login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A273593
Numbers which are equal to the sum of the aliquot parts of the aliquot parts of their Euler totient function.
0
310, 364, 5497, 11592, 40752, 494328, 3030109, 3554874, 128984839
OFFSET
1,1
COMMENTS
Fixed points of the transform n -> Sum_{i=1..k}{sigma(d_i)-d_i}, where d_i are the aliquot parts of phi(n). The first two couples connected by this transform are (109, 186) and (8826, 9198).
a(10) > 2.5*10^9. - Giovanni Resta, May 30 2016
EXAMPLE
Euler totient function of 310 is 120 and its aliquot parts are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40 and 60. Then: sigma(1) - 1 = 0, sigma(2) - 2 = 1, sigma(3) - 3 = 1, sigma(4) - 4 = 3, sigma(5) - 5 = 1, sigma(6) - 6 = 6, sigma(8) - 8 = 7, sigma(10) - 10 = 8, sigma(12) - 12 = 16, sigma(15) - 15 = 9, sigma(20) - 20 = 22, sigma(24) - 24 = 36, sigma(30) - 30 = 42, sigma(40) - 40 = 50, sigma(60) - 60 = 108 and 0 + 1 + 1 + 3 + 1 + 6 + 7 + 8 + 16 + 9 + 22 + 36 + 42 + 50 + 108 = 310.
MAPLE
with(numtheory): P:=proc(q) local a, k, n;
for n from 1 to q do a:=sort([op(divisors(phi(n)))]);
if n=add(sigma(a[k])-a[k], k=1..nops(a)-1) then print(n); fi; od; end: P(10^9);
PROG
(PARI) is(n)=my(t=eulerphi(n)); sumdiv(t, d, if(d<t, sigma(d)-d))==n \\ Charles R Greathouse IV, Jun 08 2016
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Paolo P. Lava, May 26 2016
EXTENSIONS
a(7)-a(9) from Giovanni Resta, May 30 2016
STATUS
approved