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A259934 Infinite sequence starting with a(0)=0 such that A049820(a(k)) = a(k-1) for all k>=1, where A049820(n) = n - (number of divisors of n).

%I A259934 #94 Feb 17 2025 12:13:15
%S A259934 0,2,6,12,18,22,30,34,42,46,54,58,62,70,78,90,94,102,106,114,118,121,
%T A259934 125,129,144,152,162,166,174,182,190,194,210,214,222,230,236,242,250,
%U A259934 254,270,274,282,294,298,302,310,314,330,342,346,354,358,366,374,390,394,402,410,418,426,434,442,446,462,466,474,486,494,510,522,530,546,558,562,566,574,582,590
%N A259934 Infinite sequence starting with a(0)=0 such that A049820(a(k)) = a(k-1) for all k>=1, where A049820(n) = n - (number of divisors of n).
%C A259934 Equivalently, satisfies the property: A000005(a(n)) = a(n)-a(n-1). The first differences a(n)-a(n-1) are given in A259935.
%C A259934 V. S. Guba (2015) proved that such an infinite sequence exists. Numerical evidence suggests that it may also be unique -- is it? All terms below 10^10 are defined uniquely.
%C A259934 If the current definition does not uniquely define the sequence, the "lexicographically earliest" condition may be added to make the sequence well-defined.
%C A259934 From _Vladimir Shevelev_, Jul 21 2015: (Start)
%C A259934 If a(k), a(k+1), a(k+2) is an arithmetic progression, then a(k+1) is in A175304.
%C A259934 Indeed, by the definition of this sequence, a(n)-a(n-1) = d(a(n)), for all n>=1, where d(n) = A000005(n). Hence, have a(k+1) - a(k) = a(k+2) - a(k+1) = d(a(k+1)) = d(a(k+2)). So a(k+1) + d(a(k+2)) = a(k+2) and a(k+1) + d(a(k+1)) = a(k+2).
%C A259934 Therefore, d(a(k+1) + d(a(k+1))) = d(a(k+2))= d(a(k+1)), i.e., a(k+1) is in A175304. Thus, if there are infinitely many pairs of the same consecutive terms of A259935, then A175304 is infinite (see there my conjecture). (End)
%C A259934 From _Antti Karttunen_, Nov 27 2015: (Start)
%C A259934 If multiple apparently infinite branches would occur at some point of computing, then even if the "lexicographically earliest" condition were then added to the definition, it would not help us much (when computing the sequence), as we would still not know which of the said branches were truly infinite. [See also _Max Alekseyev_'s latter Jul 9 2015 posting on SeqFan-list, where he notes the same thing.] Note that many of the derived sequences tacitly assume that the uniqueness-conjecture is true. See also comments at A262693 and A262896.
%C A259934 One sufficient (but not a necessary) condition for the uniqueness of this sequence is that the sequence A262509 has infinite number of terms. Please see further comments there.
%C A259934 The graph of sequence exhibits two markedly different slopes, depending on whether it is on the "fast lane" of A049820 (even numbers) or the "slow lane" [odd numbers, for example when traversing the 1356 odd terms from 123871 to 113569 at range a(9859) .. a(8504)]. See A263086/A263085 for the "average cumulative speed difference" between the lanes. In general, slow and fast lane stay separate, except when they terminate into one of the squares (A262514) that work as "exchange ramps", forcing the parity (and thus the speed) to change. In average, the odd squares are slightly better than the even squares in attracting lanes going towards smaller numbers (compare A263253 to A263252). The cumulative effect of this bias is that the odd terms are much rarer in this sequence than the even terms (compare A263278 to A262516).
%C A259934 (End)
%H A259934 Robert Israel, <a href="/A259934/b259934.txt">Table of n, a(n) for n = 0..64800</a>
%H A259934 M. Alekseyev et al., <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2015-July/thread.html#15037">Apparently unique infinite sequences related to the sum of divisors</a>, Discussion in SeqFan mailing list, 2015.
%H A259934 Michael De Vlieger, <a href="https://oeis.org/A263267/a263267_3.pdf">Poster illustrating A259934 and A263267</a>
%H A259934 V. S. Guba et al., <a href="https://web.archive.org/web/20231210151810/http://math.hashcode.ru/questions/66579/%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F-%D1%87%D0%B8%D1%81%D0%B5%D0%BB-%D0%BF%D0%BE%D1%81%D0%BB%D0%B5%D0%B4%D0%BE%D0%B2%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D1%8C-%D0%B8-%D0%B4%D0%B5%D0%BB%D0%B8%D1%82%D0%B5%D0%BB%D0%B8">Sequence and divisors</a>, 2015. (in Russian)
%F A259934 From _Antti Karttunen_, Nov 27 2015: (Start)
%F A259934 Other identities and observations. For all n >= 0:
%F A259934 a(n) = A262679(A262896(n)).
%F A259934 A155043(a(n)) = A262694(a(n)) = A262904(a(n)) = n.
%F A259934 A261089(n) <= a(n) <= A262503(n). [A261103 and A262506 give the distances of a(n) to these bounds.]
%F A259934 (End)
%p A259934 N:= 10^4: # to get "guaranteed unique" terms <= N
%p A259934 S:= Vector(N,datatype=integer[1]):
%p A259934 for n from N+1 to 2*N do
%p A259934   k:= n - numtheory:-tau(n);
%p A259934   if k <= N then S[k]:= S[k]+1; B[k]:= n; fi;
%p A259934 od:
%p A259934 for n from N to 3 by -1 do
%p A259934   if S[n] >= 1 then
%p A259934     k:= n - numtheory:-tau(n);
%p A259934     S[k]:= S[k]+1; B[k]:= n;
%p A259934   fi
%p A259934 od:
%p A259934 A[0]:= 0: A[1]:= 2:
%p A259934 for n from 2 do
%p A259934   b:= B[A[n-1]];
%p A259934   if b > N or S[b] > 1 then break fi;
%p A259934   A[n]:= b;
%p A259934 od:
%p A259934 seq(A[i],i=0..n-1); # _Robert Israel_, Jul 09 2015
%t A259934 NN = 10^4; (* to get "guaranteed unique" terms <= NN *)
%t A259934 Clear[A, B, S]; S[_]=0; For[n = NN+1, n <= 2*NN, n++, k = n-DivisorSigma[0, n]; If[k <= NN, S[k] = S[k]+1; B[k]=n]]; For[n=NN, n >= 3, n--, If[S[n] >= 1 , k = n-DivisorSigma[0, n]; S[k] = S[k]+1; B[k]=n]]; A[0]=0; A[1]=2; For[n=2, True, n++, b = B[A[n-1]]; If[b>NN || S[b]>1, Break[]]; A[n]=b]; Table[A[i], {i, 0, n-1}] (* _Jean-François Alcover_, Jul 22 2015, after _Robert Israel_ *)
%Y A259934 Cf. A000005, A049820, A060990, A259935 (first differences).
%Y A259934 Topmost row of A263255. Cf. also irregular tables A263267 & A263265 and array A262898.
%Y A259934 Cf. A262693 (characteristic function).
%Y A259934 Cf. A155043, A262694, A262904 (left inverses).
%Y A259934 Cf. A262514 (squares present), A263276 (their positions), A263277.
%Y A259934 Cf. A262517 (odd terms).
%Y A259934 Cf. A262509, A262510, A262897 (other subsequences).
%Y A259934 Cf. also A175304, A260257, A262680.
%Y A259934 Cf. A261089, A261103, A262503, A262506, A262516, A263279, A263280, A263085, A263086, A263253, A263257, A263278.
%Y A259934 Cf. also A262679, A262896 (see the C++ program there).
%Y A259934 No common terms with A045765 or A262903.
%Y A259934 Positions of zeros in A262522, A262695, A262696, A262697, A263254.
%Y A259934 Various metrics concerning finite side-trees: A262888, A262889, A262890.
%Y A259934 Cf. also A262891, A262892 and A262895 (cf. its graph).
%Y A259934 Cf. A260084, A260124 (variants).
%Y A259934 Cf. also A179016 (a similar "beanstalk trunk sequence" but with more tractable and regular behavior).
%K A259934 nonn,nice
%O A259934 0,2
%A A259934 _Max Alekseyev_, Jul 09 2015