A031218 Largest prime power <= n.

1, 2, 3, 4, 5, 5, 7, 8, 9, 9, 11, 11, 13, 13, 13, 16, 17, 17, 19, 19, 19, 19, 23, 23, 25, 25, 27, 27, 29, 29, 31, 32, 32, 32, 32, 32, 37, 37, 37, 37, 41, 41, 43, 43, 43, 43, 47, 47, 49, 49, 49, 49, 53, 53, 53, 53, 53, 53, 59, 59, 61, 61, 61, 64, 64, 64, 67, 67, 67, 67, 71, 71

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• changed

Comments

The length of the m-th run of {a(n)} is the length of the (m+1)-st run of A000015 for m > 1. - Colin Linzer, Mar 08 2024

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Sunben Chiu, Pinghzi Yuan, and Tao Zhou, On the greatest common divisor of binomial coefficients, Bull Korean Math. Soc. 60 (2023) 863-872
• B. Dearden, J. Iiams, and J. Metzger, Rumor Arrays, Journal of Integer Sequences, 16 (2013), #13.9.3.
• Eric Weisstein's World of Mathematics, Prime Power

Crossrefs

Cf. A000015, A000961, A065515.

Programs

Haskell

a031218 n = last $ takeWhile (<= n) a000961_list
-- Reinhard Zumkeller, Apr 25 2011

Maple

A031218 := proc(n)
    local a,pi,p,m ;
    a := 1 ;
    for pi from 1 do
        p := ithprime(pi) ;
        if p > n then
            return a;
        end if;
        for m from 0 do
            if p^m > n then
                break;
            elif p^m <= n then
                a := max(a,p^m) ;
            end if;
        end do:
    end do:
    a ;
end proc:
seq(A031218(n),n=1..40) ; # R. J. Mathar, Jul 20 2025

PARI

a(n)=if(n<1,0, while(matsize(factor(n))[1]>1,n--); n)

Python

from sympy import factorint
def A031218(n): return next(filter(lambda m:len(factorint(m))<=1, range(n,0,-1))) # Chai Wah Wu, Oct 25 2024

Formula

a(n) = n - A378457(n). - R. J. Mathar, Jul 20 2025

a(n) = A000961(A065515(n)). - Ridouane Oudra, Aug 22 2025

Extensions

More terms from Erich Friedman