A336711 Irregular triangle where the n-th row lists the positive integers k such that phi(k) = A002034(k^n).

1, 8, 9, 12, 18, 1, 24, 25, 50, 1, 32, 48, 49, 98, 1, 1, 64, 72, 121, 242, 1, 81, 96, 162, 169, 338, 1, 80, 1, 125, 250, 289, 578, 1, 128, 361, 722, 1, 192, 1, 135, 144, 216, 270, 529, 1058, 1, 160, 1, 324, 1, 175, 225, 350, 450, 841, 1682, 1, 961, 1922

Offset

1,2

Comments

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).

Links

Jinyuan Wang, Rows n = 1..359 of triangle, flattened

Bin Chen, An Equation Involving the Smarandache Function, Applied Mechanics and Materials (Volumes 204-208), 4785-4788, 2012.

Example

Triangle begins:
1 | 1, 8, 9, 12, 18;
2 | 1, 24, 25, 50;
3 | 1, 32, 48, 49, 98;
4 | 1;
5 | 1, 64, 72, 121, 242;
6 | 1, 81, 96, 162, 169, 338;
...

Prog

(PARI) s(n) = {my(s=factor(n)[, 1], k=s[#s], f=Mod(k!, n)); while(f, f*=k++); k; }
f(n) = floor(n*exp(Euler)*log(log(n^2))+2.5*n/log(log(n^2)));
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); }

Crossrefs

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

Sequence in context: A080756 A304661 A189833 * A063080 A167131 A109079

Adjacent sequences: A336708 A336709 A336710 * A336712 A336713 A336714

Keyword

nonn,tabf

Author

Jinyuan Wang, Aug 05 2020

Status

approved