A370408 -id:A370408 - cp's OEIS frontend

A371060 Lexicographically earliest sequence of distinct terms such that every triplet of successive digits (seen as side lengths) can form a triangle.

1, 2, 21, 22, 3, 4, 5, 6, 7, 8, 9, 28, 72, 65, 24, 32, 23, 31, 33, 13, 34, 25, 42, 43, 35, 36, 44, 14, 41, 441, 442, 45, 26, 52, 54, 46, 37, 53, 55, 15, 51, 551, 552, 56, 27, 62, 66, 16, 61, 661, 662, 67, 38, 63, 57, 39, 73, 64, 47, 48, 58, 49, 68, 59, 69, 74, 75, 76, 77, 17, 71, 771, 772

Offset: 1

Eric Angelini and Giorgos Kalogeropoulos, Mar 09 2024

In a triangle, the sum of any two side lengths is greater than that of the third, so that x + y > z.

			The first triplet of digits (1, 2, 2) forms an isosceles triangle with basis 1 and sides 2 and 2;
the second triplet (2, 2, 1) forms another isosceles triangle with basis 1 and sides 2 and 2;
the fifth triplet (2, 2, 3) forms another isosceles triangle with basis 3 and sides 2 and 2;
the sixth triplet (2, 3, 4) forms a scalene triangle with sides 2, 3 and 4; etc.

Eric Angelini and Giorgos Kalogeropoulos, Triangles with digits, Personal blog, March 2024.

Cf. A370408.

g[1]=1;g[2]=2;g[n_]:=g[n]=(k=1;While[MemberQ[ar=Array[g,n-1],k]|| !And@@(({a,b,c}=#;And@@{a+b>c,b+c>a,a+c>b})&/@Partition[Flatten[IntegerDigits/@Join[ar,{k}]],3,1]),k++];k);Array[g,80]

A370822 Lexicographically earliest sequence of positive integers such that all equal terms appear at mutually coprime indices.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 4, 8, 1, 9, 1, 10, 5, 11, 1, 12, 2, 13, 7, 14, 1, 15, 1, 16, 8, 17, 3, 18, 1, 19, 10, 20, 1, 21, 1, 22, 11, 23, 1, 24, 2, 25, 13, 26, 1, 27, 6, 28, 14, 29, 1, 30, 1, 31, 16, 32, 7, 33, 1, 34, 17, 35, 1, 36, 1, 37, 19

Offset: 1

Neal Gersh Tolunsky, Mar 02 2024

nonn

See A279119 for the same sequence with numbers including 0.

See A055396 for a similar sequence where all equal terms share a factor > 1.

			a(4)=2 because if we had a(4)=1, then i=2 and i=4, which are not coprime indices, would have the same value 1. So a(4)=2, which is a first occurrence.
a(9)=2 because if we had a(9)=1, i=3 and i=9, would have the same value despite not being coprime indices. a(9) can be 2 because the only other index with a 2 is a(4)=2 and 4 is coprime to 9.
a(15)=4 because 4 is the smallest value such that every previous index at which a 4 occurs is coprime to i=15. In this case, 4 has only occurred at i=8 and 8 is coprime to 15.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A100480, A279119, A370408.

from math import gcd, lcm
from itertools import combinations as C, count, islice
def agen(): # generator of terms
    yield from [1, 1, 1]
    lcms = {1: 6}
    for n in count(4):
        an = next(an for an in count(1) if an not in lcms or gcd(lcms[an], n) == 1)
        yield an
        if an not in lcms: lcms[an] = n
        else: lcms[an] = lcm(lcms[an], n)
print(list(islice(agen(), 75))) # Michael S. Branicky, Mar 02 2024

a(n) = 1 + A279119(n). - Rémy Sigrist, Mar 04 2024

a(22) and beyond from Michael S. Branicky, Mar 02 2024

cp's OEIS Frontend

A371060 Lexicographically earliest sequence of distinct terms such that every triplet of successive digits (seen as side lengths) can form a triangle.

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