cp's OEIS Frontend

A348093 Numbers k >= 1 such that there is no pair (x,y) such that x - d(x) = k or y + d(y) = k, where d = A000005 = number of divisors.

%I A348093 #23 Mar 04 2023 08:58:48
%S A348093 8,20,36,40,67,68,79,88,100,116,117,131,132,134,140,156,164,167,180,
%T A348093 185,196,204,228,244,252,268,276,284,300,308,312,321,324,341,348,370,
%U A348093 372,379,388,401,405,408,420,425,436,439,453,460,476,479
%N A348093 Numbers k >= 1 such that there is no pair (x,y) such that x - d(x) = k or y + d(y) = k, where d = A000005 = number of divisors.
%C A348093 Numbers k >= 1 such that A060990(k) + A036431(k) = 0.
%H A348093 Amiram Eldar, <a href="/A348093/b348093.txt">Table of n, a(n) for n = 1..10000</a>
%e A348093 k = 8 is a term: there are no x,y such that x - d(x) = 8, y + d(y) = 8.
%t A348093 With[{max = 480}, Complement[Range[max], Select[Union[Flatten[Table[n + DivisorSigma[0, n]*{-1, 1}, {n, 1, max + 2 + 2*Ceiling[Sqrt[2*max+4]]}]]], # <= max &]]] (* _Amiram Eldar_, Mar 04 2023 *)
%o A348093 (PARI) okp(k) = sum(i=1, k, i+numdiv(i) == k) == 0;
%o A348093 okm(k) = sum(i=1, 2*k+2, i-numdiv(i) == k) == 0;
%o A348093 isok(k) = okp(k) && okm(k); \\ _Michel Marcus_, Oct 01 2021
%Y A348093 Cf. A000005, A036431, A049820, A060990, A062249.
%Y A348093 Intersection of A036434 and A045765.
%K A348093 nonn
%O A348093 1,1
%A A348093 _Ctibor O. Zizka_, Sep 29 2021