A025479 -id:A025479 - cp's OEIS frontend

A322969 Sum of the largest exponents A025479 of the first n perfect powers > 1.

2, 5, 7, 11, 13, 16, 21, 23, 25, 31, 35, 37, 39, 42, 49, 51, 53, 55, 58, 60, 65, 73, 75, 77, 80, 82, 84, 86, 88, 97, 99, 101, 105, 107, 113, 115, 117, 119, 121, 124, 134, 136, 138, 140, 144, 147, 149, 151, 153, 155, 157, 160, 162, 164, 166, 168, 179, 181, 188

Offset: 1

Hugo Pfoertner, Jan 01 2019

nonn

			a(1) = 2 because the first perfect power 4 = 2^2,
a(2) = 5: added exponent 3 from 8 = 2^3,
a(3) = 7: added exponent 2 from 9 = 3^2,
a(4) = 11: added largest exponent 4 from 16=2^4.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A001597, A025479, A076408.

Union@ Accumulate@ Table[If[Set[e, GCD @@ #[[All, -1]]] > 1, e, 0] &@ FactorInteger@ n, {n, 4, 2400}] (* Michael De Vlieger, Jan 01 2019 *)

my(s=0); for(k=1, 3^7, if(j=ispower(k), print1(s+=j, ", ")))

A340586 Perfect powers such that the two immediately adjacent perfect powers both have a largest exponent A025479 equal to 2.

Original entry on oeis.org

8, 16, 169, 216, 343, 400, 441, 512, 625, 729, 841, 900, 1156, 1444, 1521, 1600, 1728, 1849, 1936, 2048, 2401, 2601, 2744, 2916, 3125, 3249, 3375, 3600, 3721, 3844, 4096, 4356, 4489, 4624, 4761, 4913, 5184, 5329, 5476, 5625, 5832, 6084, 6241, 6561, 6859, 7056

Offset: 1

Hugo Pfoertner, Jan 14 2021

nonn

			a(1) = 8 because its neighboring perfect powers 4 = 2^2 and 9 = 3^2 both have the largest exponent 2.
9 is not in the sequence because both exponents of the neighboring perfect powers 8 = 2^3 and 16 = 2^4 are > 2.
a(2) = 16: neighbors 9 = 3^2 and 25 = 5^2 satisfy the exponent condition.
Next excluded terms: 25 (16 = 2^4, 27 = 3^3), 27 (32 = 2^5), 32 (27 = 3^3), 36 (32 = 2^5), 49 (64 = 2^6), 64 (81 = 3^4), 81 (64 = 2^6), 100 (81 = 3^4), 121 (125 = 5^3), 125 (128 = 2^7), 128 (125 = 5^3), 144 (128 = 2^7).
a(3) = 169: neighbors 144 = 12^2 and 196 = 14^2 satisfy the exponent condition.

Cf. A000290, A001597, A025479, A111245, A153158, A340587, A340640, A340641.

a340586(limit)={my(p2=999,p1=2,n2=1,n1=4);for(n=5,limit,my(p0=ispower(n));if(p0>1,if(p2+p0==4,print1(n1,", "));n2=n1;n1=n;p2=p1;p1=p0))};
a340586(7500)

A340640 Perfect powers such that the two immediately adjacent perfect powers have at least one largest exponent A025479 greater than 2.

Original entry on oeis.org

4, 9, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 196, 225, 243, 256, 289, 324, 361, 484, 529, 576, 676, 784, 961, 1000, 1024, 1089, 1225, 1296, 1331, 1369, 1681, 1764, 2025, 2116, 2187, 2197, 2209, 2304, 2500, 2704, 2809, 3025, 3136, 3364, 3481, 3969

Offset: 1

Hugo Pfoertner, Jan 14 2021

nonn

			a(1) = 4 because the next perfect power is 8 = 2^3, i.e., its exponent is > 2.
a(2) = 9: the exponents of the neighbors 8 = 2^3 and 16 = 2^4 are both > 2.
16 is not in the sequence because both neighboring perfect powers 9 = 3^2 and 25 = 5^2 have exponents 2.
Neighbors with exponents > 2 of the next terms: a(3) = 25 (16 = 2^3), a(4) = 27 (32 = 2^5), a(5) = 32 (27 = 3^3), a(6) = 36 (32 = 2^5), a(7) = 49 (64 = 2^6), a(8) = 64 (81 = 3^4).

Cf. A000290, A001597, A025479, A111245, A153158, A340586, A340641.

a340640(limit)={my(p2=999, p1=2, n2=1, n1=4); for(n=5, limit, my(p0=ispower(n)); if(p0>1, if(p2+p0>4, print1(n1, ", ")); n2=n1; n1=n; p2=p1; p1=p0))};
a340640(5000)

A052409 a(n) = largest integer power m for which a representation of the form n = k^m exists (for some k).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Eric W. Weisstein

nonn

Greatest common divisor of all prime-exponents in canonical factorization of n for n>1: a(n)>1 iff n is a perfect power; a(A001597(k))=A025479(k). - Reinhard Zumkeller, Oct 13 2002
a(1) set to 0 since there is no largest finite integer power m for which a representation of the form 1 = 1^m exists (infinite largest m). - Daniel Forgues, Mar 06 2009
A052410(n)^a(n) = n. - Reinhard Zumkeller, Apr 06 2014
Positions of 1's are A007916. Smallest base is given by A052410. - Gus Wiseman, Jun 09 2020

			n = 72 = 2*2*2*3*3: GCD[exponents] = GCD[3,2] = 1. This is the least n for which a(n) <> A051904(n), the minimum of exponents.
For n = 10800 = 2^4 * 3^3 * 5^2, GCD[4,3,2] = 1, thus a(10800) = 1.

Daniel Forgues, Table of n, a(n) for n = 1..100000
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564
Eric Weisstein's World of Mathematics, Power
Eric Weisstein's World of Mathematics, Perfect Power
Index entries for sequences computed from exponents in factorization of n

Cf. A052410, A005361, A051903, A072411-A072414, A124010, A075802, A072776, A270492.
Apart from the initial term essentially the same as A253641.
Differs from A051904 for the first time at n=72, where a(72) = 1, while A051904(72) = 2.
Differs from A158378 for the first time at n=10800, where a(10800) = 1, while A158378(10800) = 2.
Cf. A000005, A000961, A001597, A052410, A303386, A327501.

a052409 1 = 0
a052409 n = foldr1 gcd $ a124010_row n
-- Reinhard Zumkeller, May 26 2012

# See link.
#
a:= n-> igcd(map(i-> i[2], ifactors(n)[2])[]):
seq(a(n), n=1..120);  # Alois P. Heinz, Oct 20 2019

Table[GCD @@ Last /@ FactorInteger[n], {n, 100}] (* Ray Chandler, Jan 24 2006 *)

a(n)=my(k=ispower(n)); if(k, k, n>1) \\ Charles R Greathouse IV, Oct 30 2012

from math import gcd
from sympy import factorint
def A052409(n): return gcd(*factorint(n).values()) # Chai Wah Wu, Aug 31 2022

(define (A052409 n) (if (= 1 n) 0 (gcd (A067029 n) (A052409 (A028234 n))))) ;; Antti Karttunen, Aug 07 2017

a(1) = 0; for n > 1, a(n) = gcd(A067029(n), a(A028234(n))). - Antti Karttunen, Aug 07 2017

More terms from Labos Elemer, Jun 17 2002

A025478 Least roots of perfect powers (A001597).

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 3, 2, 6, 7, 2, 3, 10, 11, 5, 2, 12, 13, 14, 6, 15, 3, 2, 17, 18, 7, 19, 20, 21, 22, 2, 23, 24, 5, 26, 3, 28, 29, 30, 31, 10, 2, 33, 34, 35, 6, 11, 37, 38, 39, 40, 41, 12, 42, 43, 44, 45, 2, 46, 3, 13, 47, 48, 7, 50, 51, 52, 14, 53, 54, 55, 5, 56, 57, 58, 15, 59, 60, 61, 62

Offset: 1

David W. Wilson

			a(5) = 2 because pp(5) = 16 = 2^4 (not 4^2 as we take the smallest base).

Daniel Forgues, Table of n, a(n) for n=1..10000

Cf. A052410 (least root), A001597 (perfect powers).

Cf. A025479 (largest exponents of perfect powers), A070228.

a025478 n = a025478_list !! (n-1)  -- a025478_list defined in A001597.
-- Reinhard Zumkeller, Mar 11 2014

pp = Select[ Range[5000], Apply[GCD, Last[ Transpose[ FactorInteger[ # ]]]] > 1 &]; f[n_] := Block[{b = 2}, While[ !IntegerQ[ Log[b, pp[[n]]]], b++ ]; b]; Join[{1}, Table[ f[n], {n, 2, 80}]]
(* Second program: *)
Prepend[DeleteCases[#, 0], 1] &@ Table[If[Set[e, GCD @@ #[[All, -1]]] > 1, Power[n, 1/e], 0] &@ FactorInteger@ n, {n, 4000}]  (* Michael De Vlieger, Apr 25 2017 *)

lista(kmax) = {my(r, e); print1(1, ", "); for(k = 1, kmax, e = ispower(k, , &r); if(e > 0, print1(r, ", ")));} \\ Amiram Eldar, Sep 07 2024

from math import gcd
from sympy import mobius, integer_nthroot, factorint
def A025478(n):
    if n == 1: return 1
    def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return integer_nthroot(kmax, gcd(*factorint(kmax).values()))[0] # Chai Wah Wu, Aug 13 2024

a(n) = A052410(A001597(n)).

(i) a(n) < n for n > 2. (ii) a(n)/n is bounded and lim sup a(n)/n must be around 0.7. (iii) Sum_{k=1..n} a(k) seems to be asymptotic to c*n^2 with c around 0.29. (iv) a(n) = 2 if n is in A070228 (proof seems self-evident), hence there is no asymptotic expression for a(n) (just the average in (iii)). - Benoit Cloitre, Oct 14 2002

Definition edited and cross-reference added by Daniel Forgues, Mar 10 2009

A076408 Sum of first n perfect powers.

Original entry on oeis.org

1, 5, 13, 22, 38, 63, 90, 122, 158, 207, 271, 352, 452, 573, 698, 826, 970, 1139, 1335, 1551, 1776, 2019, 2275, 2564, 2888, 3231, 3592, 3992, 4433, 4917, 5429, 5958, 6534, 7159, 7835, 8564, 9348, 10189, 11089, 12050, 13050, 14074, 15163, 16319, 17544

Offset: 1

Reinhard Zumkeller, Oct 09 2002

nonn

			a(8) = 1+4+8+9+16+25+27+32 = 122.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Perfect Powers.

Cf. A001597, A025479, A076407, A322969.

Accumulate[Join[{1},Select[Range[1500],GCD@@FactorInteger[#][[All,2]]>1&]]] (* Harvey P. Dale, Feb 12 2023 *)

print1(s=1,", ");for(k=2,1225,if(ispower(k),print1(s+=k,", "))) \\ Hugo Pfoertner, Jan 01 2019

A076399 Number of prime factors of n-th perfect power (with repetition).

Original entry on oeis.org

0, 2, 3, 2, 4, 2, 3, 5, 4, 2, 6, 4, 4, 2, 3, 7, 6, 2, 4, 6, 4, 5, 8, 2, 6, 3, 2, 6, 4, 4, 9, 2, 8, 4, 4, 6, 6, 2, 6, 2, 6, 10, 4, 4, 4, 8, 3, 2, 4, 4, 8, 2, 9, 6, 2, 6, 6, 11, 4, 7, 3, 2, 10, 4, 6, 4, 6, 6, 2, 8, 4, 5, 8, 4, 4, 6, 2, 8, 2, 4, 6, 12, 4, 6, 2, 6, 4, 6, 3, 2, 10, 2, 4, 6, 6, 9, 4, 6, 2, 10, 8

Offset: 1

Reinhard Zumkeller, Oct 09 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, The Number of Prime Factors on Average in Certain Integer Sequences, Journal of Integer Sequences, Vol. 25 (2022), Article 22.2.3.
Eric Weisstein's World of Mathematics, Perfect Powers.
Eric Weisstein's World of Mathematics, Prime Factor.

Cf. A001222, A025478, A025479, A076398, A076400, A083342.

a076399 n = a001222 (a025478 n) * a025479 n
-- Reinhard Zumkeller, Mar 28 2014

PrimeOmega[Select[Range[10^4], # == 1 || GCD @@ FactorInteger[#][[;; , 2]] > 1 &]] (* Amiram Eldar, Feb 18 2023 *)

is(n) = n==1 || ispower(n);
apply(bigomega, select(is, [1..5000])) \\ Amiram Eldar, Feb 18 2023

from sympy import mobius, integer_nthroot, primeomega
def A076399(n):
    def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return int(primeomega(kmax)) # Chai Wah Wu, Aug 14 2024

a(n) = A001222(A001597(n)).
a(n) = A001222(A025478(n))*A025479(n).
Sum_{A001597(k) <= x} a(k) = 2*sqrt(x)*log(log(x)) + 2*(B_2 - log(2))*sqrt(x) + O(sqrt(x)/log(x)), where B_2 = A083342 (Jakimczuk and Lalín, 2022). - Amiram Eldar, Feb 18 2023, corrected Sep 21 2024

A340642 Perfect powers such that the two immediately adjacent perfect powers both have an exponent greater than 2.

Original entry on oeis.org

4, 9, 25, 225, 676, 2116, 6724, 7921, 8100, 16641, 104329, 131044, 160801, 176400, 372100, 389376, 705600, 4096576, 7306209, 7884864, 47444544, 146385801, 254817369, 373262400, 607622500, 895804900, 1121580100, 1330936324, 1536875209, 2097182025, 2258435529, 2749953600

Offset: 1

Hugo Pfoertner, Jan 14 2021

nonn

Apparently, all known terms (checked through 10^18) are squares with maximum exponent 2, i.e., terms of A111245 (squares that are not a higher power). This would imply that of 3 immediately adjacent perfect powers, at least one is a term of A111245. Is there a known counterexample of 3 consecutive perfect powers, none of which is in A111245?

			The first terms, assuming 1 being at least a cube:
.
  n   p1  x^p1  p2  a(n)  p3  z^p3
                   =y^p2
  1  >2     1   2     4   3     8
  2   3     8   2     9   4    16
  3   4    16   2    25   3    27
  4   3   216   2   225   5   243
  5   4   625   2   676   6   729

Hugo Pfoertner, Table of n, a(n) for n = 1..181 (terms < 10^18)

Cf. A001597, A025479, A076467, A097054, A111245, A153158, A340643, A340700, A340701.

a340642(limit)={my(p2=999, p1=2, n2=1, n1=4); for(n=5, limit, my(p0=ispower(n)); if(p0>1, if(p2>2&p0>2, print1(n1,", ")); n2=n1; n1=n; p2=p1; p1=p0))};
a340642(50000000)

A093771 Perfect powers for which the exponent is a prime number: solutions to {A051409(x) is prime}.

Original entry on oeis.org

4, 8, 9, 25, 27, 32, 36, 49, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 289, 324, 343, 361, 400, 441, 484, 529, 576, 676, 784, 841, 900, 961, 1000, 1089, 1156, 1225, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849, 1936, 2025, 2048, 2116, 2187

Offset: 1

Labos Elemer, Apr 19 2004

nonn

A010051(A025479(n)) = 1. - Reinhard Zumkeller, Mar 28 2014

			All 2-,3-,5-,7th ... powers are here, 4-,6-,8th etc. powers are excluded
from A001597.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001157, A025479, A052409, A001597.

a093771 n = a093771_list !! (n-1)
a093771_list = [a001597 x | x <- [2..], a010051 (a025479 x) == 1]
-- Reinhard Zumkeller, Mar 28 2014

ffi[x_] :=Flatten[FactorInteger[x]] ep[x_] :=Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] lf[x_] :=Length[FactorInteger[x]] Do[If[PrimeQ[Apply[GCD, ep[n]]], Print[n]], {n, 2, 10000}]

is(n)=isprime(ispower(n)) \\ Charles R Greathouse IV, Oct 19 2015

GCD of prime-exponents in canonical factorization of n is prime.

A157985 Perfect powers (m^k where m is an integer and k >= 2) multiplied by -1 when m is prime for largest k (m^k thus a prime power).

Original entry on oeis.org

1, -4, -8, -9, -16, -25, -27, -32, 36, -49, -64, -81, 100, -121, -125, -128, 144, -169, 196, 216, 225, -243, -256, -289, 324, -343, -361, 400, 441, 484, -512, -529, 576, -625, 676, -729, 784, -841, 900, -961, 1000, -1024, 1089, 1156, 1225, 1296, -1331

Offset: 1

Daniel Forgues, Mar 10 2009

sign

The rather strange phrase "largest k" in the definition refers to the fact that there can be several ways to write a number in the form m^k. - N. J. A. Sloane, Jan 01 2019

Daniel Forgues, Table of n, a(n) for n=1..10000

Cf. A001597 (perfect powers), A025479 (largest exponents of perfect powers).
Cf. A025478 (least roots of perfect powers).
Cf. A157986.

a(n) = {m^k}_n * (-1)^(Pi(m) - Pi(m-1)) where {m^k}_n is the n-th perfect power with positive integer base m corresponding to largest integer exponent k and Pi(m) is the prime counting function evaluated at m.

a(n) = A001597(n) * (-1)^(Pi(m) - Pi(m-1)), with m = A001597(n)^(1/A025479(n)).

cp's OEIS Frontend

A322969 Sum of the largest exponents A025479 of the first n perfect powers > 1.

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A340586 Perfect powers such that the two immediately adjacent perfect powers both have a largest exponent A025479 equal to 2.

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A340640 Perfect powers such that the two immediately adjacent perfect powers have at least one largest exponent A025479 greater than 2.

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A052409 a(n) = largest integer power m for which a representation of the form n = k^m exists (for some k).

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A025478 Least roots of perfect powers (A001597).

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A076408 Sum of first n perfect powers.

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A076399 Number of prime factors of n-th perfect power (with repetition).

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