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A135581 The 5th divisor of numbers with 25 divisors.

%I A135581 #42 Feb 22 2025 20:03:55
%S A135581 6,8,8,15,21,11,13,27,16,35,16,27,16,27,55,27,16,16,16,65,27,16,77,16,
%T A135581 85,16,29,91,31,16,95,16,37,115,16,119,16,41,43,133,16,47,16,143,125,
%U A135581 16,125,16,53,161,16,59,16,61,125,187,16,67,16,203,125,16,209,71,16,125
%N A135581 The 5th divisor of numbers with 25 divisors.
%C A135581 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
%C A135581 This is one example of such a sequence where the divisor index is the square root of the total number of divisors (self included).
%C A135581 Other examples would be the 6th divisor of numbers with 36 divisors, 7th divisor of numbers with 49 divisors, etc.
%C A135581 Choice of the square root is arbitrary.
%C A135581 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
%H A135581 Reinhard Zumkeller, <a href="/A135581/b135581.txt">Table of n, a(n) for n = 1..10000</a>
%H A135581 Laurens Lapré, <a href="http://laurenslapre.nl/lapre_006.htm">Natural division</a>.
%H A135581 Wikipedia, <a href="http://en.wikipedia.org/wiki/Divisor_function">Divisor function</a>
%e A135581 a(1) = 6 because 6 is the 5th divisor of 1296 and 1296 is the first number with 25 divisors.
%e A135581 a(2) = 8 because 8 is the 5th divisor of 10000 and 10000 is the second number with 25 divisors.
%t A135581 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 *)
%o A135581 (Haskell)
%o A135581 a135581 n = [d | d <- [1..], a137488 n `mod` d == 0] !! 4
%o A135581 -- _Reinhard Zumkeller_, Nov 29 2011
%o A135581 (Python)
%o A135581 from math import isqrt
%o A135581 from sympy import primepi, integer_nthroot, primerange, divisors
%o A135581 def A135581(n):
%o A135581     def bisection(f,kmin=0,kmax=1):
%o A135581         while f(kmax) > kmax: kmax <<= 1
%o A135581         kmin = kmax >> 1
%o A135581         while kmax-kmin > 1:
%o A135581             kmid = kmax+kmin>>1
%o A135581             if f(kmid) <= kmid:
%o A135581                 kmax = kmid
%o A135581             else:
%o A135581                 kmin = kmid
%o A135581         return kmax
%o A135581     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])
%o A135581     return divisors(bisection(f,n,n))[4] # _Chai Wah Wu_, Feb 22 2025
%Y A135581 Cf. A005179, A137488, A200722, A201266.
%K A135581 nonn,look,easy
%O A135581 1,1
%A A135581 G. H. Ens (GerardEns(AT)gmail.com), Feb 24 2008
%E A135581 Corrected and extended by _R. J. Mathar_, Apr 21 2008. The original entries were wrong from the 16th term onwards.