A269347 -id:A269347 - cp's OEIS frontend

A271328 a(n) = A269347(3*n)/3.

1, 5, 10, 17, 28, 37, 50, 65, 82, 106, 122, 145, 170, 197, 228, 257, 294, 325, 362, 406, 442, 485, 530, 577, 628, 677, 730, 790, 842, 906, 962, 1025, 1090, 1161, 1228, 1297, 1376, 1445, 1522, 1606, 1682, 1765, 1850, 1937, 2028, 2117, 2210

Offset: 1

Alec Jones, Apr 04 2016

a(n) is equal to n^2 + 1 with predictable regularity; in particular, the values of n for which a(n) does not equal n^2 + 1 are exactly those values n for which 3n is divisible by A269347(3*m) for some m with 1 < m < n. This is in part because 1 + sum{i=1...n}(2i - 1) = n^2 + 1; when computing a(n), each term has this form except when the named condition holds.

For example, a(5) does not equal 5^2 + 1 because 3(5) is divisible by A269347(6).

Alec Jones, Table of n, a(n) for n = 1..10000

Cf. A269347, A271326.

lista(nn) = {nn *= 3; va = vector(nn); va[1] = 1; for (n=2, nn, va[n] = sum(k=1, n-1, k*((n % va[k])==0)); ); vector(#va\3, n, va[3*n]/3); } \\ Michel Marcus, Apr 04 2016

a(n) = A269347(3*n)/3.

A271326 a(n) = A269347(3n).

Original entry on oeis.org

3, 15, 30, 51, 84, 111, 150, 195, 246, 318, 366, 435, 510, 591, 684, 771, 882, 975, 1086, 1218, 1326, 1455, 1590, 1731, 1884, 2031, 2190, 2370, 2526, 2718, 2886, 3075, 3270, 3483, 3684, 3891, 4128, 4335, 4566, 4818, 5046, 5295, 5550, 5811, 6084

Offset: 1

Alec Jones, Apr 04 2016

This sequence represents the "peaks" of A269347(n), which occur for that sequence when n is divisible by 3.

Terms of this sequence are equal to 3n^2 + 3 with predictable frequency; see my comment on A271328.

Alec Jones, Table of n, a(n) for n = 1..10001

Cf. A269347, A271328

A271468 List of indices i such that A269347(3i) != 3(i^2 + 1).

Original entry on oeis.org

1, 5, 10, 15, 17, 20, 25, 28, 30, 34, 35, 37, 40, 45, 50, 51, 55, 56, 60, 65, 68, 70, 74, 75, 80, 82, 84, 85, 90, 95, 100, 102, 105, 106, 110, 111, 112, 115, 119, 120, 122, 125, 130, 135, 136, 140, 145, 148, 150, 153, 155, 160, 164, 165, 168, 170, 175, 180

Offset: 1

Peter Kagey, Apr 08 2016

nonn

Equivalently, this sequences is a list of indices i such that A271328(i) != i^2 + 1.
If k is in A271328, then k is in this sequence.
All multiples of a(k) are in the sequence for k > 1.

			A269347(3*1) = 3 != 3*(1^2 + 1) = 6 so 1 is in the sequence.
A269347(3*2) = 15 = 3*(2^2 + 1) so 2 is not in the sequence.
A269347(3*3) = 30 = 3*(3^2 + 1) so 3 is not in the sequence.
A269347(3*4) = 51 = 3*(4^2 + 1) so 4 is not in the sequence.
A269347(3*5) = 84 != 3*(5^2 + 1) = 78 so 5 is in the sequence.

Peter Kagey, Table of n, a(n) for n = 1..10000

A279967 Square array read by antidiagonals upwards in which each term is the sum of prior elements in the same row, column, diagonal, or antidiagonal that divide n; the array is seeded with an initial value a(1)=1.

Original entry on oeis.org

1, 1, 2, 2, 2, 7, 2, 9, 10, 15, 2, 10, 1, 13, 17, 8, 0, 13, 1, 14, 9, 8, 0, 13, 3, 30, 13, 10, 2, 16, 1, 23, 5, 7, 14, 15, 2, 8, 28, 32, 2, 23, 2, 9, 49, 12, 0, 48, 2, 11, 1, 20, 3, 18, 13, 28, 0, 4, 1, 56, 5, 8, 16, 35, 46, 4, 2, 6, 2, 10

Offset: 1

Alec Jones, Dec 24 2016

From Hartmut F. W. Hoft, Jan 23 2017: (Start)
Shown by induction and direct (modular) computations for
column 1: Every number is even, except for the first two 1's; in addition to row 3, value 2 occurs in rows 4*k and 4*k+1, and every value in rows 4*k+2 and 4*k+3 is divisible by 4, for all k>=1.
column 2: The first four entries, 2, 2, 9 and 10, contain the only odd number; no nonzero entry in row k>3 has 9 as a factor, and value 0 occurs in rows 4*k+1 and 4*k+2, for all k>=1.
Conjecture:
a({1, 6, 8, 9, 10, 15, 26, 45, 48, 84, 96, 112, 115, 252, 336, 343}) =
  {1, 7, 9,10, 15, 17, 30, 49, 48,104,117, 115, 122, 257, 343, 395} are the only numbers in the sequence with the property a(n) >= n (verified through n=500500, i.e., the triangle with 1000 antidiagonals).
This conjecture together with Bouniakowsky's conjecture that certain quadratic integer polynomials generate infinitely many primes (e.g. see A002496 for n^2+1 and A188382 for 2*n^2+n+1) implies that in every column in the triangle infinitely many prime sequence indices occur and therefore infinitely many 0's whenever the column contains no 1's. The proof is based on the fact that for a large enough prime sequence index p in whose prior column no 1 occurs then a(p)=0; therefore infinitely many 0's occur in that column. Obviously, once value 1 occurs in a column no 0 value can occur in a subsequent row.
Conjecture:
Every row in the triangle contains exactly two 1's.
(End)

			After 6 terms, the array looks like:
.
1   2   7
1   2
2
We have a(6) = 7 because a(1) = 1, a(3) = 2, a(4) = 2, and a(5) = 2 divide 6; 1 + 2 + 2 + 2 = 7.
From _Hartmut F. W. Hoft_, Jan 23 2017: (Start)
1   2   7  15  17   9  10  15  49  13   4  31  22
1   2  10  13  14  13  14   9  18  46  12  66
2   9   1   1  30   7   2   3  35  12   3
2  10  13   3   5  23  20  16  14  17
2   0  13  23   2   1   8  11   2
8   0   1  32  11   5   3   6
8  16  28   2  56  42   8
2   8  48   1   2 104
2   0   4  10   1
12   0   2  10
28   6   2
2  42
2
.
Expanded the triangle to the first 13 antidiagonals of the array, i.e. a(1) ... a(91), to show the start of the 2- and 0-value patterns in columns 1 and 2. The first 0 beyond column 2 is a(677) in row 27, column 11 of the triangle.
A188382(n)=2*n^2+n+1 for n>=0 are the alternate sequence indices for column 1 starting in row 1, 2*n^2+n+2 for n>=1 are the alternate sequence indices for column 2 starting in row 2, and 2*n^2+n+11 for n>=5 are the alternate sequence indices for column 11 starting in row 1.
The sequence indices in the triangle for row positions k>=1 in columns 1,..., 5 are given in sequences A000124(k), A152948(k+3), A152950(k+3), A145018(k+4) and A167499(k+4).
(End)

Peter Kagey, Table of n, a(n) for n = 1..5000

Cf. A279966 for the related sequence which counts prior terms.
Cf. A269347 for a one-dimensional version of this sequence.
Cf. also A279211, A279212.
Cf. A000124, A002496, A145018, A152948, A152950, A167499, A188382.

(*  printing of the triangle is commented out of function a279967[]  *)
pCol[{i_, j_}] := Map[{#, j}&, Range[1, i-1]]
pDiag[{i_, j_}] := If[j>=i, Map[{#, j-i+#}&, Range[1, i-1]], Map[{i-j+#, #}&, Range[1, j-1]]]
pRow[{i_, j_}] := Map[{i, #}&, Range[1, j-1]]
pAdiag[{i_, j_}] := Map[{i+j-#, #}&, Range[1, j-1]]
priorPos[{i_, j_}] := Join[pCol[{i, j}], pDiag[{i, j}], pRow[{i, j}], pAdiag[{i, j}]]
seqPos[{i_, j_}] := (i+j-2)(i+j-1)/2+j
antiDiag[k_] := Map[{k+1-#, #}&, Range[1, k]]
upperTriangle[k_] := Flatten[Map[antiDiag, Range[1, k]], 1]
a279967[k_] := Module[{ut=upperTriangle[k], ms=Table[" ", {i, 1, k}, {j, 1, k}], h, pos, val, seqL={1}}, ms[[1, 1]]=1; For[h=2, h<=Length[ut], h++, pos=ut[[h]]; val=Apply[Plus, Select[Map[ms[[Apply[Sequence, #]]]&, priorPos[pos]], #!=0 && Mod[seqPos[pos], #]==0&]]; AppendTo[seqL, val]; ms[[Apply[Sequence, pos]]]=val]; (* Print[TableForm[ms]]; *) seqL]
a279967[13] (* values in first 13 antidiagonals *)
(* Hartmut F. W. Hoft, Jan 23 2017 *)

A271503 a(1) = 1; thereafter a(n) is the product of all 0 < m < n for which a(m) divides n.

Original entry on oeis.org

1, 1, 2, 6, 2, 120, 2, 210, 2, 1890, 2, 83160, 2, 270270, 2, 4054050, 2, 275675400, 2, 1309458150, 2, 27498621150, 2, 2529873145800, 2, 15811707161250, 2, 426916093353750, 2, 49522266829035000, 2, 383797567925021250, 2, 12665319741525701250, 2

Offset: 1

Peter Kagey, Apr 08 2016

nonn

			a(1) = 1 by definition
a(2) = 1 because a(1) divides 2.
a(3) = 1 * 2 = 2 because a(1) and a(2) divide 3.
a(4) = 1 * 2 * 3 = 6 because a(1), a(2), and a(3) divide 4.
a(5) = 1 * 2 = 2 because a(1) and a(2) divide 5.

Chai Wah Wu, Table of n, a(n) for n = 1..809 (n = 1..100 from Peter Kagey)

Cf. A269347, A271504.

a = {1}; Do[AppendTo[a, Times @@ Flatten@ Position[a, m_ /; Divisible[n, m]]], {n, 2, 35}]; a (* Michael De Vlieger, Apr 09 2016 *)

from itertools import count, islice
from math import prod
from sympy import divisors
def A271503_gen(): # generator of terms
    A271503_dict = {1:1}
    yield 1
    for n in count(2):
        yield (s:=prod(A271503_dict.get(d,1) for d in divisors(n,generator=True)))
        A271503_dict[s] = A271503_dict.get(s,1)*n
A271503_list = list(islice(A271503_gen(),40)) # Chai Wah Wu, Nov 17 2022

a(2n + 1) = 2 for all n > 1.

a(n) is even for all n > 2.

A271504 With a(1) = 1, a(n) is the LCM of all 0 < m < n for which a(m) divides n.

Original entry on oeis.org

1, 1, 2, 6, 2, 60, 2, 210, 2, 630, 2, 13860, 2, 90090, 2, 90090, 2, 3063060, 2, 29099070, 2, 29099070, 2, 1338557220, 2, 3346393050, 2, 10039179150, 2, 582272390700, 2, 9025222055850, 2, 9025222055850, 2, 18050444111700, 2, 333933216066450, 2, 333933216066450

Offset: 1

Peter Kagey, Apr 08 2016

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..2309 (n = 1..100 from Peter Kagey)

Cf. A269347, A271503.

a = {1}; Do[AppendTo[a, LCM @@ Select[Range[n - 1], Divisible[n, a[[#]]] &]], {n, 2, 40}]; a (* Michael De Vlieger, Apr 08 2016 *)

a(2n + 1) = 2 for all n > 1.

a(n) is even for all n > 2.

A271774 a(1) = 1, then a(n) is the maximum of all 0 < m < n for which a(m) divides n.

Original entry on oeis.org

1, 1, 2, 3, 2, 5, 2, 7, 4, 7, 2, 11, 2, 13, 6, 13, 2, 17, 2, 19, 10, 19, 2, 23, 6, 23, 4, 27, 2, 29, 2, 31, 12, 31, 10, 33, 2, 37, 16, 37, 2, 41, 2, 43, 6, 43, 2, 47, 10, 49, 18, 47, 2, 53, 12, 53, 22, 53, 2, 59, 2, 61, 10, 61, 16, 61, 2, 67, 26, 67, 2, 71, 2

Offset: 1

Peter Kagey, Apr 14 2016

nonn

If n is an odd prime, then a(n) = 2 and a(n+1) = n. All n for which a(n) = 2 are odd primes. - Robert Israel, Apr 14 2016

			a(1) = 1 by definition.
a(2) = 1 because a(1) divides 2.
a(3) = 2 because a(2) divides 3.
a(4) = 3 because a(3) divides 4.
a(5) = 2 because a(2) divides 5.
a(6) = 5 because a(5) divides 6.
a(7) = 2 because a(2) divides 7.
a(8) = 7 because a(7) divides 8.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A088167, A269347, A271503, A271504, A271773.

A:= proc(n) option remember; local m;
    for m from n-1 by -1 do
      if n mod A(m) = 0 then return m fi
    od
end proc:
A(1):= 1:
seq(A(i),i=1..100); # Robert Israel, Apr 14 2016

a[1] = 1; a[n_] := a[n] = Block[{m = n - 1}, While[Mod[n, a[m]] > 0, m--]; m]; Array[a, 100] (* Giovanni Resta, Apr 14 2016 *)

cp's OEIS Frontend

A271328 a(n) = A269347(3*n)/3.

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A271326 a(n) = A269347(3n).

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A271468 List of indices i such that A269347(3*i) != 3*(i^2 + 1).

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A279967 Square array read by antidiagonals upwards in which each term is the sum of prior elements in the same row, column, diagonal, or antidiagonal that divide n; the array is seeded with an initial value a(1)=1.

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A271503 a(1) = 1; thereafter a(n) is the product of all 0 < m < n for which a(m) divides n.

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A271504 With a(1) = 1, a(n) is the LCM of all 0 < m < n for which a(m) divides n.

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A271774 a(1) = 1, then a(n) is the maximum of all 0 < m < n for which a(m) divides n.

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Maple

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A271468 List of indices i such that A269347(3i) != 3(i^2 + 1).