A175744 -id:A175744 - cp's OEIS frontend

A175745 Numbers with 35 divisors.

5184, 11664, 40000, 153664, 250000, 455625, 937024, 1265625, 1750329, 1827904, 1882384, 5345344, 8340544, 9529569, 10673289, 17909824, 20820969, 28344976, 37515625, 45265984, 59105344, 60886809, 73530625, 77228944, 95004009, 119946304, 143496441, 180848704

Offset: 1

Jaroslav Krizek, Aug 27 2010

nonn

Numbers of the form p^34 and p^6*q^4 (A190464), where p and q 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, A175742, A175743, A175744.

Select[Range[9000000],DivisorSigma[0,#]==35&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

is(n)=numdiv(n)==35 \\ Charles R Greathouse IV, Jun 19 2016

from sympy import primepi, integer_nthroot, primerange
def A175745(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(primepi(integer_nthroot(x//p**6,4)[0]) for p in primerange(integer_nthroot(x,6)[0]+1))+primepi(integer_nthroot(x,10)[0])-primepi(integer_nthroot(x,34)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

A000005(a(n)) = 35.

Sum_{n>=1} 1/a(n) = P(4)*P(6) - P(10) + P(34) = 0.000320676..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

Extended by T. D. Noe, May 08 2011

A175746 Numbers with 36 divisors.

Original entry on oeis.org

1260, 1440, 1800, 1980, 2016, 2100, 2340, 2400, 2700, 2772, 2940, 3060, 3150, 3168, 3276, 3300, 3420, 3528, 3744, 3840, 3900, 4140, 4284, 4410, 4500, 4704, 4788, 4860, 4896, 4950, 5100, 5148, 5220, 5292, 5376, 5472, 5580, 5600, 5700, 5796, 5850, 6468, 6624

Offset: 1

Jaroslav Krizek, Aug 27 2010

nonn

Numbers of the forms p^35, p^17*q^1, p^11*q^2, p^8*q^3, p^5*q^5, p^8*q^1*r^1, p^5*q^2*r^1, p^3*q^2*r^2 and p^2*q^2*r^1*s^1, where p, q, r, and s are distinct primes.

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

Cf. A175744, A175745.

Select[Range[10000],DivisorSigma[0,#]==36&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

is(n)=numdiv(n)==36 \\ Charles R Greathouse IV, Jun 19 2016

A000005(a(n))=36.

Extended by T. D. Noe, May 08 2011

A065985 Numbers k such that d(k) / 2 is prime, where d(k) = number of divisors of k.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 18, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, 38, 39, 44, 45, 46, 48, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 74, 75, 76, 77, 80, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 106, 111, 112, 115, 116, 117, 118, 119, 122, 123, 124, 125, 129, 133, 134

Offset: 1

Joseph L. Pe, Dec 10 2001

nonn

Numbers whose sorted prime signature (A118914) is either of the form {2*p-1} or {1, p-1}, where p is a prime. Equivalently, disjoint union of numbers of the form q^(2*p-1) where p and q are primes, and numbers of the form r * q^(p-1), where p, q and r are primes and r != q. - Amiram Eldar, Sep 09 2024

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

Cf. A000005, A118914.
Numbers with exactly 2*p divisors: A030513 (p=2), A030515 (p=3), A030628 \ {1} (p=5), A030632 (p=7), A137485 (p=11), A137489 (p=13), A175744 (p=17), A175747 (p=19).
Subsequences: A006881, A030078, A050997, A054753, A095990, A096156, A138031, A178739, A179665, A189987.

Select[Range[1, 1000], PrimeQ[DivisorSigma[0, # ] / 2] == True &]

n=0; for (m=1, 10^9, f=numdiv(m)/2; if (frac(f)==0 && isprime(f), write("b065985.txt", n++, " ", m); if (n==1000, return))) \\ Harry J. Smith, Nov 05 2009

is(n)=n=numdiv(n)/2; denominator(n)==1 && isprime(n) \\ Charles R Greathouse IV, Oct 15 2015

A274808 Numbers n such that n and n+1 both have 34 divisors.

Original entry on oeis.org

2035980763136, 218010592018431, 413918027251712, 921717810593792, 957141387771903, 1017635547447296, 1119195504115712, 1842969898713087, 2057374251679743, 2435402979278847, 2913421405257728, 3039120499474431, 3129396016513023

Offset: 1

Charles R Greathouse IV, Jul 07 2016

nonn

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

Intersection of A005237 and A175744.

PARI
```
is(n)=numdiv(n)==34 && numdiv(n+1)==34
```

has(n)=if(n%4==2, ispower(n/2, 16, &n) && isprime(n), bitand(n, 131071)==65536 && isprime(n>>16) && n>65536) \\ check if n is even with 34 divisors
list(lim)=my(v=List(), t); forprime(p=2, sqrtnint(lim\=1, 33), t=p^33; if(has(t+1), listput(v, t)); if(has(t-1), listput(v, t-1))); forprime(p=3, sqrtnint(lim\3, 16), my(p16=p^16); forprime(q=3, lim\p16, if(p==q, next); t=p16*q; if(has(t+1), listput(v, t)); if(has(t-1), listput(v, t-1)))); Set(v)

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A175745 Numbers with 35 divisors.

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A175746 Numbers with 36 divisors.

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A065985 Numbers k such that d(k) / 2 is prime, where d(k) = number of divisors of k.

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Mathematica

PARI

PARI

A274808 Numbers n such that n and n+1 both have 34 divisors.

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Author

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PARI

PARI