A111655 -id:A111655 - cp's OEIS frontend

A111653 n-th composite number appears n times.

4, 6, 6, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 21

Offset: 1

Jonathan Vos Post, Aug 12 2005

A002808(n) appears n times.

Cf. A002808, A111654, A111655.

from math import isqrt
from sympy import composite
def A111653(n): return composite(isqrt(n<<3)+1>>1) # Chai Wah Wu, Sep 06 2025

A111654 n appears n-th composite number times.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 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, 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

Offset: 1

Jonathan Vos Post, Aug 12 2005

n appears A002808(n) times.

Cf. A002808, A111653, A111655.

Module[{nn=20,comps,len},comps=Select[Range[nn],CompositeQ];len=Length[ comps];Flatten[Table[#[[1]],{#[[2]]}]&/@Thread[{Range[len],comps}]]] (* Harvey P. Dale, Aug 17 2014 *)

A111656 n-th prime appears n-th composite number times.

Original entry on oeis.org

2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19

Offset: 1

Jonathan Vos Post, Aug 12 2005

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
L. Panaitopol, Some Properties of the Series of Composed Numbers, J. Inequalities in Pure and Applied Mathematics. 2(2): Article 38, 2000.
J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6: 64-94 (1962).
Eric Weisstein's World of Mathematics, Composite Number.
Eric Weisstein's World of Mathematics, Prime Number.

Cf. A000040, A002808, A111653, A111654, A111655, A111657.

Module[{nn=20,cm,pr,len},cm=Select[Range[nn],CompositeQ];len = Length[ cm];Table[#[[1]],#[[2]]]&/@Thread[{Prime[Range[len]],cm}]]//Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 06 2018 *)

A000040(n) appears A002808(n) times.

A111657 n-th composite number appears n-th prime times.

Original entry on oeis.org

4, 4, 6, 6, 6, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15

Offset: 1

Jonathan Vos Post, Aug 12 2005

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.

L. Panaitopol, Some Properties of the Series of Composed Numbers, J. Inequalities in Pure and Applied Mathematics. 2(2): Article 38, 2000.
J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6: 64-94 (1962).
Eric Weisstein's World of Mathematics, Composite Number.
Eric Weisstein's World of Mathematics, Prime Number.

Cf. A000040, A002808, A111653, A111654, A111655, A111656.

A002808(n) appears A000040(n) times.

A175961 n-th nonprime number appears n-th nonprime numbers times.

Original entry on oeis.org

1, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15

Offset: 1

Juri-Stepan Gerasimov, Oct 31 2010

A018252(n) appears A018252(n) times; 1 appears once, n-th composite number appears n-th composite number times; 1 together with A111655.

Cf. A063905, A141468, A175944.

a(1)=1, a(n+1)=A111655(n).

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A111653 n-th composite number appears n times.

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A111654 n appears n-th composite number times.

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A111656 n-th prime appears n-th composite number times.

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A111657 n-th composite number appears n-th prime times.

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A175961 n-th nonprime number appears n-th nonprime numbers times.

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