A175428 -id:A175428 - cp's OEIS frontend

A307785 a(n) is the cubic root of A175428(n) + A175428(n+1).

2, 3, 4, 5, 6, 7, 6, 3, 4, 5, 6, 7, 6, 3, 4, 5, 6, 7, 6, 3, 4, 5, 6, 7, 6, 3, 4, 5, 6, 7, 6, 3, 4, 5, 6, 7, 6, 3, 4, 5, 6, 7, 8, 7, 4, 5, 6, 5, 2, 3, 4, 5, 6, 7, 6, 2, 3, 4, 5, 6, 7, 8, 9, 8, 5, 4, 5, 6, 7, 8, 7, 5, 6, 7, 6, 4, 5, 6, 7, 8, 9, 10, 9, 6, 5, 6, 7

Offset: 1

Rémy Sigrist, Apr 28 2019

nonn

The sum of two consecutive terms of A175428 is always a perfect cube.

			for n = 6:
- A175428(6) + A175428(7) = 135 + 208 = 343 = 7^3,
- hence a(6) = 7.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A175428.

p=1; s=0; for (n=1, 87, s+=2^p; for (v=1, oo, if (!bittest(s,v) && ispower(p+v, 3, &q), print1 (q ", "); p=v; break)))

a(n)^3 = A175428(n) + A175428(n+1).

A257218 Lexicographically earliest sequence of distinct positive integers such that gcd(a(n), a(n-1)) takes no value more than twice.

Original entry on oeis.org

1, 2, 3, 6, 4, 8, 10, 5, 15, 9, 18, 12, 16, 24, 30, 20, 40, 32, 48, 36, 27, 54, 72, 60, 45, 75, 25, 50, 70, 7, 14, 28, 42, 21, 63, 126, 84, 56, 112, 64, 96, 120, 80, 100, 150, 90, 108, 81, 162, 216, 144, 168, 140, 35, 105, 210, 180, 135, 225, 300

Offset: 1

Ivan Neretin, Apr 18 2015

Presumably a(n) is a permutation of the positive integers.
Primes seem to occur in their natural order. 31 appears as a(7060). Primes p >= 37 are not found among the first 10000 terms.
Numbers n such that a(n)=n are 1, 2, 3, 12, 306, ...
A256918(n) = gcd(a(n), a(n+1)); gcd(a(A257120(n)), a(A257120(n)+1)) = gcd(a(A257475(n)), a(A257475(n)-1)) = n. - Reinhard Zumkeller, Apr 25 2015
For p prime: A257122(p)-1 = index of the smallest multiple of p: a(A257122(p)-1) mod p = 0 and a(m) mod p > 0 for m < A257122(p)-1. - Reinhard Zumkeller, Apr 26 2015

			After a(9)=15, the values 1, 2, 3, 4, 6, and 8 are already used, while 7 is forbidden because gcd(15,7)=1 and that value of GCD has already occurred twice, at (1,2) and (2,3). The minimal value which is neither used not forbidden is 9, so a(10)=9.

Ivan Neretin and Reinhard Zumkeller, Table of n, a(n) for n = 1..25000, first 10000 terms from Ivan Neretin

Other minimal sequences of distinct positive integers that match some condition imposed on a(n) and a(n-1):
A175498 (differences are unique),
A081145 (absolute differences are unique),
A235262 (bitwise XORs are unique),
A163252 (differ by one bit in binary),
A000027 (GCD=1),
A064413 (GCD>1),
A128280 (sum is a prime),
A034175 (sum is a square),
A175428 (sum is a cube),
A077220 (sum is a triangular number),
A073666 (product plus 1 is a prime),
A081943 (product minus 1 is a prime),
A091569 (product plus 1 is a square),
A100208 (sum of squares is a prime).
Cf. A004526.
Cf. A256918, A257120, A257475, A257478, A257122 (putative inverse).
Cf. also A281978.

import Data.List (delete); import Data.List.Ordered (member)
a257218 n = a257218_list !! (n-1)
a257218_list = 1 : f 1 [2..] a004526_list where
   f x zs cds = g zs where
     g (y:ys) | cd `member` cds = y : f y (delete y zs) (delete cd cds)
              | otherwise       = g ys
              where cd = gcd x y
-- Reinhard Zumkeller, Apr 24 2015

a={1}; used=Array[0&,10000]; Do[i=1; While[MemberQ[a,i] || used[[l=GCD[a[[-1]],i]]]>=2, i++]; used[[l]]++; AppendTo[a,i], {n,2,100}]; a (* Ivan Neretin, Apr 18 2015 *)

A259429 With a(1) = 1, a(n) is the smallest number not already in the sequence such that the arithmetic mean of two neighboring terms is a cube.

Original entry on oeis.org

1, 15, 39, 89, 161, 271, 415, 17, 37, 91, 159, 273, 413, 19, 35, 93, 157, 275, 411, 21, 33, 95, 155, 277, 409, 23, 31, 97, 153, 279, 407, 25, 29, 99, 151, 281, 405, 27, 101, 149, 283, 403, 621, 65, 63, 187, 245, 5, 11, 43, 85, 165, 267, 419, 13, 3, 51, 77, 173, 259, 427, 597, 861, 163, 87, 41, 209, 223, 463, 561, 125

Offset: 1

Derek Orr, Jun 26 2015

Believed to be a permutation of the odd integers.
A259603(n) = (a(n) + a(n+1)) / 2; a(A259537(n)) = 2*n-1. - Reinhard Zumkeller, Jun 30 2015
The scatter-plot shows interesting helix-like lenticular structures. - Antti Karttunen, May 29 2016

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A259260, A086517, A175428.
Cf. A010057, A005408, A259537.
Cf. A259603, A259565, A259542.

import Data.List (delete)
a259429 n = a259429_list !! (n-1)
a259429_list = 1 : f 1 [3, 5 ..] where
   f x zs = g zs where
     g (y:ys) = if a010057 ((x + y) `div` 2) == 1
                   then y : f y (delete y zs) else g ys
-- Reinhard Zumkeller, Jun 29 2015

a = {1}; Do[k = 1; While[Or[MemberQ[a, k], !IntegerQ@ Power[Mean[{a[[i - 1]], k}], 1/3]], k++]; AppendTo[a, k], {i, 2, 120}]; a (* Michael De Vlieger, May 29 2016 *)

v=[1]; n=1; while(#v<200, s=(n+v[#v])/2; if(type(s)=="t_INT", if(ispower(s,3)&&!vecsearch(vecsort(v), n), v=concat(v, n); n=0)); n++); v

A296615 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, a(n) XOR a(n+1) is a cube (where XOR denotes the XOR binary operator).

Original entry on oeis.org

1, 9, 8, 19, 18, 26, 27, 91, 38, 39, 47, 46, 53, 52, 60, 61, 64, 65, 73, 72, 83, 82, 90, 130, 131, 139, 138, 145, 144, 152, 153, 217, 164, 124, 103, 102, 110, 111, 116, 117, 125, 165, 173, 172, 183, 182, 190, 191, 194, 195, 203, 202, 209, 208, 216, 399, 343

Offset: 1

Rémy Sigrist, Dec 17 2017

This sequence has similarities with A175428: here a(n) XOR a(n+1) is a cube, there a(n) + a(n+1) is a cube.
This sequence is conjectured to the a permutation of the natural numbers.
The first fixed points are: 1, 5676, 5677, 5698, 11677, 13226, 26943, 26946, 27575, 28039, 28569, 28625, 30127, 30162, 37660, 37661, 44672, 44673, 45934.
The scatterplot of the first terms of the sequence shows hatches (see Links section).

			The first terms, alongside a(n) XOR a(n+1), are:
  n     a(n)    a(n) XOR a(n+1)
  --    ----    ---------------
   1       1    2^3
   2       9    1^3
   3       8    3^3
   4      19    1^3
   5      18    2^3
   6      26    1^3
   7      27    4^3
   8      91    5^3
   9      38    1^3
  10      39    2^3
  11      47    1^3
  12      46    3^3
  13      53    1^3
  14      52    2^3
  15      60    1^3
  16      61    5^3
  17      64    1^3
  18      65    2^3
  19      73    1^3
  20      72    3^3

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 50000 terms

Cf. A000578, A175428.

seen = 0; unseen = 1
other(p) = seen += 2^p; while (bittest(seen, unseen), unseen++); \
for (v=unseen, oo, if (!bittest(seen, v) && ispower(bitxor(p,v),3), return (v)))
for (n=1, 57, v=if (n==1, 1, other(v)); print1 (v ", "))

cp's OEIS Frontend

A307785 a(n) is the cubic root of A175428(n) + A175428(n+1).

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A257218 Lexicographically earliest sequence of distinct positive integers such that gcd(a(n), a(n-1)) takes no value more than twice.

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A259429 With a(1) = 1, a(n) is the smallest number not already in the sequence such that the arithmetic mean of two neighboring terms is a cube.

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A296615 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, a(n) XOR a(n+1) is a cube (where XOR denotes the XOR binary operator).

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PARI