A137488 -id:A137488 - cp's OEIS frontend

A189991 Numbers with prime factorization p^4*q^4.

1296, 10000, 38416, 50625, 194481, 234256, 456976, 1185921, 1336336, 1500625, 2085136, 2313441, 4477456, 6765201, 9150625, 10556001, 11316496, 14776336, 17850625, 22667121, 29986576, 35153041, 45212176, 52200625, 54700816, 57289761, 68574961, 74805201

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 03 2011

nonn

The primes p and q must be distinct, or else the product has factorization p^8 (or q^8, for that matter).

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

Cf. A006881, A137488, A179671, A189990.

Cf. A085964, A085968, A085986.

lst = {}; Do[If[Sort[Last/@FactorInteger[n]] == {4, 4}, Print[n]; AppendTo[lst, n]], {n,55000000}]; lst (* Orlovsky *)
lim = 10^8; pMax = PrimePi[(lim/16)^(1/4)]; Select[Union[Flatten[Table[Prime[i]^4 Prime[j]^4, {i, 2, pMax}, {j, i - 1}]]], # <= lim &] (* Alonso del Arte, May 03 2011 *)
With[{nn=30},Take[Union[Times@@@(Subsets[Prime[Range[nn]],{2}]^4)],nn]] (* Harvey P. Dale, Mar 05 2015 *)

list(lim)=my(v=List(),t);forprime(p=2, lim^(1/8), t=p^4;forprime(q=p+1, (lim\t)^(1/4), listput(v,t*q^4))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A189991(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 int(n+x+(t:=primepi(s:=isqrt(y:=integer_nthroot(x,4)[0])))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

Sum_{n>=1} 1/a(n) = (P(4)^2 - P(8))/2 = (A085964^2 - A085968)/2 = 0.000933..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

a(n) = A006881(n)^4 = A085986(n)^2. - Chai Wah Wu, Mar 27 2025

A135581 The 5th divisor of numbers with 25 divisors.

Original entry on oeis.org

6, 8, 8, 15, 21, 11, 13, 27, 16, 35, 16, 27, 16, 27, 55, 27, 16, 16, 16, 65, 27, 16, 77, 16, 85, 16, 29, 91, 31, 16, 95, 16, 37, 115, 16, 119, 16, 41, 43, 133, 16, 47, 16, 143, 125, 16, 125, 16, 53, 161, 16, 59, 16, 61, 125, 187, 16, 67, 16, 203, 125, 16, 209, 71, 16, 125

Offset: 1

G. H. Ens (GerardEns(AT)gmail.com), Feb 24 2008

n=1 means the first number that has 25 divisors (1296), 6 is the 5th divisor of 1296. The second number with 25 divisors is 10000 and its 5th divisor is 8
This is one example of such a sequence where the divisor index is the square root of the total number of divisors (self included).
Other examples would be the 6th divisor of numbers with 36 divisors, 7th divisor of numbers with 49 divisors, etc.
Choice of the square root is arbitrary.
All but 16 primes {2, 3, 5, 7, 17, 19, 23, 83, 89, 97, 101, 103, 107, 109, 113} are in this sequence; p^3 and p^4 are in this sequence for all prime p; pq is in this sequence for all prime p and q with p < q < p^2. No other terms are members. - Charles R Greathouse IV, Nov 28 2011

			a(1) = 6 because 6 is the 5th divisor of 1296 and 1296 is the first number with 25 divisors.
a(2) = 8 because 8 is the 5th divisor of 10000 and 10000 is the second number with 25 divisors.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Laurens Lapré, Natural division.
Wikipedia, Divisor function

Cf. A005179, A137488, A200722, A201266.

a135581 n = [d | d <- [1..], a137488 n `mod` d == 0] !! 4
-- Reinhard Zumkeller, Nov 29 2011

upto=10^10;With[{max1=Ceiling[Power[upto, (4)^-1]],max2=Ceiling[ Power[ upto, (24)^-1]]},Take[Divisors[#][[5]]&/@Select[Union[Join[ Range[ max2]^24, Times@@@(Subsets[Range[max1],{2}]^4)]],DivisorSigma[0,#] == 25&], Ceiling[max1/4]]] (* Harvey P. Dale, Nov 25 2011 *)

from math import isqrt
from sympy import primepi, integer_nthroot, primerange, divisors
def A135581(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 int(n+x+(t:=primepi(s:=isqrt(y:=integer_nthroot(x,4)[0])))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)))-primepi(integer_nthroot(x,24)[0])
    return divisors(bisection(f,n,n))[4] # Chai Wah Wu, Feb 22 2025

Corrected and extended by R. J. Mathar, Apr 21 2008. The original entries were wrong from the 16th term onwards.

A175755 Numbers with 49 divisors.

Original entry on oeis.org

46656, 1000000, 7529536, 11390625, 85766121, 113379904, 308915776, 1291467969, 1544804416, 1838265625, 3010936384, 3518743761, 9474296896, 17596287801, 27680640625, 34296447249, 38068692544, 56800235584, 75418890625, 107918163081, 164206490176, 208422380089

Offset: 1

Jaroslav Krizek, Aug 27 2010

nonn

Numbers of the forms p^48 and p^6*q^6, where p and q are distinct primes.

			a(1) = A114334(49); a(2) = A159765(49).

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

Cf. A000005, A139575, A175754, A201266, A137488, A135581.

a175755 n = a175755_list !! (n-1)
a175755_list = m (map (^ 48) a000040_list) (map (^ 6) a006881_list) where
   m xs'@(x:xs) ys'@(y:ys) | x < y = x : m xs ys'
                           | otherwise = y : m xs' ys
-- Reinhard Zumkeller, Nov 29 2011

Select[Range[100000000],DivisorSigma[0,#]==48&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

is(n)=numdiv(n)==49 \\ Charles R Greathouse IV, Jun 19 2016

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A175755(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 int(n+x+(t:=primepi(s:=isqrt(y:=integer_nthroot(x,6)[0])))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1))-primepi(integer_nthroot(x,48)[0]))
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

A000005(a(n)) = 49.

Sum_{n>=1} 1/a(n) = (P(6)^2 - P(12))/2 + P(48) = 0.0000226806..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

Extended by T. D. Noe, May 08 2011

A336595 Numbers whose number of divisors is divisible by 5.

Original entry on oeis.org

16, 48, 80, 81, 112, 144, 162, 176, 208, 240, 272, 304, 324, 336, 368, 400, 405, 432, 464, 496, 512, 528, 560, 567, 592, 624, 625, 648, 656, 688, 720, 752, 784, 810, 816, 848, 880, 891, 912, 944, 976, 1008, 1040, 1053, 1072, 1104, 1134, 1136, 1168, 1200, 1232

Offset: 1

Amiram Eldar, Jul 26 2020

nonn

The asymptotic density of this sequence is 1 - zeta(5)/zeta(4) = 0.0419426259... (Sathe, 1945).

			16 is a term since A000005(16) = 5 is divisible by 5.

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 63.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical Notes, XIII. A Sequal to Note IV, Elemente der Mathematik, Vol. 18 (1963), pp. 8-11.
S. S. Pillai, On a congruence property of the divisor function, J. Indian Math. Soc. (N. S.), Vol. 6, (1942), pp. 118-119.
L. G. Sathe, On a congruence property of the divisor function, American Journal of Mathematics, Vol. 67, No. 3 (1945), pp. 397-406.

Cf. A000005, A008587, A059269, A336596.

Cf. A030514, A030628, A113849, A178739, A179665, A030633, A030638, A137488, A137493, A175745.

q:= n-> is(irem(numtheory[tau](n), 5)=0):
select(q, [$1..1300])[];  # Alois P. Heinz, Jul 26 2020

Select[Range[1300], Divisible[DivisorSigma[0, #], 5] &]

A030514 UNION A030628 \ {1} UNION A030633 UNION A030638 UNION A137488 UNION A137493 UNION A175745 UNION A175749 UNION A175752 UNION A175756 UNION ... - R. J. Mathar, May 05 2023

A175750 Numbers with 42 divisors.

Original entry on oeis.org

2880, 4032, 4800, 6336, 7488, 9408, 9792, 10944, 11200, 13248, 14580, 15552, 15680, 16704, 17600, 17856, 20412, 20800, 21312, 23232, 23328, 23616, 24768, 27072, 27200, 30400, 30528, 32076, 32448, 33984, 34496, 35136, 36450, 36800, 37908, 38592, 38720, 40768

Offset: 1

Jaroslav Krizek, Aug 27 2010

nonn

Numbers of the forms p^41, p^20*q^1, p^13*q^2, p^6*q^5 and p^6*q^2*r^1 (A179703), 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. A137484, A137488.

Select[Range[50000],DivisorSigma[0,#]==42&] (* Vladimir Joseph Stephan Orlovsky, May 04 2011 *)

is(n)=numdiv(n)==42 \\ Charles R Greathouse IV, Jun 19 2016

A000005(a(n))=42.

Extended by T. D. Noe, May 08 2011

A175756 Numbers with 50 divisors.

Original entry on oeis.org

6480, 9072, 14256, 16848, 22032, 24624, 29808, 30000, 37584, 40176, 41472, 47952, 53136, 55728, 60912, 68688, 70000, 76464, 79056, 86832, 92016, 94608, 101250, 102384, 107568, 110000, 115248, 115344, 125712, 130000, 130896, 133488

Offset: 1

Jaroslav Krizek, Aug 27 2010

nonn

Numbers of the forms p^49, p^24*q^1, p^9*q^4 and p^4*q^4*r^1 (A190012), 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. A137484, A137488, A175750.

Select[Range[200000], DivisorSigma[0, #] == 50 &] (* Vladimir Joseph Stephan Orlovsky, May 04 2011 *)

is(n)=numdiv(n)==50 \\ Charles R Greathouse IV, Jun 19 2016

A000005(a(n))=50.

cp's OEIS Frontend

A189991 Numbers with prime factorization p^4*q^4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A135581 The 5th divisor of numbers with 25 divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Python

Extensions

A175755 Numbers with 49 divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

Extensions

A336595 Numbers whose number of divisors is divisible by 5.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A175750 Numbers with 42 divisors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A175756 Numbers with 50 divisors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula