A362405 - OEIS

%I #12 Apr 20 2023 13:20:38

%S 1638,1848,3798,11448,16854,26910,35574,37248,57120,69678,69822,85848,

%T 94248,110526,208848,272214,305046,310248,335478,335479,368448,573048,

%U 580680,687240,1017126,1154270,1230606,1289358,1423248,1467414,1697808,1718880,1776750,1777248

%N Numbers k such that k, k+1 and k+2 are all in A362401.

%C Up to 10^8, k = 335478 is the only number k such that k, k+1, k+2 and k+3 are all in A362401. Are there any other such terms?

%H Amiram Eldar, <a href="/A362405/b362405.txt">Table of n, a(n) for n = 1..132</a> (terms below 10^8)

%e 1638 is a term since 1638, 1639 and 1640 are all in the range of A162296: A162296(1053) = 1638, A162296(576) = 1639 and A162296(1636) = 1640.

%t s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1)]; s[1] = 0; seq[max_] := Module[{v = Select[Union[Array[s, max]], 0 < # <= max &], w, i, j}, i = Position[Differences[v], 1] // Flatten; w = v[[i]]; j = Position[Differences[w], 1] // Flatten; w[[j]]]; seq[10^6]

%o (PARI) s(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((p^(e + 1) - 1)/(p - 1))) -  prod(i = 1, #f~, f[i, 1] + 1);}

%o lista(kmax) = {my(v = select(x -> (x < kmax), Set(vector(kmax, k, s(k))))); for(k=1, #v-2, if(v[k+1] - v[k] == 1 && v[k+2] - v[k+1] == 1, print1(v[k], ", ")));}

%Y Subsequence of A362401 and A362404.

%Y Cf. A162296.

%K nonn

%O 1,1

%A _Amiram Eldar_, Apr 18 2023