A083375 -id:A083375 - cp's OEIS frontend

A138607 List first A008578(1) odd numbers, then first A008578(2) even numbers, then the next A008578(3) odd numbers, then the next A008578(4) even numbers, etc.

1, 2, 4, 3, 5, 7, 6, 8, 10, 12, 14, 9, 11, 13, 15, 17, 19, 21, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73

Offset: 1

Ctibor O. Zizka, May 14 2008

A permutation of numbers.

			Let
  S1={1}
  S2={2,4}
  S3={3,5,7}
  S4={6,8,10,12,14}
  S5={9,11,13,15,17,19,21}
  S6={16,18,20,22,24,26,28,30,32,34,36}
  ...
then S1, S2, S3, S4, S5, S6,... gives this sequence.

Index entries for sequences that are permutations of the natural numbers

Inverse: A166014. Cf. A138606, A138608, A138609, A138612.

If n < 3, a(n) = n. If n-2 = A007504(A083375(n-2)), then a(n) = a(n-1-A000040(A083375(n-2)))+2, otherwise a(n) = a(n-1)+2. - Antti Karttunen, Oct 05 2009.

Edited, extended, and offset changed from 0 to 1 by Antti Karttunen, Oct 05 2009

A350174 For k = 0, 1, 2, 3, ... write k prime(k+1) times.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Michel Marcus, Dec 18 2021

a(n) = k is the largest k with sum of primes A007504(k) <= n. - Kevin Ryde, Apr 19 2022

J.-P. Delahaye, Des suites fractales d’entiers, Pour la Science, No. 531 January 2022. Sequence g) p. 82.

Cf. A000040, A007504.

Essentially the same as A083375.

a:=[];
for n from 0 to 10 do a:=[op(a), seq(n,i=1..ithprime(n+1))]; od:
a; # N. J. A. Sloane, Dec 18 2021

from itertools import count, islice, chain
from sympy import prime
def A350174gen(): return chain.from_iterable([k]*prime(k+1) for k in count(0))
A350174_list = list(islice(A350174gen(),50)) # Chai Wah Wu, Dec 19 2021

a(n) = A083375(n+1) - 1. - Peter Munn, May 26 2023

A366066 a(n) is the largest positive integer k such that n can be expressed as the sum of k distinct positive integers that are coprime to each other.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 2, 3, 3, 3, 4, 3, 4, 3, 4, 4, 4, 5, 4, 5, 4, 5, 4, 5, 5, 5, 5, 5, 6, 5, 6, 5, 6, 5, 6, 6, 6, 6, 6, 6, 6, 7, 6, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 7, 8, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 8, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Yifan Xie, Sep 28 2023

The indices at which k first appears, for k >= 0: 1, 3, 6, 11, 18, 29, 42, 59, 78 (A014284). Such n's are expressed as the sum of 1 and the first primes.
Runs with length >= 2 start at numbers k^2 - 1 (k >= 2).
If there are terms between runs of k and k+1, these two numbers occur alternately. Suppose that m is such a term that is b(m) terms after the first occurrence of k+1; if b(m) is odd, there are at least two even numbers in the expression of n as the sum of k+1 integers, which are not coprime to each other, so a(m) = k.

			For n = 11, 1+2+3+5=11; so a(11) = 4.
For n = 12, 1+4+7=12; so a(12) = 3.

Yifan Xie, Table of n, a(n) for n = 0..9999

Cf. A000040, A000290, A014284, A083375.

lista(nn) = v=[0]; f=[7, 12, 14, 19, 21, 23, 30, 32, 34, 43, 45, 47, 60, 62, 79]; for(n=1, nn, for(i=1, prime(n), v=concat(v, n))); for(n=1, 15, v[f[n]+1]=v[f[n]+1]-1); v;

a(n) = A083375(n) - 1 if and only if n = 7, 12, 14, 19, 21, 23, 30, 32, 34, 43, 45, 47, 60, 62, 79; otherwise, a(n) = A083375(n).

cp's OEIS Frontend

A138607 List first A008578(1) odd numbers, then first A008578(2) even numbers, then the next A008578(3) odd numbers, then the next A008578(4) even numbers, etc.

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A350174 For k = 0, 1, 2, 3, ... write k prime(k+1) times.

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A366066 a(n) is the largest positive integer k such that n can be expressed as the sum of k distinct positive integers that are coprime to each other.

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