cp's OEIS Frontend

A158918 n, ps(n), ps^2(n), ..., ps^9(n) forms an increasing ps-sequence of length 10.

%I A158918 #16 Jun 01 2025 17:01:12
%S A158918 12900,737100,772176,832050,844032,844992,864976,872208,879984,887088,
%T A158918 926400,939900,954828,960372,962724,964800,967500,969444,972804,
%U A158918 973296,975828,976144,980000,982044,984064
%N A158918 n, ps(n), ps^2(n), ..., ps^9(n) forms an increasing ps-sequence of length 10.
%C A158918 ps (read phi-sigma) is the consecution of the sigma (sum of divisors) and Euler totient function.
%C A158918 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.
%C A158918 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.
%C A158918 In fact ps-sequences end very fast in cycles, often cycles of small length.
%C A158918 I didn't find an increasing ps-sequence of length > 12.
%e A158918 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;
%e A158918 s(14400) = 127*13*31, ps(14400) = 126*12*30 = 2^4*3^4*5*7 = 45360; ...
%o A158918 (Sage) #(not fast!)
%o A158918 def phi(L):
%o A158918     m = 1
%o A158918     for l in L:
%o A158918         m = m * (l[0]-1)
%o A158918         for i in (1..l[1]-1):
%o A158918             m = m * l[0]
%o A158918     return m
%o A158918 def sigma(L):
%o A158918     m = 1
%o A158918     for l in L:
%o A158918         q = 1
%o A158918         for i in (0..l[1]):
%o A158918             q = q * l[0]
%o A158918             m = m * (q-1) / (l[0]-1)
%o A158918     return m
%o A158918 cc = 8
%o A158918 START = 2
%o A158918 END = 10000000
%o A158918 for f0 in (START..END):
%o A158918     c = 0
%o A158918     f = f0
%o A158918     Lf = list(factor(f))
%o A158918     s = sigma(Lf)
%o A158918     Ls = list(factor(s))
%o A158918     f1 = phi(Ls)
%o A158918     while f < f1:
%o A158918         c = c + 1
%o A158918         f = f1
%o A158918         Lf = list(factor(f))
%o A158918         s = sigma(Lf)
%o A158918         Ls = list(factor(s))
%o A158918         f1 = phi(Ls)
%o A158918     if c > cc:
%o A158918         print(c, ":", f0)
%K A158918 nonn
%O A158918 1,1
%A A158918 _Matthijs Coster_, Mar 30 2009