A025473 - OEIS

%I #68 Aug 16 2024 08:37:19

%S 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,

%T 2,67,71,73,79,3,83,89,97,101,103,107,109,113,11,5,127,2,131,137,139,

%U 149,151,157,163,167,13,173,179,181,191,193,197,199,211,223,227,229,233,239

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

%C 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

%C "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

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

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

%H David Wasserman, <a href="/A025473/b025473.txt">Table of n, a(n) for n = 1..1000</a>

%H OEIS Wiki, <a href="/wiki/LCM_numeral_system">LCM numeral system</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic Polynomial</a>

%F a(n) = A006530(A000961(n)) = A020639(A000961(n)). - _David Wasserman_, Feb 16 2006

%F From _Reinhard Zumkeller_, Jun 26 2011: (Start)

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

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

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

%F a(1) = A051451(1) ; for n > 1, a(n) = A051451(n)/A051451(n-1). - _Peter Munn_, Aug 11 2024

%p cvm := proc(n, level) local f,opf; if n < 2 then RETURN() fi;

%p f := ifactors(n); opf := op(1,op(2,f)); if nops(op(2,f)) > 1 or

%p op(2,opf) <= level then RETURN() fi; op(1,opf) end:

%p A025473_list := n -> [1,seq(cvm(i,0),i=1..n)];

%p A025473_list(240); # _Peter Luschny_, Sep 21 2011

%t 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 *)

%t Join[{1}, First@ First@# & /@ FactorInteger@ Select[Range@ 240, PrimePowerQ]] (* _Robert G. Wilson v_, Aug 17 2017 *)

%o (Sage)

%o def A025473_list(n) :

%o     R = [1]

%o     for i in (2..n) :

%o         if i.is_prime_power() :

%o             R.append(prime_divisors(i)[0])

%o     return R

%o A025473_list(239) # _Peter Luschny_, Feb 07 2012

%o (Haskell)

%o a025473 = a020639 . a000961 -- _Reinhard Zumkeller_, Aug 14 2013

%o (PARI) print1(1); for(n=2,1e3, if(isprimepower(n,&p), print1(", "p))) \\ _Charles R Greathouse IV_, Apr 28 2014

%o (Python)

%o from sympy import primepi, integer_nthroot, primefactors

%o def A025473(n):

%o     if n == 1: return 1

%o     def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))

%o     m, k = n, f(n)

%o     while m != k:

%o         m, k = k, f(k)

%o     return primefactors(m)[0] # _Chai Wah Wu_, Aug 15 2024

%Y Cf. A013595, A020500, A025476.

%Y Cf. A000961, A014963, A051451.

%K easy,nonn,nice

%O 1,2

%A _David W. Wilson_, Dec 11 1999

%E Offset corrected by _David Wasserman_, Dec 22 2008