A262890 -id:A262890 - cp's OEIS frontend

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).

0, 2, 6, 12, 18, 22, 30, 34, 42, 46, 54, 58, 62, 70, 78, 90, 94, 102, 106, 114, 118, 121, 125, 129, 144, 152, 162, 166, 174, 182, 190, 194, 210, 214, 222, 230, 236, 242, 250, 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

Offset: 0

Max Alekseyev, Jul 09 2015

Equivalently, satisfies the property: A000005(a(n)) = a(n)-a(n-1). The first differences a(n)-a(n-1) are given in A259935.
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.
If the current definition does not uniquely define the sequence, the "lexicographically earliest" condition may be added to make the sequence well-defined.
From Vladimir Shevelev, Jul 21 2015: (Start)
If a(k), a(k+1), a(k+2) is an arithmetic progression, then a(k+1) is in A175304.
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).
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)
From Antti Karttunen, Nov 27 2015: (Start)
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.
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.
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).
(End)

Robert Israel, Table of n, a(n) for n = 0..64800
M. Alekseyev et al., Apparently unique infinite sequences related to the sum of divisors, Discussion in SeqFan mailing list, 2015.
Michael De Vlieger, Poster illustrating A259934 and A263267
V. S. Guba et al., Sequence and divisors, 2015. (in Russian)

Cf. A000005, A049820, A060990, A259935 (first differences).
Topmost row of A263255. Cf. also irregular tables A263267 & A263265 and array A262898.
Cf. A262693 (characteristic function).
Cf. A155043, A262694, A262904 (left inverses).
Cf. A262514 (squares present), A263276 (their positions), A263277.
Cf. A262517 (odd terms).
Cf. A262509, A262510, A262897 (other subsequences).
Cf. also A175304, A260257, A262680.
Cf. A261089, A261103, A262503, A262506, A262516, A263279, A263280, A263085, A263086, A263253, A263257, A263278.
Cf. also A262679, A262896 (see the C++ program there).
No common terms with A045765 or A262903.
Positions of zeros in A262522, A262695, A262696, A262697, A263254.
Various metrics concerning finite side-trees: A262888, A262889, A262890.
Cf. also A262891, A262892 and A262895 (cf. its graph).
Cf. A260084, A260124 (variants).
Cf. also A179016 (a similar "beanstalk trunk sequence" but with more tractable and regular behavior).

N:= 10^4: # to get "guaranteed unique" terms <= N
S:= Vector(N,datatype=integer[1]):
for n from N+1 to 2*N do
  k:= n - numtheory:-tau(n);
  if k <= N then S[k]:= S[k]+1; B[k]:= n; fi;
od:
for n from N to 3 by -1 do
  if S[n] >= 1 then
    k:= n - numtheory:-tau(n);
    S[k]:= S[k]+1; B[k]:= n;
  fi
od:
A[0]:= 0: A[1]:= 2:
for n from 2 do
  b:= B[A[n-1]];
  if b > N or S[b] > 1 then break fi;
  A[n]:= b;
od:
seq(A[i],i=0..n-1); # Robert Israel, Jul 09 2015

NN = 10^4; (* to get "guaranteed unique" terms <= NN *)
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 *)

From Antti Karttunen, Nov 27 2015: (Start)
Other identities and observations. For all n >= 0:
a(n) = A262679(A262896(n)).
A155043(a(n)) = A262694(a(n)) = A262904(a(n)) = n.
A261089(n) <= a(n) <= A262503(n). [A261103 and A262506 give the distances of a(n) to these bounds.]
(End)

A262697 a(n)=0 if n is in A259934, otherwise number of nodes (including leaves and the node n itself) in that finite subtree whose root is n and edge-relation is defined by A049820(child) = parent.

Original entry on oeis.org

0, 6, 0, 3, 2, 2, 0, 1, 1, 38, 3, 37, 0, 1, 2, 33, 2, 32, 0, 1, 1, 30, 0, 29, 1, 1, 3, 28, 1, 26, 0, 24, 2, 1, 0, 23, 1, 1, 16, 21, 1, 2, 0, 1, 2, 18, 0, 17, 13, 1, 1, 16, 1, 14, 0, 1, 1, 13, 0, 10, 11, 9, 0, 1, 1, 8, 1, 1, 1, 6, 0, 4, 10, 3, 1, 1, 23, 2, 0, 1, 2, 22, 4, 20, 9, 1, 3, 19, 1, 5, 0, 13, 2, 4, 0, 11, 8, 10, 1, 3, 1, 2, 0, 1, 6, 9, 0, 8, 1, 1, 2, 6, 1, 1, 0, 3, 1, 1, 0, 2, 5, 0, 12, 1

Offset: 0

Antti Karttunen, Oct 04 2015

nonn

			For n=1, its transitive closure (as defined by edge-relation A049820(child) = parent) is the union of {1} itself together with all its descendants, together {1, 3, 4, 5, 7, 8}. We see that there are no other nodes in a subtree whose root is 1, because A049820(3) = 3 - d(3) = 1, A049820(4) = 1, A049820(5) = 3, A049820(7) = 5, A049820(8) = 4 and both 7 and 8 are terms of A045765. Thus a(1) = 6.
For n=9, its transitive closure is {9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 27, 29, 31, 33, 35, 36, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 79}, containing 38 terms, thus a(9) = 38.

Antti Karttunen, Table of n, a(n) for n = 0..17724

Cf. A000005, A045765, A049820, A060990, A082284, A259934, A262686, A262693.
Cf. A262679, A262522, A262695, A262696, A262890.
Cf. also A213727, A227643, A255327.

If A262693(n) = 1 [when n is in A259934],
  then a(n) = 0,
otherwise, if A060990(n) = 0 [when n is one of the leaves, A045765],
  then a(n) = 1,
otherwise:
  a(n) = 1 + Sum_{k = A082284(n) .. A262686(n)} [A049820(k) = n] * a(k).
(In the last clause [ ] stands for Iverson bracket, giving as its result 1 only when A049820(k) = n, and 0 otherwise).

A262896 If n is in A262892, a(n) = A259934(n), otherwise the largest term in A045765 from which A259934(n) can be reached by iterating A049820, without visiting any other (larger) term of A259934.

Original entry on oeis.org

8, 2, 79, 12, 18, 40, 30, 140, 42, 52, 54, 66, 68, 123, 98, 90, 94, 116, 106, 126, 164, 121, 369, 133, 156, 168, 180, 184, 280, 229, 190, 194, 210, 218, 252, 246, 236, 242, 272, 254, 312, 324, 300, 364, 298, 302, 372, 356, 334, 342, 346, 354, 439, 366, 374, 390, 672, 414, 410, 438, 426, 460, 442, 452, 470, 466, 564, 496, 494, 524, 627, 530, 546, 558, 562, 566, 574, 592, 859, 660, 606, 642, 708, 650

Offset: 0

Antti Karttunen, Oct 06 2015

nonn

a(n) is the largest leaf-node among the finite subtrees branching from node n in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent, and A259934(n) itself if it is one of the nonbranching nodes (A262897).

Note that without (so far undetected) regularity in A262509, there is no a priori upper bound for the value of a(n), and for some n this might not even be finite, if it happens that contrary to its conjectured nature, A259934 is not the unique infinite component, but just the lexicographically earliest instance of multiple infinite branches of the tree. In that case we might consider this sequence to be well-defined only up to the least such node branching to multiple infinite components, or alternatively, we might mark the nonfinite values at those points with -1.

Antti Karttunen, Table of n, a(n) for n = 0..10000
Max Alekseyev & Antti Karttunen, Standalone C++-program for computing this sequence

Cf. A000005, A049820, A082284, A259934, A262509, A262522, A262679, A262686, A262890, A262892, A262897, A262904, A263081.

(define (A262896 n) (let ((t (A259934 n))) (let loop ((m t) (k (A262686 t))) (cond ((<= k t) m) ((= t (A049820 k)) (loop (max m (A262522 k)) (- k 1))) (else (loop m (- k 1)))))))

a(n) = max(A259934(n), Max_{k = A082284(A259934(n)) .. A262686(A259934(n))} [A049820(k) = A259934(n)] * A262522(k)).
(Here [ ] stands for Iverson bracket, giving as its result 1 only when A049820(k) = A259934(n), and 0 otherwise).
Other identities. For all n >= 0:
A262904(a(n)) = n. [A262904 works as a left inverse for this sequence.]
A259934(n) = A262679(a(n)).
For all n >= 1:
a(A262892(n)) = A259934(A262892(n)) = A262897(n).

A262892 Indices of nonbranching nodes in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent.

Original entry on oeis.org

1, 3, 4, 6, 8, 10, 15, 16, 18, 21, 30, 31, 32, 36, 37, 39, 44, 45, 49, 50, 51, 53, 54, 55, 58, 60, 62, 65, 68, 71, 72, 73, 74, 75, 76, 80, 83, 84, 90, 91, 93, 96, 109, 112, 117, 122, 123, 124, 126, 127, 131, 134, 135, 137, 141, 142, 144, 145, 147, 149, 152, 154, 155, 161, 162, 165, 166, 170, 178, 187, 189, 190, 191, 193, 195, 199, 201, 203, 205, 211, 212, 213, 219, 223, 225

Offset: 1

Antti Karttunen, Oct 04 2015

nonn

Positions of zeros in A262890. Positions of ones in A262891.

Sequence b(n) = A262508(n)-1 = 9235, 9236, 9237, 9246, 9247, 9329, 9330, 9352, 9355, ..., is a subsequence.

Antti Karttunen, Table of n, a(n) for n = 1..4085

Cf. A049820, A259934, A262890, A262891, A262896.
Cf. A262897 (nonbranching nodes themselves).
Cf. A262508.

Other identities. For all n >= 1:

A262897(n) = A259934(a(n)) = A262896(a(n)).

A262888 a(n) = total number of nodes in the finite subtrees branching "left" (to the "smaller side") from node n in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent.

Original entry on oeis.org

6, 0, 41, 0, 0, 5, 0, 16, 0, 2, 0, 0, 1, 24, 4, 0, 0, 0, 0, 0, 0, 0, 105, 2, 0, 0, 0, 3, 18, 7, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 13, 1, 0, 0, 0, 0, 6, 1, 0, 0, 0, 47, 0, 0, 0, 90, 0, 0, 5, 0, 0, 0, 1, 0, 0, 12, 0, 0, 3, 61, 0, 0, 0, 0, 0, 0, 1, 117, 7, 0, 2, 10, 0, 0, 1, 23, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 568, 0, 1, 1, 4, 0, 5, 9, 0, 0, 0, 0, 0, 8, 0, 1, 1, 0, 2, 10, 1, 1, 0

Offset: 0

Antti Karttunen, Oct 04 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8107

Cf. A000005, A049820, A082284, A259934, A262686, A262697, A262889, A262890, A262894.

Cf. also A255328.

(define (A262888 n) (let ((t (A259934 n))) (let loop ((s 0) (k (A259934 (+ 1 n)))) (cond ((<= k t) s) ((= t (A049820 k)) (loop (+ s (A262697 k)) (- k 1))) (else (loop s (- k 1)))))))

a(n) = sum_{k = A082284(A259934(n)) .. A259934(n+1)} [A049820(k) = A259934(n)] * A262697(k).
(Here [ ] stands for Iverson bracket, giving as its result 1 only when A049820(k) = A259934(n), and 0 otherwise).
Other identities. For all n >= 0:
A262890(n) = a(n) + A262889(n).

A262889 a(n) = total number of nodes in the finite subtrees branching "right" (to the "larger side") from node n in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 1, 13, 0, 0, 0, 1, 1, 2, 2, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 5, 0, 4, 0, 1, 7, 0, 0, 7, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 5, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 3, 0, 22, 1, 0, 1, 2, 0, 6, 0, 0, 0, 0, 0, 0

Offset: 0

Antti Karttunen, Oct 04 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8107

Cf. A000005, A049820, A082284, A259934, A262686, A262697, A262888, A262890, A262894.

Cf. also A255329.

(define (A262889 n) (let ((t (A259934 n)) (u (A259934 (+ 1 n)))) (let loop ((s 0) (k (A262686 t))) (cond ((<= k u) s) ((= t (A049820 k)) (loop (+ s (A262697 k)) (- k 1))) (else (loop s (- k 1)))))))

a(n) = sum_{k = A259934(n+1) .. A262686(A259934(n))} [A049820(k) = A259934(n)] * A262697(k).
(Here [ ] stands for Iverson bracket, giving as its result 1 only when A049820(k) = A259934(n), and 0 otherwise).
Other identities. For all n >= 0:
A262890(n) = A262888(n) + a(n).

cp's OEIS Frontend

A262893 Partial sums of A262890.

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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).

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A262697 a(n)=0 if n is in A259934, otherwise number of nodes (including leaves and the node n itself) in that finite subtree whose root is n and edge-relation is defined by A049820(child) = parent.

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A262896 If n is in A262892, a(n) = A259934(n), otherwise the largest term in A045765 from which A259934(n) can be reached by iterating A049820, without visiting any other (larger) term of A259934.

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A262892 Indices of nonbranching nodes in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent.

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A262888 a(n) = total number of nodes in the finite subtrees branching "left" (to the "smaller side") from node n in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent.

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A262889 a(n) = total number of nodes in the finite subtrees branching "right" (to the "larger side") from node n in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent.

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