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A045765 k - d(k) never takes these values, where d(k) = A000005(k).

%I A045765 #36 Jul 26 2025 10:59:50
%S A045765 7,8,13,19,20,24,25,28,33,36,37,40,43,49,50,52,55,56,63,64,66,67,68,
%T A045765 74,75,79,85,88,98,100,103,108,109,112,113,116,117,123,124,126,131,
%U A045765 132,133,134,136,140,143,145,150,153,156,159,160,163,164,167,168
%N A045765 k - d(k) never takes these values, where d(k) = A000005(k).
%C A045765 Complement of A236562. - _Jaroslav Krizek_, Feb 09 2014
%C A045765 Positions of zeros in A060990, leaf-nodes in the tree generated by edge-relation A049820(child) = parent. - _Antti Karttunen_, Oct 06 2015
%C A045765 Since A000005(x) <= 1 + x/2, k is in the sequence if there are no x <= 2*(k+1) with k = x - d(x). - _Robert Israel_, Oct 12 2015
%C A045765 This can be improved as: k is in the sequence if there are no x <= k + A002183(2+A261100(k)) with k = x - d(x). Cf. also A070319, A262686. - _Antti Karttunen_, Oct 12 2015
%C A045765 Luca (2005) proved that this seqeunce is infinite. - _Amiram Eldar_, Jul 26 2025
%H A045765 Antti Karttunen, <a href="/A045765/b045765.txt">Table of n, a(n) for n = 1..10000</a>
%H A045765 Florian Luca, <a href="https://doi.org/10.1007/s10474-005-0010-8">On numbers not of the form n-omega(n)</a>, Acta Mathematica Hungarica, Vol. 106, No. 1 (2005), pp. 117-135.
%p A045765 N:= 1000: # to get all terms <= N
%p A045765 sort(convert({$1..N} minus {seq(x - numtheory:-tau(x), x=1..2*(1+N))},list)); # _Robert Israel_, Oct 12 2015
%t A045765 lim = 10000; Take[Complement[Range@ lim, Sort@ DeleteDuplicates@ Table[n - DivisorSigma[0, n], {n, lim}]], 57] (* _Michael De Vlieger_, Oct 13 2015 *)
%o A045765 (PARI)
%o A045765 allocatemem((2^31)+(2^30));
%o A045765 uplim = 36756720 + 640; \\ = A002182(53) + A002183(53).
%o A045765 v060990 = vector(uplim);
%o A045765 for(n=3, uplim, v060990[n-numdiv(n)]++);
%o A045765 A060990 = n -> if(!n,2,v060990[n]);
%o A045765 uplim2 = 36756720;
%o A045765 n=0; k=1; while(n <= uplim2, if(0==A060990(n), write("b045765_big.txt", k, " ", n); k++); n++;);
%o A045765 \\ _Antti Karttunen_, Oct 09 2015
%o A045765 (Scheme)
%o A045765 (define A045765 (ZERO-POS 1 1 A060990))
%o A045765 ;; Using also IntSeq-library of _Antti Karttunen_, Oct 06 2015
%Y A045765 Top row of A262898.
%Y A045765 Cf. A000005, A002183, A049820, A060990, A070319, A236562, A236565, A261100, A262511, A262686, A262901, A262902, A262903, A262909, A263081.
%Y A045765 Cf. A263091 (primes in this sequence), A263095 (squares).
%Y A045765 Cf. A259934 (gives the infinite trunk of the same tree, conjectured to be unique).
%K A045765 nonn
%O A045765 1,1
%A A045765 _David W. Wilson_