A025476 -id:A025476 - cp's OEIS frontend

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

David W. Wilson, Dec 11 1999

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

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

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

Cf. A013595, A020500, A025476.

Cf. A000961, A014963, A051451.

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

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

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 *)

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

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

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

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

Offset corrected by David Wasserman, Dec 22 2008

A048148 Distinct elements of A045948.

Original entry on oeis.org

1, 2, 4, 12, 24, 120, 360, 720, 5040, 10080, 30240, 332640, 1663200, 3326400, 43243200, 129729600, 259459200, 4410806400, 30875644800, 586637251200, 1173274502400, 26985313555200, 134926567776000, 404779703328000, 11738611396512000, 363896953291872000

Offset: 1

Labos Elemer

nonn

A001222(a(n)) = n - 1. - Eric Desbiaux, Apr 05 2018

Chai Wah Wu, Table of n, a(n) for n = 1..412 (terms 1..237 from Michael De Vlieger)

Cf. A003418, A034386, A002110, A000142, A049536.

Map[Function[m, Exp[Sum[MangoldtLambda@ n, {n, m}]]/Product[x, {x, Prime@ Range@ PrimePi@ m}]], Select[Range[10^3], Or[# == 1, And[PrimePowerQ@ #, ! PrimeQ@ #]] &] ] (* Michael De Vlieger, Aug 01 2017, after Fred Daniel Kline at A045948 *)

# uses program from A025476
from math import prod
def A048148(n): return prod(A025476(i) for i in range(1,n)) # Chai Wah Wu, Aug 15 2024

A025476(n) = a(n+1) / a(n). - Eric Desbiaux, Jun 22 2013

More terms from James Sellers, May 03 2000

More terms from Michael De Vlieger, Aug 01 2017

A216153 The partial products of a(n) are the distinct values of the exponential of the von Mangoldt function modified by restricting the divisors to prime divisors (A205957).

Original entry on oeis.org

1, 2, 6, 4, 3, 10, 24, 14, 15, 8, 54, 40, 21, 22, 96, 5, 26, 9, 56, 900, 16, 33, 34, 35, 216, 38, 39, 160, 1764, 88, 135, 46, 384, 7, 250, 51, 104, 486, 55, 224, 57, 58, 7200, 62, 189, 32, 65, 4356, 136, 69, 4900, 864, 74, 375, 152, 77, 6084, 640, 27, 82

Offset: 1

Peter Luschny, Sep 02 2012

The partial products of a(n) are A216152(n) which are the distinct values of the 'prime lcm(n)' A205957.
Let b(n) denote the nonprime numbers A018252(n).
If n = 1 then a(n) = b(n) = 1
else if a(n) < b(n) then
a(n) is a cototient of consecutive pure powers of primes (A053211),
b(n) is a prime power with exponent > 1 (A025475),
b(n)/a(n) is a prime root of n-th nontrivial prime power (A025476);
else if a(n) > b(n) then
b(n) is a number which is neither a prime power nor a semiprime (A102467);
else if a(n) = b(n) then
a(n) is the product of two distinct primes (A006881).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Peter Luschny, The von Mangoldt Transformation.

Cf. A205957, A205959, A216152.

A205957[n_] := Exp[-Sum[ MoebiusMu[p]*Log[k/p], {k, 1, n}, {p, FactorInteger[k][[All, 1]]}]]; nonPrime[1] = 1; nonPrime[n_] := Which[k0 = k /. FindRoot[ n + PrimePi[k] == k , {k, n}] // Floor; n+PrimePi[k0] == k0, k0 , n+PrimePi[k0+1] == k0+1, k0+1, n+PrimePi[k0+2] == k0+2, k0+2, True, k0]; a[1] = 1; a[n_] := A205957[nonPrime[n]] / A205957[nonPrime[n-1]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jun 27 2013 *)

def A216153(n):
    if n == 1 : return 1
    return A205957(A018252(n))/A205957(A018252(n-1))

a(n) = A205957(A018252(n))/A205957(A018252(n-1)) for n > 1, a(1) = 1.

A068389 Differences between the primes generating the n-th prime power.

Original entry on oeis.org

0, 1, -1, 3, -2, -1, 5, -5, 1, 8, -6, -3, 11, -10, -1, 15, -10, 12, -17, 21, -18, -2, 26, 2, -29, 9, 26, 4, 2, -41, 1, 10, 34, -40, 46, -48, 54, 2, -59, 65, -50, 54, 2, 6, -76, 16, 64, 6, -87, 95, 4, 2, 4, 2, -86, 90, -102, -6, 122, -125, 5, 124, 6, 2, -136

Offset: 0

Jon Perry, Mar 04 2002

sign

			The first prime powers are 4, 8, 9, 16, 25, ... These are generated by 2, 2, 3, 2, 5, ... (A025476) and the differences are 0, 1, -1, 3, ...

Cf. A025476.

More terms from Sean A. Irvine, Feb 13 2024

cp's OEIS Frontend

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

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A048148 Distinct elements of A045948.

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A216153 The partial products of a(n) are the distinct values of the exponential of the von Mangoldt function modified by restricting the divisors to prime divisors (A205957).

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Author

Keywords

Comments

Links

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Programs

Mathematica

Sage

Formula

A068389 Differences between the primes generating the n-th prime power.

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