A089088 -id:A089088 - cp's OEIS frontend

A112988 Position of n-th prime in A089088.

2, 5, 9, 12, 19, 23, 30, 34, 41, 52, 55, 65, 73, 77, 85, 95, 105, 110, 121, 128, 133, 143, 151, 162, 175, 182, 187, 195, 200, 208, 231, 239, 249, 253, 271, 276, 286, 298, 306, 318, 328, 332, 350, 354, 362, 366, 387, 408, 416, 420, 427, 439, 443, 461, 472, 483

Offset: 1

Reinhard Zumkeller, Oct 08 2005

A089088(a(n)) = A000040(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A089088, A112976.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a112988 = (+ 1) . fromJust . (`elemIndex` a089088_list) . a000040
-- Reinhard Zumkeller, Feb 27 2013

A112990 Inverse of A089088.

Original entry on oeis.org

1, 2, 5, 3, 9, 4, 12, 6, 7, 8, 19, 10, 23, 11, 13, 14, 30, 15, 34, 16, 17, 18, 41, 20, 21, 22, 24, 25, 52, 26, 55, 27, 28, 29, 31, 32, 65, 33, 35, 36, 73, 37, 77, 38, 39, 40, 85, 42, 43, 44, 45, 46, 95, 47, 48, 49, 50, 51, 105, 53, 110, 54, 56, 57, 58, 59, 121, 60, 61, 62, 128

Offset: 1

Reinhard Zumkeller, Oct 08 2005

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for sequences that are permutations of the natural numbers

Cf. A112978.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a112990 = (+ 1) . fromJust . (`elemIndex` a089088_list)
-- Reinhard Zumkeller, Feb 27 2013

A373938 Number of k in the range s(1), ..., s(n-1) such that gcd(k, s(n)) > 1, where s = A089088.

Original entry on oeis.org

0, 0, 1, 2, 1, 3, 2, 4, 1, 7, 6, 1, 6, 7, 11, 11, 8, 10, 1, 15, 4, 12, 1, 8, 15, 21, 15, 12, 16, 1, 10, 23, 18, 1, 14, 23, 29, 23, 20, 22, 1, 31, 6, 29, 18, 27, 35, 14, 31, 20, 28, 1, 43, 30, 1, 26, 31, 16, 45, 35, 24, 45, 47, 36, 1, 34, 39, 16, 53, 47, 26, 40, 1, 59, 20, 42, 1, 30, 47, 65, 18

Offset: 0

Scott R. Shannon, Jun 23 2024

nonn

In the terms studied a(n) = 1 if s(n) is 4 or a prime.

			a(5) = 5 as A089088(5) = 8 and 8 shares a factor m > 1 with three previous terms, A089088(1) = 2, A089088(2) = 4, and A089088(3) = 6.

Scott R. Shannon, Table of n, a(n) for n = 0..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 2..2^14.

Cf. A089088, A373880, A373390, A064413.

nn = 120; s[0] = 1; s[1] = p = 2; s[2] = i = 4; s[3] = j = 6;
Do[If[PrimeQ[j/2],
    k = Prime[p++],
    k = If[PrimeQ[j], i, j] + 1; While[PrimeQ[k], k++]];
  Set[{s[n], i, j}, {k, j, k}], {n, 4, nn}];
s = Array[s, nn];
{0}~Join~Table[Function[{m, w}, Count[w, _?(! CoprimeQ[#, m[[1]]] &)]] @@
TakeDrop[#, -1] &@  s[[;; n]], {n, nn}] (* Michael De Vlieger, Jun 23 2024 *)

A337136 a(1) = 1, a(2) = 2; for n > 1, a(n) = smallest positive number which has not yet appeared which has a common factor with a(n-2) + a(n-1).

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 8, 7, 9, 10, 19, 29, 12, 41, 53, 14, 67, 15, 16, 31, 47, 13, 18, 62, 20, 22, 21, 43, 24, 134, 26, 25, 17, 27, 11, 28, 30, 32, 34, 33, 201, 36, 39, 35, 37, 38, 40, 42, 44, 46, 45, 49, 48, 97, 50, 51, 101, 52, 54, 56, 55, 57, 58, 23, 60, 83, 65

Offset: 1

Scott R. Shannon, Aug 18 2020

Conjecture: This is a permutation of the natural numbers. Up to 1 million terms the only fixed points are 1,2,3,6,9,10, and it is likely that there are no others.

			a(5) = 4 as a(3) + a(4) = 3 + 5 = 8, and 4 is the smallest number that shares a common factor with 8 that has not yet appeared.
a(11) = 19 as a(9) + a(10) = 9 + 10 = 19, and as 19 is prime, a(11) must be the smallest multiple of 19 that has not yet appeared, being 19 in this case.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..1024, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue, highlighting in light blue those numbers in the last mentioned category whose prime power exponents all exceed 1.
Scott R. Shannon, Image of the first 1000000 terms. The green line is y=x.

Cf. A064413, A336957, A098550, A089088, A251622.

A251622 a(1) = 1; a(2) = 2; for n > 2, a(n) = smallest number not already used which shares a factor with a(n-1) or a(n-2).

Original entry on oeis.org

1, 2, 4, 6, 3, 8, 9, 10, 5, 12, 14, 7, 16, 18, 15, 20, 21, 22, 11, 24, 26, 13, 28, 30, 25, 27, 33, 36, 32, 34, 17, 38, 19, 40, 35, 42, 39, 44, 45, 46, 23, 48, 50, 51, 52, 54, 56, 49, 58, 29, 60, 55, 57, 63, 66, 62, 31, 64, 68, 70, 65, 72, 69, 74, 37, 76, 78, 75, 80

Offset: 1

Franklin T. Adams-Watters, Dec 06 2014

nonn

This appears certainly to be a permutation of the positive integers.
I believe the arguments we used to show that the EKG sequence (A064413) is a permutation of the natural numbers apply here with almost no change. - N. J. A. Sloane, Jan 02 2015
For n > 2: gcd(a(n),a(n-1)*a(n-2)) > 1. - Reinhard Zumkeller, Apr 05 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, Exper. Math. 11 (2002), 437-446.
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG Sequence

Cf. A064413, A089088, A098550.

Cf. A256628 (conjectured inverse).

a251622 n = a251622_list !! (n-1)
a251622_list = 1 : 2 : f 1 2 [3..] where
   f u v xs = g xs where
     g (w:ws) = if gcd w u > 1 || gcd w v > 1
                   then w : f v w (delete w xs) else g ws
-- Reinhard Zumkeller, Apr 05 2015

a[1] = 1; a[2] = 2;
a[n_] := a[n] = For[k = 1, True, k++, If[FreeQ[Array[a, n-1], k], If[!CoprimeQ[k, a[n-1]] || !CoprimeQ[k, a[n-2]], Return[k]]]];
Array[a, 100] (* Jean-François Alcover, Sep 05 2018 *)

invecn(v,k,x)=for(i=1,k,if(v[i]==x,return(i)));0
alist(n)=local(v=vector(n),x);v[1]=1;v[2]=2;for(k=3,n,x=3;while(invecn(v,k-1,x)||(gcd(v[k-1],x)==1&&gcd(v[k-2],x)==1),x++);v[k]=x);v

A351001 a(0) = 0, a(1) = 1; for n > 1, a(n) is the smallest positive number which has not appeared which has a common factor with a(n-2) + a(n-1) but does not equal a(n-2) + a(n-1).

Original entry on oeis.org

0, 1, 2, 6, 4, 5, 3, 10, 26, 8, 12, 14, 13, 9, 11, 15, 16, 62, 18, 20, 19, 21, 22, 86, 24, 25, 7, 28, 30, 29, 118, 27, 35, 31, 32, 33, 39, 34, 146, 36, 38, 37, 40, 42, 41, 166, 23, 45, 17, 44, 122, 46, 48, 47, 50, 194, 52, 51, 206, 514, 54, 56, 55, 57, 49, 53, 58, 60, 59, 63, 61, 64

Offset: 0

Scott R. Shannon, Jan 28 2022

nonn

This is a permutation of the natural numbers. Up to 500000 terms the fixed points are 0, 1, 2, 4, 5, 15, 16, 18, 21, 22, 24, 25, 29, and it is likely no more exist.

			a(3) = 6 as a(1)+a(2) = 3, 6 does not equal 3, and gcd(3,6) > 1.
a(4) = 4 as a(2)+a(3) = 8, 4 does not equal 8, and gcd(8,4) > 1.

Scott R. Shannon, Image of the first 500000 terms. The green line is y = n.

Cf. A337136, A064413, A336957, A098550, A089088, A251622.

s = {0, 1, 2}; u = 3; c[] = 0; Set[{i, j}, s[[-2 ;; -1]]]; Array[Set[c[s[[#]]], #] &, Length[s]]; s~Join~Reap[Do[Set[k, u]; While[Nand[c[k] == 0, GCD[i + j, k] > 1, i + j != k], k++]; Sow[k]; Set[c[k], n]; If[k == u, While[c[u] == 1, u++]]; i = j; j = k, {n, Length[s] + 1, 2^10}]][[-1, -1]] (* _Michael De Vlieger, Jan 28 2022 *)

A112975 Lexicographically earliest permutation of the natural numbers such that each term has a common divisor with at least two earlier terms, a(n)=n for n<4.

Original entry on oeis.org

1, 2, 3, 6, 4, 8, 9, 10, 12, 14, 15, 5, 16, 18, 20, 21, 7, 22, 24, 25, 26, 27, 28, 30, 32, 33, 11, 34, 35, 36, 38, 39, 13, 40, 42, 44, 45, 46, 48, 49, 50, 51, 17, 52, 54, 55, 56, 57, 19, 58, 60, 62, 63, 64, 65, 66, 68, 69, 23, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86

Offset: 1

Reinhard Zumkeller, Oct 08 2005

nonn

Inverse: A112978; A112977(n) = a(a(n));
For n>3: n is prime iff a(n) < a(n-1); a(A112976(n)) = A000040(n).
For n>8, a(n) can be described as follows: all composite numbers in natural order, with primes inserted so that every prime p immediately follows 3p. - Ivan Neretin, Apr 26 2015

Cf. A089088.

A373998 a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 5; for n > 4, a(n) is the smallest unused positive number that is coprime to a(n-1) and a(n-2) but has a common factor with at least one of a(1)...a(n-3).

Original entry on oeis.org

1, 2, 3, 5, 4, 9, 25, 8, 21, 55, 16, 7, 11, 6, 35, 121, 12, 49, 65, 18, 77, 13, 10, 27, 91, 20, 33, 119, 26, 15, 17, 14, 39, 85, 22, 57, 115, 28, 19, 23, 24, 95, 143, 32, 45, 133, 34, 69, 125, 38, 51, 145, 44, 63, 29, 40, 81, 161, 50, 87, 169, 46, 75, 187, 52, 93, 175, 58, 31, 99, 56, 155

Offset: 1

Scott R. Shannon, Jun 24 2024

nonn

For the terms studied, and similar to A373390 and A089088, the terms are concentrated along distinct lines of different gradient, four in this sequence, and like those sequences, the lowermost line is composed only of primes.
Note that in A089088, A373390, and this sequence, a(n) is coprime to zero, one and two previous terms, and the corresponding sequence terms are concentrated along two, three and four lines respectively.
The fixed points are 1, 2, 3, 8, 45, 51, and it is likely no more exist. For the terms studied the primes appear in their natural order. The sequence is conjectured to be a permutation of the positive integers.

			a(7) = 25 as 25 is the smallest unused number that is coprime to a(5) = 4 and a(6) = 9 while sharing a factor with a(4) = 5.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 50000 terms. Numbers with one, two, three, four, or five and more prime factors, counted with multiplicity, are show as red, yellow, green, blue and violet respectively.

Cf. A373999, A373390, A089088, A064413, A098550, A373880.

from math import gcd, lcm
from itertools import count, islice
def agen(): # generator of terms
    alst = [1, 2, 3, 5]
    yield from alst
    aset, LCM, mink = set(alst), lcm(*alst[:-2]), 4
    while True:
        an = next(k for k in count(mink) if k not in aset and 1 == gcd(k, alst[-1]) == gcd(k, alst[-2]) and gcd(k, LCM) > 1)
        LCM = lcm(LCM, alst[-2])
        alst.append(an)
        aset.add(an)
        while mink in aset: mink += 1
        yield an
print(list(islice(agen(), 72))) # Michael S. Branicky, Jun 24 2024

A350927 a(1)=1, a(2)=2; for n > 2, a(n) is the smallest unused positive number such that gcd(a(n-1) * |a(n-1) - a(n-2)|, a(n)) > 1.

Original entry on oeis.org

1, 2, 4, 6, 3, 9, 8, 10, 5, 15, 12, 14, 7, 21, 16, 18, 20, 22, 11, 33, 24, 26, 13, 39, 27, 28, 30, 25, 35, 32, 34, 17, 51, 36, 38, 19, 57, 40, 42, 44, 46, 23, 69, 45, 48, 50, 52, 54, 56, 49, 63, 58, 29, 87, 60, 55, 65, 62, 31, 93, 64, 66, 68, 70, 72, 74, 37, 111, 75, 76, 78, 80, 82, 41, 123, 81

Offset: 1

Scott R. Shannon, Jan 28 2022

nonn

This is likely a permutation of the natural numbers. The sequence shows similar behavior to the EKG sequence A064413. In the first 500000 terms the fixed points are 1, 2, 77, 221, and it is likely no more exist.

			a(4) = 6 as a(3)*|a(3)-a(2)| = 4*2 = 8, 6 has not been used and gcd(6,8) > 1.
a(8) = 10 as a(7)*|a(7)-a(6)| = 8*1 = 8, 10 has not been used and gcd(10,8) > 1.

Scott R. Shannon, Image of the first 10000 terms. The green line is y = n.

Cf. A351001, A337136, A064413, A336957, A098550, A089088, A251622.

A373999 a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 5, a(5) = 7; for n > 5, a(n) is the smallest unused positive number that is coprime to a(n-1), a(n-2) and a(n-3) but has a common factor with at least one of a(1)...a(n-4).

Original entry on oeis.org

1, 2, 3, 5, 7, 4, 9, 25, 49, 8, 27, 55, 91, 16, 51, 11, 13, 10, 17, 21, 121, 20, 169, 57, 77, 32, 65, 19, 33, 14, 85, 247, 69, 22, 35, 221, 23, 6, 95, 119, 143, 12, 115, 133, 187, 18, 125, 161, 209, 24, 145, 217, 253, 26, 15, 29, 31, 28, 39, 185, 289, 38, 63, 37, 155, 34, 81, 203, 205, 44

Offset: 1

Scott R. Shannon, Jun 24 2024

nonn

Initially the terms agree with the observation noted in A373998 and are concentrated predominantly along five lines of different gradient, with the primes forming the lowermost line. However this pattern is disrupted by the second lowest line showing a repetitive discontinuous jump to lower values which also interrupts the third middle line. An examination of the terms shows this is due to the appearance of three consecutive terms which are not divisible by 2 or 3. This allows subsequent terms to be a low multiple of 2 and 3, forming numbers which are less than the most recently appearing prime values. Also of note is after approximately 85000 terms the upper two lines merge; it is assumed that the remaining four lines continue in the above pattern as n grows arbitrarily large, although this is unknown.

Other than the first three terms the fixed points in the first 100000 terms are 35, 63, 219, 231, 1407, 2967, 3003, 6555, 14007, 14031, 32103, 77343, although it is likely more exist. For the terms studied the primes appear in their natural order. The sequence is conjectured to be a permutation of the positive integers.

			a(9) = 49 as 49 is the smallest unused number that is coprime to a(6) = 4, a(7) = 9, and a(8) = 25, while sharing a factor with a(5) = 7.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 100000 terms. Numbers with one, two, three, four, or five and more prime factors, counted with multiplicity, are show as red, yellow, green, blue and violet respectively.

Cf. A373998, A373390, A089088, A064413, A098550, A373880.

from math import gcd, lcm
from itertools import count, islice
def agen(): # generator of terms
    alst = [1, 2, 3, 5, 7]
    yield from alst
    aset, LCM, mink = set(alst), lcm(*alst[:-3]), 4
    while True:
        an = next(k for k in count(mink) if k not in aset and all(1 == gcd(k, m) for m in alst[-3:]) and gcd(k, LCM) > 1)
        LCM = lcm(LCM, alst[-3])
        alst.append(an)
        aset.add(an)
        while mink in aset: mink += 1
        yield an
print(list(islice(agen(), 70))) # Michael S. Branicky, Jun 24 2024

cp's OEIS Frontend

A112988 Position of n-th prime in A089088.

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A112990 Inverse of A089088.

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A373938 Number of k in the range s(1), ..., s(n-1) such that gcd(k, s(n)) > 1, where s = A089088.

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A337136 a(1) = 1, a(2) = 2; for n > 1, a(n) = smallest positive number which has not yet appeared which has a common factor with a(n-2) + a(n-1).

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A251622 a(1) = 1; a(2) = 2; for n > 2, a(n) = smallest number not already used which shares a factor with a(n-1) or a(n-2).

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A351001 a(0) = 0, a(1) = 1; for n > 1, a(n) is the smallest positive number which has not appeared which has a common factor with a(n-2) + a(n-1) but does not equal a(n-2) + a(n-1).

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Mathematica

A112975 Lexicographically earliest permutation of the natural numbers such that each term has a common divisor with at least two earlier terms, a(n)=n for n<4.

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A373998 a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 5; for n > 4, a(n) is the smallest unused positive number that is coprime to a(n-1) and a(n-2) but has a common factor with at least one of a(1)...a(n-3).

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A350927 a(1)=1, a(2)=2; for n > 2, a(n) is the smallest unused positive number such that gcd(a(n-1) * |a(n-1) - a(n-2)|, a(n)) > 1.

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