A327250 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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.