A243700 -id:A243700 - cp's OEIS frontend

A244010 Partial sums of A243700.

1, 4, 6, 11, 20, 27, 35, 48, 63, 74, 88, 104, 130, 154, 195, 224, 242, 270, 290, 320, 342, 374, 399, 432, 475, 520, 551, 588, 638, 690, 744, 800, 858, 893, 980, 1018, 1073, 1140, 1180, 1240, 1312, 1356, 1419, 1496, 1575, 1622, 1692, 1741, 1862, 1950, 2003, 2132, 2226, 2322, 2420, 2520, 2579

Offset: 1

N. J. A. Sloane, Jun 17 2014

nonn

Max Alekseyev, Table of n, a(n) for n = 1..100000

Cf. A243700, A244011.

A244011 The quotient when A243700(n) is calculated.

Original entry on oeis.org

1, 2, 2, 4, 7, 5, 6, 10, 13, 8, 11, 14, 20, 18, 32, 22, 15, 21, 16, 23, 17, 25, 19, 26, 33, 35, 24, 29, 39, 41, 43, 45, 46, 28, 68, 30, 44, 53, 31, 48, 59, 34, 51, 62, 63, 36, 56, 38, 95, 69, 42, 101, 74, 75, 77, 78, 47, 71, 82, 84, 50, 126, 90, 52, 80, 55, 81, 151, 57, 86, 99, 161, 60, 93, 61, 94

Offset: 1

N. J. A. Sloane, Jun 17 2014

nonn

Max Alekseyev, Table of n, a(n) for n = 1..30532

Cf. A243700, A244010.

a(n) = A244010(A243700(n))/A243700(n).

A243864 Positive integers that do not appear in A243700.

Original entry on oeis.org

4, 6, 10, 12, 17, 19, 21, 23, 27, 34, 36, 39, 42, 46, 48, 51, 57, 61, 64, 66, 69, 73, 75, 78, 81, 83, 85, 90, 92, 95, 99, 104, 108, 111, 114, 116, 119, 124, 126, 133, 135, 137, 140, 145, 148, 150, 152, 157, 160, 163, 165, 168, 170, 173, 176, 178, 186, 191, 195, 198, 200, 204, 206, 209, 212, 216, 221, 223, 225, 230, 237, 239

Offset: 1

Max Alekseyev, Jun 13 2014

nonn

A positive integer m belongs to this sequence iff A243700(1) + A243700(2) + ... + A243700(m) is not 0 modulo m.

Based on an analysis by John Pezzullo of the b-file of 30542 terms it appears that a(n) is growing like c*n, where c is approximately 3; see the link. - Alexander R. Povolotsky, Jun 19 2014

Max Alekseyev, Table of n, a(n) for n = 1..30542 (contains all terms below 10^5)
Éric Angelini, Franklin Adams-Watters, Max Alekseyev, A. E. Povolotsky, N. J. A. Sloane, and R. G. Wilson v, a(n) divides the sum of the first a(n) terms of T, Various postings to the old Sequence Fans Mailing List, assembled by _N. J. A. Sloane_, Dec 24 2024
John Pezzullo, Data points approximation analysis.
Hugo Pfoertner, Table of n, a(n) for n = 1..100000
Hugo Pfoertner, Plot of a(n) - (3.33879*n^0.998152 + 3.36), fit of 100000 terms.
Hugo Pfoertner, Plot of a(n) - (3.2418*n^1.00065 + 307.6), fit of 530000 terms.

Cf. A243700.

A244016 Numbers in A243700 sorted into increasing order.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 13, 14, 15, 16, 18, 20, 22, 24, 25, 26, 28, 29, 30, 31, 32, 33, 35, 37, 38, 40, 41, 43, 44, 45, 47, 49, 50, 52, 53, 54, 55, 56, 58, 59, 60, 62, 63, 65, 67, 68, 70, 71, 72, 74, 76, 77, 79, 80, 82, 84, 86, 87, 88, 89, 91, 93, 94, 96, 97, 98, 100

Offset: 1

Max Alekseyev and N. J. A. Sloane, Jun 19 2014

nonn

Cf. A243700, A243864, A244010, A244011, A244017, A244018.

A244017 The quotients when A243700 is calculated (A244011), but shown in the order corresponding to the sorted sequence A244016.

Original entry on oeis.org

1, 2, 2, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87

Offset: 1

Max Alekseyev and N. J. A. Sloane, Jun 19 2014

nonn

Cf. A243700, A243864, A244010, A244011, A244016, A244018.

a(n) = A244010(A244016(n))/A244016(n).

A019444 a_1, a_2, ..., is a permutation of the positive integers such that the average of each initial segment is an integer, using the greedy algorithm to define a_n.

Original entry on oeis.org

1, 3, 2, 6, 8, 4, 11, 5, 14, 16, 7, 19, 21, 9, 24, 10, 27, 29, 12, 32, 13, 35, 37, 15, 40, 42, 17, 45, 18, 48, 50, 20, 53, 55, 22, 58, 23, 61, 63, 25, 66, 26, 69, 71, 28, 74, 76, 30, 79, 31, 82, 84, 33, 87, 34, 90, 92, 36, 95, 97, 38, 100, 39, 103, 105, 41, 108, 110, 43, 113

Offset: 1

R. K. Guy and Tom Halverson (halverson(AT)macalester.edu)

Self-inverse when considered as a permutation or function, i.e., a(a(n)) = n. - Howard A. Landman, Sep 25 2001
That each initial segment has an integer average is trivially equivalent to the sum of the first n elements always being divisible by n. - Franklin T. Adams-Watters, Jul 07 2014
Also, a lexicographically minimal sequence of distinct positive integers such that all values of a(n)-n are also distinct. - Ivan Neretin, Apr 18 2015
Comments from N. J. A. Sloane, Mar 29 2025 (Start):
Let d(n) = number of 1 <= i <= n such that a(i) < i. The d(i) sequence begins 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, ..., and appears to be A060144 without its initial term.
Let r(n) = 1 if a(n) < a(n+1), otherwise 0, and let f(n) = 1 if a(n) > a(n+1), otherwise 0. Then R = partial sums of r(n) and F = partial sums of f(n) count the rises and falls, respectively, in the present sequence. It appears that R and F are essentially A060143 and A060144 (again).
If a(n) is the k-th term in a monotonically strictly increasing rum of terms, set R(n) = k. It appears that the sequence  R(n), n>=1, is essentially A270788.
For other sequences derived from the present one, see A382162, A382168, and A382169.
(End)

Muharem Avdispahić and Faruk Zejnulahi, An integer sequence with a divisibility property, Fibonacci Quarterly, Vol. 58:4 (2020), 321-333.

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10000
Éric Angelini, Franklin Adams-Watters, Max Alekseyev, A. E. Povolotsky, N. J. A. Sloane, and R. G. Wilson v, a(n) divides the sum of the first a(n) terms of T, Various postings to the old Sequence Fans Mailing List, assembled by _N. J. A. Sloane_, Dec 24 2024
Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 18.
Math Forum, Problem of the Week 818
Jeffrey Shallit, Proving properties of some greedily-defined integer recurrences via automata theory, arXiv:2308.06544 [cs.DM], August 12 2023.
A. Shapovalov, Problem M1517 (in Russian), Kvant 5 (1995), 20-21. English translation appeared in Quantum problem M185, Sept/October 1996 (beware, file is 75Mb).
B. J. Venkatachala, A curious bijection on natural numbers, JIS 12 (2009) 09.8.1.
Index entries for sequences that are permutations of the natural numbers

Cf. A019445, A019446, A243700, A001622, A060143, A060144, A270788.

A244018 Numbers missing from A244017.

Original entry on oeis.org

3, 9, 12, 27, 37, 40, 49, 54, 58, 70, 85, 88, 106, 109, 119, 122, 125, 128, 132, 141, 154, 160, 164, 166, 171, 173, 176, 186, 193, 211, 213, 227, 238, 243, 245, 265, 277, 280, 283, 296, 301, 304, 318, 324, 326, 333, 343, 346

Offset: 1

Max Alekseyev and N. J. A. Sloane, Jun 19 2014

nonn

Cf. A243700, A243864, A244010, A244011, A244016, A244017.

A244671 The lexicographically earliest increasing sequence such that a(n) divides the sum of the first a(n) terms.

Original entry on oeis.org

1, 3, 5, 6, 10, 11, 12, 13, 14, 15, 20, 22, 24, 26, 28, 29, 30, 31, 32, 48, 49, 55, 56, 60, 61, 67, 68, 72, 89, 93, 97, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 141, 161, 162, 163, 164, 165, 166, 175, 188, 189, 190, 191, 222, 269

Offset: 1

Jaroslav Krizek, Jul 04 2014

nonn

Partial sums give A244672.

A244672(a(n)) / a(n) = integer.

			a(1) = 1 because 1 divides the first term (1/1=1); a(2) cannot be 2 because 2 does not divide the sum of the first 2 terms (3/2 is not an integer), a(2) must be 3; if a(2) = 3 then a(3) must be 5 (5 is the smallest number > a(2) such that the sum of the first 3 terms (i.e. 9) is divisible by a(2) = 3); if a(4) = 6 (holds 6 > a(3)), a(5) must be 10 (10 is the smallest number > a(4) such that the sum of first 5 terms (i.e. 25) is divisible by a(3) = 5); etc...

Cf. A243700, A244672.

N:= 1000: # to get the first N terms
A:= {1,3}: s:= 4:
for n from 3 to N do
  if member(n,A,'p') then
     r:= A[n-1]+1 + (-s-A[n-1]-1 mod A[p])
  else
     r:= A[n-1]+1
  fi;
  A:= A union {r};
  s:= s + r;
od:
A; # Robert Israel, Jul 06 2014

cp's OEIS Frontend

A244010 Partial sums of A243700.

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A244011 The quotient when A243700(n) is calculated.

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Formula

A243864 Positive integers that do not appear in A243700.

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A244016 Numbers in A243700 sorted into increasing order.

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A244017 The quotients when A243700 is calculated (A244011), but shown in the order corresponding to the sorted sequence A244016.

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Author

Keywords

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Formula

A019444 a_1, a_2, ..., is a permutation of the positive integers such that the average of each initial segment is an integer, using the greedy algorithm to define a_n.

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Mathematica

PARI

PARI

Formula

A244018 Numbers missing from A244017.

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A244671 The lexicographically earliest increasing sequence such that a(n) divides the sum of the first a(n) terms.

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Maple