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

1,2

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)

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 *)

Prog

(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

Crossrefs

Cf. A000005, A006218, A066843.

Sequence in context: A098422 A191066 A087762 * A160422 A213782 A269428

Adjacent sequences: A195346 A195347 A195348 * A195350 A195351 A195352

Keyword

nonn

Author

Carl Najafi, Sep 16 2011

Status

approved