login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A128608 a(n)=sigma(A128607(n)), where A128607(n) is the sequence of perfect (or pure) powers such that a(n) is a perfect power. 3
1, 121, 400, 961, 116281, 2989441, 7958041, 361722361, 962922961, 1902442689, 1891467081, 5168743489, 4755619521, 5215583961, 6835486329, 7496615889, 13884144561, 13884144561, 35018011161, 120776405841, 120776405841, 230195565369, 253358202409, 171651947481 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Denote by egcd(n) the gcd of all the powers in the prime factorization of n. In our context, a square has egcd=2, a cube has egcd=3 and so on. The only elements n in the sequence for which egcd(n)>2 are 81 and 343. Are there any others? Conjecture I: egcd(A128607(n))=2 for all n>2. Let a(n)=sigma(A128607(n)). Note that A128607(11)=1857437604=(2^2)*(3^2)*(11^2)*(653^2) has a(11)=5168743489=(7^3)*(13^3)*(19^3). Any other cubes or higher egcd's in this sequence? Conjecture II: egcd(a(n))=2 for all n ne 11.
LINKS
EXAMPLE
a(2) = sigma(A128607(2)) = sigma(343) = 1+7+7^2+7^3 = 400 = 2^4*5^2.
MAPLE
N:= 10^13: # to get all terms where A128607(n) <= N
pows:= {1, seq(seq(n^k, n = 2 .. floor(N^(1/k))), k = 2 .. floor(log[2](N)))}:
filter:= proc(n) local s, F;
s:= numtheory:-sigma(n);
F:= map(t -> t[2], ifactors(s)[2]);
igcd(op(F)) >= 2
end proc:
filter(1):= true: A128608:= sort(convert((filter, pows), list)):
map(numtheory:-sigma, A128608); # Robert Israel, Feb 14 2016
MATHEMATICA
M = 10^13; (* to get all terms where A128607(n) <= M *)
pows = {1, Table[Table[n^k, {n, 2, Floor[M^(1/k)]}], {k, 2, BitLength[M]-1}]} // Flatten // Union;
okQ[n_] := Module[{s, F}, s = DivisorSigma[1, n]; F = FactorInteger[s][[All, 2]]; GCD @@ F >= 2];
okQ[1] = True;
DivisorSigma[1, #]& /@ Select[pows, okQ] (* Jean-François Alcover, Feb 09 2023, after Robert Israel *)
CROSSREFS
Sequence in context: A253321 A253328 A257035 * A326710 A144719 A222551
KEYWORD
nonn
AUTHOR
Walter Kehowski, Mar 20 2007
EXTENSIONS
1, 13884144561, 35018011161, 120776405841, added by Zak Seidov, Feb 14 2016
Edited by Robert Israel, Feb 14 2016
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 23 13:11 EDT 2024. Contains 371913 sequences. (Running on oeis4.)