A137487 -id:A137487 - 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

A163569 Numbers of the form p^3q^2r where p, q and r are three distinct primes.

Original entry on oeis.org

360, 504, 540, 600, 756, 792, 936, 1176, 1188, 1224, 1350, 1368, 1400, 1404, 1500, 1656, 1836, 1960, 2052, 2088, 2200, 2232, 2250, 2484, 2600, 2646, 2664, 2904, 2952, 3096, 3132, 3348, 3384, 3400, 3500, 3800, 3816, 3996, 4056, 4116, 4248, 4312, 4392

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 31 2009

nonn

There is no constraint on which of the three primes is the largest or smallest.

			360=2^3*3^2*5. 504=2^3*3^2*7. 1188=2^2*3^3*11.

Subsequence of A137487. - R. J. Mathar, Aug 01 2009

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,2,3}; Select[Range[5000], f]

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\12)^(1/3), t1=p^3;forprime(q=2, sqrt(lim\t1), if(p==q, next);t2=t1*q^2;forprime(r=2, lim\t2, if(p==r||q==r, next);listput(v,t2*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A163569(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(x//(p**3*q**2)) for p in primerange(integer_nthroot(x,3)[0]+1) for q in primerange(isqrt(x//p**3)+1))+sum(primepi(integer_nthroot(x//p**3,3)[0]) for p in primerange(integer_nthroot(x,3)[0]+1))+sum(primepi(isqrt(x//p**4)) for p in primerange(integer_nthroot(x,4)[0]+1))+sum(primepi(x//p**5) for p in primerange(integer_nthroot(x,5)[0]+1))-(primepi(integer_nthroot(x,6)[0])<<1)
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

A274362 Numbers n such that n and n+1 both have 24 divisors.

Original entry on oeis.org

5984, 11780, 20349, 22815, 24794, 26144, 27675, 29799, 31724, 33579, 33824, 34335, 34748, 36764, 37323, 37664, 38324, 38367, 38675, 38709, 40544, 41624, 42020, 44505, 44954, 47564, 47684, 48950, 50024, 51204, 52155, 52767, 53703, 53955, 54495, 55419

Offset: 1

Charles R Greathouse IV, Jun 19 2016

nonn

Goldston-Graham-Pintz-Yildirim prove that this sequence is infinite; in particular infinitely often a(k) = A189982(n) = A189982(n+1) - 1. In fact, their proof shows that at least one of the residue classes 355740n + 47480, 889350n + 118700, or 592900n + 79134 contains infinitely many terms of this sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yıldırım, Small gaps between almost primes, the parity problem, and some conjectures of Erdos on consecutive integers, arXiv:0803.2636 [math.NT] (2008).

Intersection of A005237 and A137487.

Cf. A000005, A274357, A189982.

Reap[For[k = 1, k < 56000, k++, If[DivisorSigma[0, k] == DivisorSigma[0, k + 1] == 24, Sow[k]]]][[2, 1]] (* Jean-François Alcover, Dec 16 2018 *)

PARI
```
is(n)=numdiv(n)==24 && numdiv(n+1)==24
```

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A137486 Numbers with 23 divisors.

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A163569 Numbers of the form p^3q^2r where p, q and r are three distinct primes.

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A274362 Numbers n such that n and n+1 both have 24 divisors.

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cp's OEIS Frontend

A137486 Numbers with 23 divisors.

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A163569 Numbers of the form p^3*q^2*r where p, q and r are three distinct primes.

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Python

A274362 Numbers n such that n and n+1 both have 24 divisors.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A163569 Numbers of the form p^3q^2r where p, q and r are three distinct primes.