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Irregular triangle where the n-th row lists the positive integers k such that phi(k) = A002034(k^n).
2

%I #19 Aug 13 2020 12:19:09

%S 1,8,9,12,18,1,24,25,50,1,32,48,49,98,1,1,64,72,121,242,1,81,96,162,

%T 169,338,1,80,1,125,250,289,578,1,128,361,722,1,192,1,135,144,216,270,

%U 529,1058,1,160,1,324,1,175,225,350,450,841,1682,1,961,1922

%N Irregular triangle where the n-th row lists the positive integers k such that phi(k) = A002034(k^n).

%C If k = Product_{i=1..t} p_i^e_i and phi(k) = S(k^n), then (p-1)*p^(e-1)*phi(k/p) = S(p^e), where p^e = Max_{i=1..t} S(p_i^e_i) and S(n) = A002034(n).

%H Jinyuan Wang, <a href="/A336711/b336711.txt">Rows n = 1..359 of triangle, flattened</a>

%H Bin Chen, <a href="https://doi.org/10.4028/www.scientific.net/AMM.204-208.4785">An Equation Involving the Smarandache Function</a>, Applied Mechanics and Materials (Volumes 204-208), 4785-4788, 2012.

%e Triangle begins:

%e 1 | 1, 8, 9, 12, 18;

%e 2 | 1, 24, 25, 50;

%e 3 | 1, 32, 48, 49, 98;

%e 4 | 1;

%e 5 | 1, 64, 72, 121, 242;

%e 6 | 1, 81, 96, 162, 169, 338;

%e ...

%o (PARI) s(n) = {my(s=factor(n)[, 1], k=s[#s], f=Mod(k!, n)); while(f, f*=k++); k; }

%o f(n) = floor(n*exp(Euler)*log(log(n^2))+2.5*n/log(log(n^2)));

%o row(n) = {my(k, t, v=List([1])); while(4*n*(t++)>=2^t, forprime(p=2, if(t<3, t*n+1, sqrtnint(t*n, t-2)), for(m=1, f(1+s(p^(t*n))/(p-1)/p^(t-1)), if(eulerphi(k=m*p^t)==s(k^n), listput(v, k))))); Set(v); }

%Y Cf. A000010, A002034, A337029 (row lengths).

%K nonn,tabf

%O 1,2

%A _Jinyuan Wang_, Aug 05 2020