409, 24336, 26896, 29241, 29584, 30625, 35344, 39204, 41616, 42849, 44944, 48400, 51984

Offset: 1

Michel Marcus, Oct 07 2012

nonn

The ratio d(k^2)/d(k) is: 1 for the number 1, 3 for numbers of the form p^4*q^2, 5 for numbers of the form p^4*q^2*r^2 (p, q, r being different primes).
Primes can't be in the sequence. A prime p has two divisors, while p^2 has three divisors: 1, p, p^2. - Alonso del Arte, Oct 07 2012
All the terms are squares since d(m) is odd if and only if m is a square, so d(k^2) is odd and since d(k)|d(k^2), d(k) is also odd, so k is a square. The ratio d(k^2)/d(k) can take values other than 1, 3, and 5: 1587600 is the least term with a ratio 9, and 192099600 is the least term with a ratio 15. - Amiram Eldar, May 23 2020
From Bernard Schott, May 29 2020 and Nov 22 2020: (Start)
This sequence comes from the 3rd problem, proposed by Belarus, during the 39th International Mathematical Olympiad in 1998 at Taipei (Taiwan) [see the link IMO].
If the prime signature of k is (u_1, u_2, ... , u_q) then d(k^2)/d(k) = Product_{i=1..q} (2*u_i+1)/(u_i+1). Two results:
1) If k is a term such that d(k^2)/d(k) = m, then all numbers that have the same prime signature of k are also terms and give the same ratio (see examples below).
2) The set of the integer values of the ratio d(k^2)/d(k) is exactly the set of all positive odd integers (see Marcin E. Kuczma reference).
Some examples:
For numbers with prime signature = (4, 2) (A189988), the ratio is 3 and the smallest such integer is 144 = 2^4 * 3^2.
For numbers with prime signature = (4, 2, 2) (A179746), the ratio is 5 and the smallest such integer is 3600 = 2^4 * 3^2 * 5^2.
For numbers with prime signature = (4, 4, 2, 2) the ratio is 9 and the smallest such integer is 1587600 = 2^4 * 3^4 * 5^2 * 7^2.
For numbers with prime signature = (8, 4, 4, 2, 2) the ratio is 17 and the smallest such integer is 76839840000 = 2^8 * 3^4 * 5^4 * 7^2 * 11^2 (found by David A. Corneth with other prime signatures). (End)

			d(1^2)/d(1) = d(1)/d(1) = 1 an integer, so 1 belongs to the sequence.
144^2 has 45 divisors: 1, 2, 3, 4, 6, 8, 9, 12, ..., 20736, while 144 has 15 divisors: 1, 2, 3, 4, 6, 8, 9, 12, ..., 144; 45/15 = 3 and so 144 is in the sequence.

Marcin E. Kuczma, International Mathematical Olympiads, 1986-1999, The Mathematical Association of America, 2003, pages 134-135.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..100 from Paolo P. Lava)
The IMO Compendium, Problem 3, 39th IMO 1998.
Kin Y. Li, Problem 3, Mathematical Excalibur, Vol. 4, No. 3, Jan.-Mar. 1999.
Index to sequences related to Olympiads.

Cf. A000005, A003593, A025487, A048691.
Subsequences: A189988 (d(k^2)/d(k) = 3), A179746 (d(k^2)/d(k) = 5).
Cf. A339055 (values taken by d(a(n)^2)/d(a(n))), A339056 (smallest k such that d(k^2)/d(k) = n-th odd).

Select[Range[1000], IntegerQ[DivisorSigma[0, #^2]/DivisorSigma[0, #]] &] (* Alonso del Arte, Oct 07 2012 *)
Select[Range[228]^2, Divisible[DivisorSigma[0, #^2], DivisorSigma[0, #]] &] (* Amiram Eldar, May 23 2020 *)

dn2dn(n)= {for (i=1, n, if (denominator(numdiv(i^2)/numdiv(i))==1, print1(i,", ");););}

A190106 Numbers with prime factorization p^2q^3r^3 where p, q, and r are distinct primes.

Original entry on oeis.org

5400, 9000, 10584, 13500, 24696, 26136, 36504, 37044, 49000, 62424, 68600, 77976, 95832, 114264, 121000, 143748, 158184, 165375, 169000, 171500, 181656, 207576, 231525, 237276, 266200, 289000, 295704, 332024, 353736, 361000, 363096

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 04 2011

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, List of prime signatures, 2010.
Index to sequences related to prime signature.

Cf. A179691, A179698, A179746, A189991.

Cf. A085548, A085541, A085965, A085966, A085968.

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,3,3};Select[Range[500000],f]

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\4)^(1/6), t1=p^3;forprime(q=p+1, (lim\t1)^(1/3), t2=t1*q^3;forprime(r=2, sqrt(lim\t2), if(p==r||q==r, next);listput(v,t2*r^2)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A190106(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x+sum((t:=primepi(s:=isqrt(y:=integer_nthroot(x//r**2,3)[0])))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(isqrt(x)+1))+sum(primepi(integer_nthroot(x//p**5,3)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))-primepi(integer_nthroot(x,8)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Sum_{n>=1} 1/a(n) = P(2)*P(3)^2/2 - P(2)*P(6)/2 - P(3)*P(5) + P(8) = 0.00085907862422456410530..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024

A175752 Numbers with 45 divisors.

Original entry on oeis.org

3600, 7056, 8100, 15876, 17424, 19600, 20736, 22500, 24336, 39204, 41616, 48400, 51984, 54756, 67600, 76176, 86436, 93636, 94864, 99225, 104976, 115600, 116964, 121104, 122500, 132496, 138384, 144400, 147456, 160000, 171396, 197136

Offset: 1

Jaroslav Krizek, Aug 27 2010

nonn

Numbers of the forms p^44, p^14*q^2, p^8*q^4 (squares of A189988) and p^4*q^2*r^2 (A179746), where p, q, and r are distinct primes.

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, Index entries for number of divisors

Cf. A000005, A139574, A175751.

Select[Range[400000],DivisorSigma[0,#]==45&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

is(n)=numdiv(n)==45 \\ Charles R Greathouse IV, Jun 19 2016

A000005(a(n)) = 45.

Sum_{n>=1} 1/a(n) = (P(2)^2*P(4) - P(4)^2)/2 - P(2)*P(6) + P(8) + P(2)*P(14) - P(16) + P(4)*P(8) - P(12) + P(44) = 0.00133023..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

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A217584 Numbers k such that d(k^2)/d(k) is an integer, where d(k) is the number of divisors of k.

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A190106 Numbers with prime factorization p^2q^3r^3 where p, q, and r are distinct primes.

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A175752 Numbers with 45 divisors.

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cp's OEIS Frontend

A217584 Numbers k such that d(k^2)/d(k) is an integer, where d(k) is the number of divisors of k.

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A190106 Numbers with prime factorization p^2*q^3*r^3 where p, q, and r are distinct primes.

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Mathematica

PARI

Python

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A175752 Numbers with 45 divisors.

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Author

Keywords

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Programs

Mathematica

PARI

Formula

A190106 Numbers with prime factorization p^2q^3r^3 where p, q, and r are distinct primes.