A045765 - OEIS

%I #36 Jul 26 2025 10:59:50

%S 7,8,13,19,20,24,25,28,33,36,37,40,43,49,50,52,55,56,63,64,66,67,68,

%T 74,75,79,85,88,98,100,103,108,109,112,113,116,117,123,124,126,131,

%U 132,133,134,136,140,143,145,150,153,156,159,160,163,164,167,168

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

%C Complement of A236562. - _Jaroslav Krizek_, Feb 09 2014

%C Positions of zeros in A060990, leaf-nodes in the tree generated by edge-relation A049820(child) = parent. - _Antti Karttunen_, Oct 06 2015

%C 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 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 Luca (2005) proved that this seqeunce is infinite. - _Amiram Eldar_, Jul 26 2025

%H Antti Karttunen, <a href="/A045765/b045765.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 N:= 1000: # to get all terms <= N

%p sort(convert({$1..N} minus {seq(x - numtheory:-tau(x), x=1..2*(1+N))},list)); # _Robert Israel_, Oct 12 2015

%t lim = 10000; Take[Complement[Range@ lim, Sort@ DeleteDuplicates@ Table[n - DivisorSigma[0, n], {n, lim}]], 57] (* _Michael De Vlieger_, Oct 13 2015 *)

%o (PARI)

%o allocatemem((2^31)+(2^30));

%o uplim = 36756720 + 640; \\ = A002182(53) + A002183(53).

%o v060990 = vector(uplim);

%o for(n=3, uplim, v060990[n-numdiv(n)]++);

%o A060990 = n -> if(!n,2,v060990[n]);

%o uplim2 = 36756720;

%o n=0; k=1; while(n <= uplim2, if(0==A060990(n), write("b045765_big.txt", k, " ", n); k++); n++;);

%o \\ _Antti Karttunen_, Oct 09 2015

%o (Scheme)

%o (define A045765 (ZERO-POS 1 1 A060990))

%o ;; Using also IntSeq-library of _Antti Karttunen_, Oct 06 2015

%Y Top row of A262898.

%Y Cf. A000005, A002183, A049820, A060990, A070319, A236562, A236565, A261100, A262511, A262686, A262901, A262902, A262903, A262909, A263081.

%Y Cf. A263091 (primes in this sequence), A263095 (squares).

%Y Cf. A259934 (gives the infinite trunk of the same tree, conjectured to be unique).

%K nonn

%O 1,1

%A _David W. Wilson_