cp's OEIS Frontend

A000430 Primes and squares of primes.

%I A000430 #68 Aug 09 2024 15:14:23
%S A000430 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,
%T A000430 79,83,89,97,101,103,107,109,113,121,127,131,137,139,149,151,157,163,
%U A000430 167,169,173,179,181,191,193,197,199,211,223
%N A000430 Primes and squares of primes.
%C A000430 Also numbers n such that the product of proper divisors is < n.
%C A000430 See A050216 for lengths of blocks of consecutive primes. - _Reinhard Zumkeller_, Sep 23 2011
%C A000430 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
%C A000430 Called multiplicatively deficient numbers by Chau (2004). - _Amiram Eldar_, Jun 29 2022
%D A000430 F. Smarandache, Definitions solved and unsolved problems, conjectures and theorems in number theory and geometry, edited by M. Perez, Xiquan Publishing House 2000
%D A000430 F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.
%H A000430 T. D. Noe, <a href="/A000430/b000430.txt">Table of n, a(n) for n = 1..1000</a>
%H A000430 William Chau, <a href="https://www.jstor.org/stable/24340619">The tau, sigma, rho functions, and some related numbers</a>, Pi Mu Epsilon Journal, Vol. 11, No. 10 (Spring 2004), pp. 519-534; <a href="http://www.pme-math.org/journal/issues/PMEJ.Vol.11.No10.pdf">entire issue</a>.
%H A000430 F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/Sequences-book.pdf">Sequences of Numbers Involved in Unsolved Problems</a>.
%F A000430 A084114(a(n)) = 0, see also A084110. - _Reinhard Zumkeller_, May 12 2003
%F A000430 A109810(a(n)) = 2. - _Reinhard Zumkeller_, May 24 2010
%F A000430 A010051(a(n)) + A010055(a(n))*A064911(a(n)) = 1;
%F A000430 A056595(a(n)) = 1. - _Reinhard Zumkeller_, Aug 15 2011
%F A000430 A032741(a(n)) = A046951(a(n)); A293575(a(n)) = 0. - _Juri-Stepan Gerasimov_, Oct 14 2017
%F A000430 The number of terms not exceeding x is N(x) ~ (x + 2*sqrt(x))/log(x) (Chau, 2004). - _Amiram Eldar_, Jun 29 2022
%t A000430 nn = 223; t = Union[Prime[Range[PrimePi[nn]]], Prime[Range[PrimePi[Sqrt[nn]]]]^2] (* _T. D. Noe_, Apr 11 2011 *)
%t A000430 Module[{upto=250,prs},prs=Prime[Range[PrimePi[upto]]];Select[Join[ prs,prs^2], #<=upto&]]//Sort (* _Harvey P. Dale_, Oct 08 2016 *)
%o A000430 (Haskell)
%o A000430 a000430 n = a000430_list !! (n-1)
%o A000430 a000430_list = m a000040_list a001248_list where
%o A000430    m (x:xs) (y:ys) | x < y = x : m xs (y:ys)
%o A000430                    | x > y = y : m (x:xs) ys
%o A000430 -- _Reinhard Zumkeller_, Sep 23 2011
%o A000430 (PARI) is(n)=isprime(n) || (issquare(n,&n) && isprime(n)) \\ _Charles R Greathouse IV_, Sep 04 2013
%o A000430 (Python)
%o A000430 from math import isqrt
%o A000430 from sympy import primepi
%o A000430 def A000430(n):
%o A000430     def f(x): return n+x-primepi(x)-primepi(isqrt(x))
%o A000430     m, k = n, f(n)
%o A000430     while m != k:
%o A000430         m, k = k, f(k)
%o A000430     return int(m) # _Chai Wah Wu_, Aug 09 2024
%Y A000430 Union of A000040 and A001248.
%Y A000430 Cf. A007422, A010051, A010055, A032741, A046951, A050216, A056595, A058080, A064911, A084110, A084114, A293575.
%K A000430 nonn,easy,nice
%O A000430 1,1
%A A000430 R. Muller