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”).

A158918
n, ps(n), ps^2(n), ..., ps^9(n) forms an increasing ps-sequence of length 10,
0
12900, 737100, 772176, 832050, 844032, 844992, 864976, 872208, 879984, 887088, 926400, 939900, 954828, 960372, 962724, 964800, 967500, 969444, 972804, 973296, 975828, 976144, 980000, 982044, 984064
OFFSET
1,1
COMMENTS
ps (read phi-sigma) is the consecution of the sigma (sum of divisors) and Euler totient function.
The problem raises from the aliquot sequences. In the case of aliquot sequences we calculate the next element by calculating the sum of the divisors, and afterwards subtract the original number n.
Instead of subtract n we apply the totient function in order to get ps-sequences. Fast decreases can appear if the sum of divisors consist of many different small primes.
In fact ps-sequences end very fast in cycles, often cycles of small length.
I didn't find an increasing ps-sequence of length > 12.
EXAMPLE
12900 = 2^2*3*5^2*43, s(12900) = 7*4*31*44, ps(12900) = 6*30*8*10 = 2^6*3^2*5^2 = 14400;
s(14400) = 127*13*31, ps(14400) = 126*12*30 = 2^4*3^4*5*7 = 45360; ...
PROG
(Sage) #(not fast!)
def phi(L):
m = 1
for l in L:
m = m * (l[0]-1)
for i in (1..l[1]-1):
m = m * l[0]
return m
def sigma(L):
m = 1
for l in L:
q = 1
for i in (0..l[1]):
q = q * l[0]
m = m * (q-1) / (l[0]-1)
return m
cc = 8
START = 2
END = 10000000
for f0 in (START..END):
c = 0
f = f0
Lf = list(factor(f))
s = sigma(Lf)
Ls = list(factor(s))
f1 = phi(Ls)
while f < f1:
c = c + 1
f = f1
Lf = list(factor(f))
s = sigma(Lf)
Ls = list(factor(s))
f1 = phi(Ls)
if c > cc:
print(c, ":", f0)
CROSSREFS
Sequence in context: A184453 A233936 A360836 * A185885 A212079 A162895
KEYWORD
nonn
AUTHOR
Matthijs Coster, Mar 30 2009
STATUS
approved