A327250 - OEIS

%I #18 Apr 06 2023 06:35:58

%S 3,80,175,272,492,860,943,6556,6867,7104,7215,14672,17459,21804,22672,

%T 24435,24476,26128,30899,34595,39215,41327,45548,49468,56563,57075,

%U 63440,63744,67123,72556,78524,87615,90243,104111,109939,113283,113296,115344,121539,131651

%N Numbers k such that s(k) = s(k+1), where s(k) is A059975.

%C Madeleine Farris named these numbers "Euler-totient Ruth-Aaron numbers" (in analogy to the Ruth-Aaron numbers, A039752). She proved that the number of terms <= x is O(x*(log(log(x))^4)/(log(x))^2) and that the sum of their reciprocals is bounded.

%H Amiram Eldar, <a href="/A327250/b327250.txt">Table of n, a(n) for n = 1..10000</a>

%H Madeleine Farris, <a href="https://repository.wellesley.edu/thesiscollection/649/">Ruth-Aaron Numbers: An Exploration in Analytic Number Theory</a> (thesis), Wellesley College, 2019.

%e 3 is in the sequence since A059975(3) = A059975(4) = 2.

%t f[p_,e_] := e * (p-1); a[n_] := Plus @@ (f @@@ FactorInteger[n]); aQ[n_] := a[n] == a[n+1]; Select[Range[10^5], aQ]

%o (PARI) s(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i, 2] * (f[i, 1] - 1));}

%o lista(kmax) = {my(s1 = s(1), s2); for(k=2, kmax, s2 = s(k); if(s1 == s2, print1(k-1, ", ")); s1 = s2);} \\ _Amiram Eldar_, Apr 06 2023

%Y Cf. A059975, A006145, A039752.

%K nonn

%O 1,1

%A _Amiram Eldar_, Sep 15 2019