A025473 a(1) = 1; for n > 1, a(n) = prime root of n-th prime power (A000961).

1, 2, 3, 2, 5, 7, 2, 3, 11, 13, 2, 17, 19, 23, 5, 3, 29, 31, 2, 37, 41, 43, 47, 7, 53, 59, 61, 2, 67, 71, 73, 79, 3, 83, 89, 97, 101, 103, 107, 109, 113, 11, 5, 127, 2, 131, 137, 139, 149, 151, 157, 163, 167, 13, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Author

David W. Wilson, Dec 11 1999

Keywords

• easy
• nonn
• nice

Comments

This sequence is related to the cyclotomic sequences A013595 and A020500, leading to the procedure used in the Mathematica program. - Roger L. Bagula, Jul 08 2008

"LCM numeral system": a(n+1) is radix for index n, n >= 0; a(-n+1) is 1/radix for index n, n < 0. - Daniel Forgues, May 03 2014

This is the LCM-transform of A000961; same as A014963 with all 1's (except a(1)) removed. - David James Sycamore, Jan 11 2024

References

• Paul J. McCarthy, Algebraic Extensions of Fields, Dover books, 1976, pages 40, 69

Links

• David Wasserman, Table of n, a(n) for n = 1..1000
• OEIS Wiki, LCM numeral system
• Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Crossrefs

Cf. A013595, A020500, A025476.

Cf. A000961, A014963, A051451.

Programs

Haskell

a025473 = a020639 . a000961 -- Reinhard Zumkeller, Aug 14 2013

Maple

cvm := proc(n, level) local f,opf; if n < 2 then RETURN() fi;
f := ifactors(n); opf := op(1,op(2,f)); if nops(op(2,f)) > 1 or
op(2,opf) <= level then RETURN() fi; op(1,opf) end:
A025473_list := n -> [1,seq(cvm(i,0),i=1..n)];
A025473_list(240); # Peter Luschny, Sep 21 2011

Mathematica

a = Join[{1}, Flatten[Table[If[PrimeQ[Apply[Plus, CoefficientList[Cyclotomic[n, x], x]]], Apply[Plus, CoefficientList[Cyclotomic[n, x], x]], {}], {n, 1, 1000}]]] (* Roger L. Bagula, Jul 08 2008 *)
Join[{1}, First@ First@# & /@ FactorInteger@ Select[Range@ 240, PrimePowerQ]] (* Robert G. Wilson v, Aug 17 2017 *)

PARI

print1(1); for(n=2,1e3, if(isprimepower(n,&p), print1(", "p))) \\ Charles R Greathouse IV, Apr 28 2014

Python

from sympy import primepi, integer_nthroot, primefactors
def A025473(n):
    if n == 1: return 1
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return primefactors(m)[0] # Chai Wah Wu, Aug 15 2024

Sage

def A025473_list(n) :
    R = [1]
    for i in (2..n) :
        if i.is_prime_power() :
            R.append(prime_divisors(i)[0])
    return R
A025473_list(239) # Peter Luschny, Feb 07 2012

Formula

a(n) = A006530(A000961(n)) = A020639(A000961(n)). - David Wasserman, Feb 16 2006

From Reinhard Zumkeller, Jun 26 2011: (Start)

A000961(n) = a(n)^A025474(n).

A056798(n) = a(n)^(2*A025474(n)).

A192015(n) = A025474(n)*a(n)^(A025474(n)-1). (End)

a(1) = A051451(1) ; for n > 1, a(n) = A051451(n)/A051451(n-1). - Peter Munn, Aug 11 2024

Extensions

Offset corrected by David Wasserman, Dec 22 2008