A195349 Numbers n such that Sum_{k=1..n} d(k) divides Product_{k=1..n} d(k), where d(k) is the number of divisors of k.

1, 7, 19, 41, 57, 64, 68, 133, 145, 149, 164, 235, 267, 291, 317, 336, 358, 419, 433, 503, 528, 566, 599, 612, 659, 726, 801, 927, 1017, 1035, 1077, 1118, 1190, 1206, 1213, 1281, 1297, 1309, 1320, 1323, 1367, 1446, 1473, 1485, 1516, 1595, 1611, 1634, 1941

Offset: 1

Author

Carl Najafi, Sep 16 2011

Keywords

• nonn

Comments

d(k) is sometimes called tau(k) or sigma_0(k). Is this sequence infinite?

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..2000 (term 1..500 from Harvey P. Dale)

Crossrefs

Cf. A000005, A006218, A066843.

Programs

Mathematica

t = {}; a = 0; b = 1; Do[a = a + DivisorSigma[0, n]; b = b*DivisorSigma[0, n]; If[Mod[b, a] == 0, AppendTo[t, n]], {n, 2000}]; t (* T. D. Noe, Sep 16 2011 *)
With[{c=DivisorSigma[0,Range[2000]]},Position[Thread[{FoldList[ Times,c], Accumulate[ c]}],?(Divisible[#[[1]],#[[2]]]&),1,Heads->False]] // Flatten (* _Harvey P. Dale, Apr 14 2019 *)

Python

from sympy import divisor_count
A195349_list, s, p = [], 0, 1
for k in range(1,10**4):
    d = divisor_count(k)
    s += d
    p *= d
    if p % s == 0:
        A195349_list.append(k) # Chai Wah Wu, Oct 09 2021