A000430 Primes and squares of primes.

2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223

Offset: 1

Author

R. Muller

Keywords

• nonn
• easy
• nice

Comments

Also numbers n such that the product of proper divisors is < n.

See A050216 for lengths of blocks of consecutive primes. - Reinhard Zumkeller, Sep 23 2011

Numbers q > 1 such that d(q) < 4. Numbers k such that the number of ways of writing k = m + t is equal to the number of ways of writing k = r*s, where m|t and r|s. - Juri-Stepan Gerasimov, Oct 14 2017

Called multiplicatively deficient numbers by Chau (2004). - Amiram Eldar, Jun 29 2022

References

• F. Smarandache, Definitions solved and unsolved problems, conjectures and theorems in number theory and geometry, edited by M. Perez, Xiquan Publishing House 2000
• F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• William Chau, The tau, sigma, rho functions, and some related numbers, Pi Mu Epsilon Journal, Vol. 11, No. 10 (Spring 2004), pp. 519-534; entire issue.
• F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.

Crossrefs

Union of A000040 and A001248.

Cf. A007422, A010051, A010055, A032741, A046951, A050216, A056595, A058080, A064911, A084110, A084114, A293575.

Programs

Haskell

a000430 n = a000430_list !! (n-1)
a000430_list = m a000040_list a001248_list where
   m (x:xs) (y:ys) | x < y = x : m xs (y:ys)
                   | x > y = y : m (x:xs) ys
-- Reinhard Zumkeller, Sep 23 2011

Mathematica

nn = 223; t = Union[Prime[Range[PrimePi[nn]]], Prime[Range[PrimePi[Sqrt[nn]]]]^2] (* T. D. Noe, Apr 11 2011 *)
Module[{upto=250,prs},prs=Prime[Range[PrimePi[upto]]];Select[Join[ prs,prs^2], #<=upto&]]//Sort (* Harvey P. Dale, Oct 08 2016 *)

PARI

is(n)=isprime(n) || (issquare(n,&n) && isprime(n)) \\ Charles R Greathouse IV, Sep 04 2013

Python

from math import isqrt
from sympy import primepi
def A000430(n):
    def f(x): return n+x-primepi(x)-primepi(isqrt(x))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return int(m) # Chai Wah Wu, Aug 09 2024

Formula

A084114(a(n)) = 0, see also A084110. - Reinhard Zumkeller, May 12 2003

A109810(a(n)) = 2. - Reinhard Zumkeller, May 24 2010

A010051(a(n)) + A010055(a(n))*A064911(a(n)) = 1;

A056595(a(n)) = 1. - Reinhard Zumkeller, Aug 15 2011

A032741(a(n)) = A046951(a(n)); A293575(a(n)) = 0. - Juri-Stepan Gerasimov, Oct 14 2017

The number of terms not exceeding x is N(x) ~ (x + 2*sqrt(x))/log(x) (Chau, 2004). - Amiram Eldar, Jun 29 2022