A253415 -id:A253415 - cp's OEIS frontend

A253425 Lengths of runs of identical terms in A253415.

1, 1, 6, 11, 18, 12, 5, 22, 91, 143, 1, 93, 370, 182, 20, 20, 315, 332, 973, 157, 1223, 1807, 325, 4044, 7412, 11211, 4600, 2176, 14848, 4659, 3123, 10852, 1678, 20862, 3348

Offset: 1

Reinhard Zumkeller, Dec 31 2014

Cf. A253415, A095258.

import Data.List (group)
a253425 n = a253425_list !! (n-1)
a253425_list = map length $ group a253415_list

c[] = 0; c[1] = j = 1; u = 2; s = 3; Most@ Tally[#][[All, -1]] &@ Reap[Do[d = Divisors[s]; k = 1; While[c[d[[k]]] > 0, k++]; Set[k, d[[k]]]; Set[c[k], i]; If[k == u, While[c[u] > 0, u++]]; Sow[u]; j = k; s += k, {i, 2, 2^12}]][[-1, -1]] (* _Michael De Vlieger, Jan 23 2022 *)

from itertools import islice
from sympy import divisors
def A253425_gen(): # generator of terms
    bset, l, m, s = {1}, 0, 2, 3
    while True:
        for d in divisors(s):
            if d not in bset:
                bset.add(d)
                if m in bset:
                    yield l
                    l = 1
                    while m in bset:
                        m += 1
                else:
                    l += 1
                s += d
                break
A253425_list = list(islice(A253425_gen(),20)) # Chai Wah Wu, Jan 25 2022

a(14)-a(35) from Michael De Vlieger, Jan 23 2022

A095258 a(n+1) is the smallest divisor of (2 + sum of first n terms) but not occurring earlier; a(1) = 1.

Original entry on oeis.org

1, 3, 2, 4, 6, 9, 27, 18, 8, 5, 17, 34, 68, 12, 24, 10, 25, 11, 13, 23, 7, 47, 94, 235, 15, 16, 32, 48, 51, 289, 578, 102, 36, 26, 73, 219, 30, 20, 14, 46, 50, 470, 60, 40, 146, 21, 49, 28, 113, 29, 19, 35, 42, 54, 64, 22, 77, 329, 84, 56, 292, 365, 65, 37, 131, 38, 33

Offset: 1

Reinhard Zumkeller, May 31 2004

Conjecture: integer permutation with inverse A095259: a(A095259(n))=A095259(a(n))=n. - Comment revised: Reinhard Zumkeller, Dec 31 2014
A095260(n) = a(a(n)).
First fixed points: 1, 4, 54, 416, ...
A253415(n) = smallest missing number within the first n terms. - Reinhard Zumkeller, Dec 31 2014

Rémy Sigrist, Table of n, a(n) for n = 1..10000 (first 797 terms from Reinhard Zumkeller)
Michael De Vlieger, Log-log scatterplot of a(n) for n = 1..2^16.
Michael De Vlieger, Annotated log-log scatterplot of a(n) for n = 1..2^14 showing records in red, local minima in blue, primes in magenta, prime powers not prime in gold.
Index entries for sequences that are permutations of the natural numbers

Cf. A253415.

import Data.List (delete)
a095258 n = a095258_list !! (n-1)
a095258_list = 1 : f [2..] 1 where
   f xs z = g xs where
     g (y:ys) = if mod z' y > 0 then g ys else y : f (delete y xs) (z + y)
     z' = z + 2
-- Reinhard Zumkeller, Dec 31 2014

c[] = 0; j = c[1] = 1; s = 3; {j}~Join~Reap[Do[d = Divisors[s]; k = 1; While[c[d[[k]]] > 0, k++]; Set[k, d[[k]]]; Set[c[k], i]; Sow[k]; j = k; s += k, {i, 2, 80}]][[-1, -1]] (* _Michael De Vlieger, Jan 23 2022 *)

from itertools import islice
from sympy import divisors
def A095258_gen(): # generator of terms
    bset, s = {1}, 3
    yield 1
    while True:
        for d in divisors(s):
            if d not in bset:
                yield d
                bset.add(d)
                s += d
                break
A095258_list = list(islice(A095258_gen(),30)) # Chai Wah Wu, Jan 25 2022

A253443 Smallest missing number within the first n terms in A109890.

Original entry on oeis.org

4, 4, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 34, 37, 37, 37, 37, 37

Offset: 4

Reinhard Zumkeller, Jan 01 2015

nonn

A253584(n) occurs exactly A253444(n) times.

Reinhard Zumkeller, Table of n, a(n) for n = 4..10000

Cf. A095258, A095259, A253444 (run lengths), A253584 (range), A253415.

import Data.List (insert)
a253443 n = a253443_list !! (n-4)
a253443_list = f (4, []) 6 where
   f (m,ys) z = g $ dropWhile (< m) $ a027750_row' z where
     g (d:ds) | elem d ys = g ds
              | otherwise = m : f (ins [m, m+1 ..] (insert d ys)) (z + d)
     ins (u:us) vs'@(v:vs) = if u < v then (u, vs') else ins us vs
-- Reinhard Zumkeller, Jan 03 2015

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A253425 Lengths of runs of identical terms in A253415.

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A095258 a(n+1) is the smallest divisor of (2 + sum of first n terms) but not occurring earlier; a(1) = 1.

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A253443 Smallest missing number within the first n terms in A109890.

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Author

Keywords

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Haskell