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

1,1

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.

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.

Example

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

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]

Prog

(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

Crossrefs

Cf. A059975, A006145, A039752.

Sequence in context: A265592 A105065 A011187 * A197248 A183265 A233124

Adjacent sequences: A327247 A327248 A327249 * A327251 A327252 A327253

Keyword

nonn

Author

Amiram Eldar, Sep 15 2019

Status

approved