A137485 -id:A137485 - cp's OEIS frontend

A137486 Numbers with 23 divisors.

4194304, 31381059609, 2384185791015625, 3909821048582988049, 81402749386839761113321, 3211838877954855105157369, 1174562876521148458974062689, 13569980418174090907801371961

Offset: 1

R. J. Mathar, Apr 22 2008

Maple implementation: see A030513.

22nd powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. - Omar E. Pol, May 06 2008

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

Cf. A137485 (22 divs), A137487 (24 divs).

Prime[Range[15]]^22 (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

a(n)=prime(n)^22 \\ Charles R Greathouse IV, Jun 19 2016

A000005(a(n))=23.

a(n)=A000040(n)^(23-1)=A000040(n)^(22). - Omar E. Pol, May 06 2008

A137484 Numbers with 21 divisors.

Original entry on oeis.org

576, 1600, 2916, 3136, 7744, 10816, 18225, 18496, 23104, 33856, 35721, 53824, 61504, 62500, 87616, 88209, 107584, 118336, 123201, 140625, 141376, 179776, 210681, 222784, 238144, 263169, 287296, 322624, 341056, 385641, 399424, 440896

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^20 or p^2*q^6 (A189990) where p and q are distinct primes. - R. J. Mathar, Mar 01 2010

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000005, A030513, A030638 (20 divisors), A137485 (22 divisors), A189990.

Select[Range[450000],DivisorSigma[0,#]==21&] (* Vladimir Joseph Stephan Orlovsky, May 03 2011 *)

is(n)=numdiv(n)==21 \\ Charles R Greathouse IV, Jun 19 2016

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A137484(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(isqrt(x//p**6)) for p in primerange(integer_nthroot(x,6)[0]+1))+primepi(integer_nthroot(x,8)[0])-primepi(integer_nthroot(x,20)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

A000005(a(n)) = 21.

Sum_{n>=1} 1/a(n) = P(2)*P(6) - P(8) + P(20) = 0.00365945..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

A166546 Natural numbers n such that d(n) + 1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80, 82, 83, 84, 85, 86, 87, 89, 90

Offset: 1

Giovanni Teofilatto, Oct 16 2009

nonn

Natural numbers n such that d(d(n)+1)= 2. - Giovanni Teofilatto, Oct 26 2009

The complement is the union of A001248, A030514, A030516, A030626, A030627, A030629, A030631, A030632, A030633 etc. - R. J. Mathar, Oct 26 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..6955

Cf. A000005.

Cf. A073915. - R. J. Mathar, Oct 26 2009

[n: n in [1..100] | IsPrime(NumberOfDivisors(n)+1)]; // Vincenzo Librandi, Jan 20 2019

Select[Range@90, PrimeQ[DivisorSigma[0, #] + 1] &] (* Vincenzo Librandi, Jan 20 2019 *)

isok(n) = isprime(numdiv(n)+1); \\ Michel Marcus, Jan 20 2019

{1} U A000040 U A030513 U A030515 U A030628 U A030630 U A030634 U A030636 U A137485 U A137491 U A137493 U ... . - R. J. Mathar, Oct 26 2009

A274366 Numbers n such that n and n+1 both have 22 divisors.

Original entry on oeis.org

2200933376, 3724751871, 4982377472, 5055007743, 5828903936, 6506195967, 9771369471, 12238318592, 13810439168, 15213325311, 15503492096, 15624424448, 17027310591, 20703583232, 22590198783, 23194860543, 27596727296, 28274019327, 30136306688, 30450801663

Offset: 1

Charles R Greathouse IV, Jun 19 2016

nonn

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

Intersection of A005237 and A137485.

PARI
```
is(n)=numdiv(n)==22 && numdiv(n+1)==22
```

list(lim)=my(v=List(), t); forprime(p=2, sqrtnint(lim\=1, 21), t=p^21; if(numdiv(t+1)==22, listput(v, t)); if(numdiv(t-1)==22, listput(v, t-1))); forprime(p=3, sqrtnint(lim\3, 10), my(p10=p^10); forprime(q=3, lim\p10, if(p==q, next); t=p10*q; if(numdiv(t+1)==22, listput(v, t)); if(numdiv(t-1)==22, listput(v, t-1)))); Set(v)

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

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A137486 Numbers with 23 divisors.

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A137484 Numbers with 21 divisors.

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A166546 Natural numbers n such that d(n) + 1 is prime.

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A274366 Numbers n such that n and n+1 both have 22 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

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