A327250 Numbers k such that s(k) = s(k+1), where s(k) is A059975.

3, 80, 175, 272, 492, 860, 943, 6556, 6867, 7104, 7215, 14672, 17459, 21804, 22672, 24435, 24476, 26128, 30899, 34595, 39215, 41327, 45548, 49468, 56563, 57075, 63440, 63744, 67123, 72556, 78524, 87615, 90243, 104111, 109939, 113283, 113296, 115344, 121539, 131651

Offset: 1

Author

Amiram Eldar, Sep 15 2019

Keywords

• nonn

Comments

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.

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Madeleine Farris, Ruth-Aaron Numbers: An Exploration in Analytic Number Theory (thesis), Wellesley College, 2019.

Crossrefs

Cf. A059975, A006145, A039752.

Programs

Mathematica

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

PARI

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