A259935 -id:A259935 - cp's OEIS frontend

A260257 Numbers n such that A259935(n) = A259935(n-1).

4, 12, 14, 23, 29, 30, 35, 37, 45, 54, 58, 59, 60, 61, 62, 75, 77, 78, 79, 80, 87, 88, 92, 98, 115, 116, 117, 118, 119, 122, 123, 125, 127, 144, 147, 154, 155, 158, 159, 160, 163, 171, 175, 179, 183, 184, 190, 192, 206, 207, 217, 221, 222, 225, 228, 232, 234, 236, 238, 242, 249, 255, 256, 257, 258, 260, 268, 269, 270, 271, 272, 278, 281, 284, 288, 294, 301

Offset: 1

Vladimir Shevelev, Jul 21 2015

nonn

For every n, A259934(a(n)-2), A259934(a(n)-1), A259934(a(n)) is arithmetic progression, such that A259934(a(n)-1) is in A175304 (see comment in A259934).

Cf. A175304, A259934, A259935.

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).

Original entry on oeis.org

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)

A262694 Least monotonic left inverse for A259934.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20

Offset: 0

Antti Karttunen, Oct 04 2015

nonn

Each n occurs A259935(n+1) times.

Antti Karttunen, Table of n, a(n) for n = 0..65538

Cf. A259934, A259935.

One less than the partial sums of A262693.

Other identities. For all n >= 0:

a(A259934(n)) = n.

A278232 a(n) = Least number with the prime signature of A259934(n).

Original entry on oeis.org

2, 6, 12, 12, 6, 30, 6, 30, 6, 24, 6, 6, 30, 30, 60, 6, 30, 6, 30, 6, 4, 8, 6, 144, 24, 48, 6, 30, 30, 30, 6, 210, 6, 30, 30, 12, 12, 24, 6, 120, 6, 30, 60, 6, 6, 30, 6, 210, 60, 6, 30, 6, 30, 30, 210, 6, 30, 30, 30, 30, 30, 30, 6, 210, 6, 30, 96, 30, 210, 60, 30, 210, 60, 6, 6, 30, 30, 30, 30, 30, 420, 30, 60, 30, 6, 30, 30, 24, 120, 30, 12, 12, 30, 6, 30, 6

Offset: 1

Antti Karttunen, Nov 22 2016

nonn

Terms and the b-file computed with the help of b-file provided for A259934 by Robert Israel.

Antti Karttunen, Table of n, a(n) for n = 1..64800

Cf. A046523, A259934, A259935.
Cf. A263276 (positions of squares).
Cf. also A278262.

(define (A278232 n) (A046523 (A259934 n)))

a(n) = A046523(A259934(n)).

A260084 Infinite sequence starting with a(0)=0 such that A(a(k)) = a(k-1) for all k>=1, where A(n) = n - A037445(n).

Original entry on oeis.org

0, 2, 6, 10, 14, 18, 22, 30, 34, 42, 46, 54, 58, 62, 70, 78, 82, 90, 94, 102, 106, 114, 118, 122, 130, 138, 142, 146, 154, 158, 162, 166, 174, 182, 190, 194, 210, 214, 222, 230, 238, 242, 250, 254, 270, 274, 278, 286, 294, 298, 302, 310, 314, 330, 334, 342, 346, 354, 358, 366, 374, 390, 394, 402, 410, 418, 426, 434, 442

Offset: 0

Vladimir Shevelev, Jul 15 2015

nonn

The first infinitary analog (see also A260124) of A259934 (see comment there). Using Guba's method (2015) one can prove that such an infinite sequence exists.
All the first differences are powers of 2 (A260085). The infinitary case is interesting because here we have at least two analogs of sequences A259934, A259935 (respectively A260084, A260124 and A260085, A260123).
It is a corollary of the fact that all terms of A037445, except for n=1, are even (powers of 2). Therefore, in the analogs of A259934 we can begin with not only 0,2 (as in this sequence), but also with 0,1 (as in A260124). Then this sequence contains only the even terms, while A260124 - only the odd ones.
A generalization. For an even m, the multiplication of A260124 by 2^m and 2^(m+1) gives two infinite solutions of the system of equations for integer x_n, n>=1: A037445(x_1 + ... + x_n) = x_n/2^A005187(m), n>=1. In particular, for m=0, we obtain A260124 and A260084.

Cf. A037445, A259934, A259935, A260085, A260124.

a(n) = 2 * A260124(n).

A260123 The second infinite sequence of positive integers such that a(n) = A037445(a(1)+a(2)+...+a(n)) for all n>=1 (see also A260085).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 2, 4, 4, 2, 4, 2, 4, 2, 4, 2, 2, 4, 4, 2, 2, 4, 2, 2, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 2, 8, 2, 2, 4, 4, 2, 2, 4, 2, 8, 2, 4, 2, 4, 2, 4, 4, 8, 2, 4, 4, 4, 4, 4, 4, 2, 8, 2, 4, 2, 4, 4, 8, 2, 4, 2, 4, 2, 2, 4, 4, 2, 2, 4, 2, 8, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 2, 8, 2, 2, 8

Offset: 1

Vladimir Shevelev, Jul 17 2015

nonn

The second after A260085 infinitary analog of A259935 (see comment there). Using Guba's method (2015) one can prove that such an infinite sequence exists.

All terms are powers of 2.

Cf. A037445, A259935, A260085.

A286540 a(n) = gcd(A259934(n), A259934(n-1)) = A009191(A259934(n)).

Original entry on oeis.org

2, 2, 6, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 1, 1, 1, 3, 8, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 6, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 8, 8, 24, 18, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 6

Offset: 1

Antti Karttunen, May 21 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..64800 (computed from the b-file of A259934 provided by Robert Israel)

Cf. A009191, A259934, A259935.

(define (A286540 n) (gcd (A259934 n) (A259934 (- n 1))))
(define (A286540 n) (gcd (A259934 n) (A259935 n)))

a(n) = gcd(A259934(n), A259934(n-1)) = gcd(A259934(n), A259935(n)).

a(n) = A009191(A259934(n)).

A260085 Infinite sequence of positive integers such that a(n) = A037445(a(1)+a(2)+...+a(n)) for all n>=1.

Original entry on oeis.org

2, 4, 4, 4, 4, 4, 8, 4, 8, 4, 8, 4, 4, 8, 8, 4, 8, 4, 8, 4, 8, 4, 4, 8, 8, 4, 4, 8, 4, 4, 4, 8, 8, 8, 4, 16, 4, 8, 8, 8, 4, 8, 4, 16, 4, 4, 8, 8, 4, 4, 8, 4, 16, 4, 8, 4, 8, 4, 8, 8, 16, 4, 8, 8, 8, 8, 8, 8, 4, 16, 4, 8, 4, 8, 8, 16, 4, 8, 4, 8, 4, 4, 8, 8, 4, 4, 8, 4, 16, 8, 8, 4, 16, 4, 8, 8, 8, 4, 8, 4, 16, 4, 4, 16

Offset: 1

Vladimir Shevelev, Jul 15 2015

nonn

The first infinitary analog (see also A260123) of A259935 (see comment there). Using Guba's method (2015) one can prove that such an infinite sequence exists.

All terms are powers of 2.

Cf. A037445, A259935, A260084, A260123.

a(n) = 2 * A260123(n).

cp's OEIS Frontend

A260257 Numbers n such that A259935(n) = A259935(n-1).

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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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A262694 Least monotonic left inverse for A259934.

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A278232 a(n) = Least number with the prime signature of A259934(n).

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A260084 Infinite sequence starting with a(0)=0 such that A(a(k)) = a(k-1) for all k>=1, where A(n) = n - A037445(n).

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A260123 The second infinite sequence of positive integers such that a(n) = A037445(a(1)+a(2)+...+a(n)) for all n>=1 (see also A260085).

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A286540 a(n) = gcd(A259934(n), A259934(n-1)) = A009191(A259934(n)).

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A260085 Infinite sequence of positive integers such that a(n) = A037445(a(1)+a(2)+...+a(n)) for all n>=1.

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