A072777 -id:A072777 - cp's OEIS frontend

A375142 Numbers whose powerful part (A057521) is a power of a squarefree number that is larger than 1 (A072777).

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164

Offset: 1

Amiram Eldar, Aug 01 2024

Subsequence of A013929 and first differs from it at n = 27: A013929(27) = 72 = 2^3 * 3^2 is not a term of this sequence.
Numbers whose prime factorization has one distinct exponent that does not equal 1.
Numbers that are a product of a squarefree number (A005117) and a power of a different squarefree number that is not squarefree.
The asymptotic density of this sequence is Sum_{k>=2} (d(k)-1)/zeta(2) = 0.36113984820338109927..., where d(k) = zeta(k) * Product_{p prime} (1 + Sum_{i=k+1..2*k-1} (-1)^i/p^i), if k is even, and d(k) = (zeta(2*k)/zeta(k)) * Product_{p prime} (1 + 2/p^k + Sum_{i=k+1..2*k-1} (-1)^(i+1)/p^i) if k is odd > 1.

			12 = 2^2 * 3 is a term because its powerful part, 4 = 2^2, is a power of a squarefree number, 2, that is larger than 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical notes. III. Certain equally distributed sets of integers, Pacific Journal of Mathematics, Vol. 12, No. 1 (1962), pp. 77-84.
Index entries for sequences computed from exponents in factorization of n.

Cf. A005117, A057521.
Subsequence of A013929.
Subsequences: A067259, A072777, A190641, A336591.

q[n_] := Count[Union[FactorInteger[n][[;; , 2]]], _?(# > 1 &)] == 1; Select[Range[200], q]

is(k) = {my(e = select(x -> (x > 1), Set(factor(k)[,2]))); #e == 1;}

A368251 The number of nonsquarefree divisors of n that are powers of squarefree numbers (A072777).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 19 2023

nonn

First differs from A046660 and A066301 at n = 36, and from A183094 at n = 72.

Let b(n, k) be the sequence that counts the divisors of n that are k-th powers of squarefree numbers. Then, b(n, 1) = A034444(n), b(n, 2) = A323308(n), b(n, 3) = A368248(n). b(n, k) is multiplicative with b(p^e, k) = 2 if e >= k, and 1 otherwise. The asymptotic mean of b(n, k) for k >= 2 is lim_{m->oo} (1/m) * Sum_{n=1..m} b(n, k) = zeta(k)/zeta(2*k). Since a(n) = Sum_{k>=2} (b(n, k) - 1), the formula for the asymptotic mean of this sequence follows (see the Formula section).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A048105, A072777, A327527, A368250.
Cf. A034444, A323308, A368248.
Cf. A046660, A066301, A183094.

a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, 1 + Total[2^Accumulate[Count[e, #] & /@ Range[Max[e], 1, -1]] - 1] - 2^Length[e]]; a[1] = 0; Array[a, 100]

a(n) = {my(f = factor(n), e, m, h, c); if(n == 1, 0, e = f[,2]; m = vecmax(e); h = vector(m); for(i = 1,m, c = 0; for(j = 1, #e, if(e[j] == (m+1-i), c++)); h[i] = c); for(i = 2, m, h[i] += h[i-1]); for(i = 1, m, h[i] = 2^h[i]-1); 1 + vecsum(h) - 1<<#e);}

a(n) = A327527(n) - A034444(n).
a(n) = 0 if and only if n is squarefree (A005117).
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=2} (zeta(k)/zeta(2*k) - 1) = 0.848633... (A368250).

A001597 Perfect powers: m^k where m > 0 and k >= 2.

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, 1369, 1444, 1521, 1600, 1681, 1728, 1764

Offset: 1

N. J. A. Sloane

Might also be called the nontrivial powers. - N. J. A. Sloane, Mar 24 2018
See A175064 for number of ways to write a(n) as m^k (m >= 1, k >= 1). - Jaroslav Krizek, Jan 23 2010
a(1) = 1, for n >= 2: a(n) = numbers m such that sum of perfect divisors of x = m has no solution. Perfect divisor of n is divisor d such that d^k = n for some k >= 1. a(n) for n >= 2 is complement of A175082. - Jaroslav Krizek, Jan 24 2010
A075802(a(n)) = 1. - Reinhard Zumkeller, Jun 20 2011
Catalan's conjecture (now a theorem) is that 1 occurs just once as a difference, between 8 and 9.
For a proof of Catalan's conjecture, see the paper by Metsänkylä. - L. Edson Jeffery, Nov 29 2013
m^k is the largest number n such that (n^k-m)/(n-m) is an integer (for k > 1 and m > 1). - Derek Orr, May 22 2014
From Daniel Forgues, Jul 22 2014: (Start)
a(n) is asymptotic to n^2, since the density of cubes and higher powers among the squares and higher powers is 0. E.g.,
  a(10^1) = 49 (49% of 10^2),
  a(10^2) = 6400 (64% of 10^4),
  a(10^3) = 804357 (80.4% of 10^6),
  a(10^4) = 90706576 (90.7% of 10^8),
  a(10^n) ~ 10^(2n) - o(10^(2n)). (End)
A proper subset of A001694. - Robert G. Wilson v, Aug 11 2014
a(10^n): 1, 49, 6400, 804357, 90706576, 9565035601, 979846576384, 99066667994176, 9956760243243489, ... . - Robert G. Wilson v, Aug 15 2014

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 66.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section D9.
René Schoof, Catalan's Conjecture, Springer-Verlag, 2008, p. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David W. Wilson, Table of n, a(n) for n = 1..10000.
Abdelkader Dendane, Power (Exponential) Calculator.
H. W. Gould, Problem H-170, Fib. Quart., 8 (1970), p. 268, problem H-170.
Werner Hürlimann, A First Digit Theorem for Powers of Perfect Powers, Communications in Mathematics and Applications Vol. 5, No. 3 (2014), pp. 91-99.
Rafael Jakimczuk, On the Distribution of Perfect Powers, Journal of Integer Sequences, Vol. 14 (2011), Article 11.8.5.
Rafael Jakimczuk, Asymptotic Formulae for the n-th Perfect Power, Journal of Integer Sequences, Vol. 15 (2012), Article 12.5.5.
Holly Krieger and Brady Haran, Catalan's Conjecture, Numberphile video (2018).
Tauno Metsänkylä, Catalan's conjecture: another old Diophantine problem solved, Bull. Amer. Math. Soc. (NS), Vol. 41, No. 1 (2004), pp. 43-57.
Preda Mihǎilescu, Primary Cyclotomic Units and a Proof of Catalan's Conjecture, Journal für die Reine und Angewandte Mathematik, vol. 27 (2004), pp. 167-195.
Donald J. Newman, A Problem Seminar, Springer; see Problem #72.
M. A. Nyblom, A Counting Function for the Sequence of Perfect Powers, Austral. Math. Soc. Gaz. 33 (2006), 338-343.
Hugo Pfoertner, 1010196 perfect powers up to 10^12, compressed 7z archive, 3.3 MB (2023).
Alf van der Poorten, Remarks on the sequence of 'perfect' powers.
Michel Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 2003-2004.
Michel Waldschmidt, Lecture on the abc conjecture and some of its consequences, Abdus Salam School of Mathematical Sciences (ASSMS), Lahore, 6th World Conference on 21st Century Mathematics 2013.
Eric Weisstein's World of Mathematics, Perfect Power.

Complement of A007916.
Subsequence of A072103; A072777 is a subsequence.
Union of A075109 and A075090.
Cf. A023055, A023057, A025478, A070428, A072102, A074981, A076404, A239728, A239870, A097054, A089579, A089580.
There are four different sequences which may legitimately be called "prime powers": A000961 (p^k, k >= 0), A246655 (p^k, k >= 1), A246547 (p^k, k >= 2), A025475 (p^k, k=0 and k >= 2), and which are sometimes confused with the present sequence.
First differences give A053289.

import Data.Map (singleton, findMin, deleteMin, insert)
a001597 n = a001597_list !! (n-1)
(a001597_list, a025478_list, a025479_list) =
   unzip3 $ (1, 1, 2) : f 9 (3, 2) (singleton 4 (2, 2)) where
   f zz (bz, ez) m
    | xx < zz = (xx, bx, ex) :
                f zz (bz, ez+1) (insert (bx*xx) (bx, ex+1) $ deleteMin m)
    | xx > zz = (zz, bz, 2) :
                f (zz+2*bz+1) (bz+1, 2) (insert (bz*zz) (bz, 3) m)
    | otherwise = f (zz+2*bz+1) (bz+1, 2) m
    where (xx, (bx, ex)) = findMin m  --  bx ^ ex == xx
-- Reinhard Zumkeller, Mar 28 2014, Oct 04 2012, Apr 13 2012

[1] cat [n : n in [2..1000] | IsPower(n) ];

isA001597 := proc(n)
    local e ;
    e := seq(op(2,p),p=ifactors(n)[2]) ;
    return ( igcd(e) >=2 or n =1 ) ;
end proc:
A001597 := proc(n)
    option remember;
    local a;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA001597(a) then
                return a ;
            end if;
         end do;
    end if;
end proc:
seq(A001597(n),n=1..70) ; # R. J. Mathar, Jun 07 2011
N:= 10000: # to get all entries <= N
sort({1,seq(seq(a^b, b = 2 .. floor(log[a](N))), a = 2 .. floor(sqrt(N)))}); # Robert FERREOL, Jul 18 2023

min = 0; max = 10^4;  Union@ Flatten@ Table[ n^expo, {expo, Prime@ Range@ PrimePi@ Log2@ max}, {n, Floor[1 + min^(1/expo)], max^(1/expo)}] (* T. D. Noe, Apr 18 2011; slightly modified by Robert G. Wilson v, Aug 11 2014 *)
perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; Select[Range@ 1765, perfectPowerQ] (* Ant King, Jun 29 2013; slightly modified by Robert G. Wilson v, Aug 11 2014 *)
nextPerfectPower[n_] := If[n == 1, 4, Min@ Table[ (Floor[n^(1/k)] + 1)^k, {k, 2, 1 + Floor@ Log2@ n}]]; NestList[ nextPerfectPower, 1, 55] (* Robert G. Wilson v, Aug 11 2014 *)
Join[{1},Select[Range[2000],GCD@@FactorInteger[#][[All,2]]>1&]] (* Harvey P. Dale, Apr 30 2018 *)

{a(n) = local(m, c); if( n<2, n==1, c=1; m=1; while( cMichael Somos, Aug 05 2009 */

is(n)=ispower(n) || n==1 \\ Charles R Greathouse IV, Sep 16 2015

list(lim)=my(v=List(vector(sqrtint(lim\=1),n,n^2))); for(e=3,logint(lim,2), for(n=2,sqrtnint(lim,e), listput(v,n^e))); Set(v) \\ Charles R Greathouse IV, Dec 10 2019

from sympy import perfect_power
def ok(n): return n==1 or perfect_power(n)
print([m for m in range(1, 1765) if ok(m)]) # Michael S. Branicky, Jan 04 2021

import sympy
class A001597() :
    def _init_(self) :
        self.a = [1]
    def at(self, n):
        if n <= len(self.a):
            return self.a[n-1]
        else:
            cand = self.at(n-1)+1
            while sympy.perfect_power(cand) == False:
                cand += 1
            self.a.append(cand)
            return cand
a001597 = A001597()
for n in range(1,20):
    print(a001597.at(n)) # R. J. Mathar, Mar 28 2023

from sympy import mobius, integer_nthroot
def A001597(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 kmax # Chai Wah Wu, Aug 13 2024

def A001597_list(n) :
    return [k for k in (1..n) if k.is_perfect_power()]
A001597_list(1764) # Peter Luschny, Feb 03 2012

Goldbach showed that Sum_{n >= 2} 1/(a(n)-1) = 1.
Formulas from postings to the Number Theory List by various authors, 2002:
Sum_{i >= 2} Sum_{j >= 2} 1/i^j = 1;
Sum_{k >= 2} 1/(a(k)+1) = Pi^2 / 3 - 5/2;
Sum_{k >= 2} 1/a(k) = Sum_{n >= 2} mu(n)(1- zeta(n)) approx = 0.87446436840494... See A072102.
For asymptotics see Newman.
For n > 1: gcd(exponents in prime factorization of a(n)) > 1, cf. A124010. - Reinhard Zumkeller, Apr 13 2012
a(n) ~ n^2. - Thomas Ordowski, Nov 04 2012
a(n) = n^2 - 2*n^(5/3) - 2*n^(7/5) + (13/3)*n^(4/3) - 2*n^(9/7) + 2*n^(6/5) - 2*n^(13/11) + o(n^(13/11)) (Jakimczuk, 2012). - Amiram Eldar, Jun 30 2023

Minor corrections from N. J. A. Sloane, Jun 27 2010

A072774 Powers of squarefree numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97

Offset: 1

Reinhard Zumkeller, Jul 10 2002

nonn

Essentially the same as A062770. - R. J. Mathar, Sep 25 2008
Numbers m such that in canonical prime factorization all prime exponents are identical: A124010(m,k) = A124010(m,1) for k = 2..A000005(m). - Reinhard Zumkeller, Apr 06 2014
Heinz numbers of uniform partitions. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 16 2018

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

Complement of A059404.
Cf. A072775, A072776, A072777 (subsequence), A005117, A072778, A124010, A329332 (tabular arrangement), A384667 (characteristic function).
A subsequence of A242414.
Cf. A000005, A000009, A000837, A007916, A013661, A047966, A052409, A052410, A072774, A078374, A289023, A289509, A300486, A302491, A302796, A302979.

import Data.Map (empty, findMin, deleteMin, insert)
import qualified Data.Map.Lazy as Map (null)
a072774 n = a072774_list !! (n-1)
(a072774_list, a072775_list, a072776_list) = unzip3 $
   (1, 1, 1) : f (tail a005117_list) empty where
   f vs'@(v:vs) m
    | Map.null m || xx > v = (v, v, 1) :
                             f vs (insert (v^2) (v, 2) m)
    | otherwise = (xx, bx, ex) :
                  f vs' (insert (bx*xx) (bx, ex+1) $ deleteMin m)
    where (xx, (bx, ex)) = findMin m
-- Reinhard Zumkeller, Apr 06 2014

isA := n -> n=1 or is(1 = nops({seq(p[2], p in ifactors(n)[2])})):
select(isA, [seq(1..97)]);  # Peter Luschny, Jun 10 2025

Select[Range[100], Length[Union[FactorInteger[#][[All, 2]]]] == 1 &] (* Geoffrey Critzer, Mar 30 2015 *)

is(n)=ispower(n,,&n); issquarefree(n) \\ Charles R Greathouse IV, Oct 16 2015

from math import isqrt
from sympy import mobius, integer_nthroot
def A072774(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))-1
    def f(x): return n-2+x-sum(g(integer_nthroot(x,k)[0]) for k in range(1,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 kmax # Chai Wah Wu, Aug 19 2024

a(n) = A072775(n)^A072776(n).
Sum_{n>=1} 1/a(n)^s = 1 + Sum_{k>=1} (zeta(k*s)/zeta(2*k*s)-1) for s > 1. - Amiram Eldar, Mar 20 2025
a(n)/n ~ Pi^2/6 (A013661). - Friedjof Tellkamp, Jun 09 2025

A100778 Integer powers of primorial numbers.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 30, 32, 36, 64, 128, 210, 216, 256, 512, 900, 1024, 1296, 2048, 2310, 4096, 7776, 8192, 16384, 27000, 30030, 32768, 44100, 46656, 65536, 131072, 262144, 279936, 510510, 524288, 810000, 1048576, 1679616, 2097152, 4194304, 5336100

Offset: 1

Amarnath Murthy, Nov 28 2004

Smallest squarefree numbers or their powers with distinct prime signatures. Or least numbers with prime signatures (p*q*r*...)^k, where p,q,r,... are primes and k is a whole number.
Also Heinz numbers of uniform integer partitions whose union is an initial interval of positive integers. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). The sequence of all uniform integer partitions whose Heinz numbers belong to the sequence begins: (1), (11), (12), (111), (1111), (123), (11111), (1122), (111111), (1111111), (1234), (111222), (11111111), (111111111), (112233), (1111111111). - Gus Wiseman, Dec 26 2018
From Amiram Eldar, Sep 26 2023: (Start)
Intersection of A025487 and A072774.
The distinct terms of A046523(A072774(n)) in ascending orders.
The k-th power of the n-th primorial number, A002110(n)^k, has (k+1)^n divisors which are the set of the (k+1)-free prime(n)-smooth numbers. (End)

			10 is not a term as 6 is a member with the same prime signature 10 > 6.
216 is a term as 216 = (2*3)^3. 243 is not a term as 32 represents that prime signature.

David A. Corneth, Table of n, a(n) for n = 1..8606 (terms <= 10^1000)

Cf. A000961, A001597, A002110, A007947, A025487, A046523, A055932, A056239, A057588, A072774, A072777, A112798, A304250, A319169, A322792, A322793.

unintQ[n_]:=And[SameQ@@Last/@FactorInteger[n],Length[FactorInteger[n]]==PrimePi[FactorInteger[n][[-1,1]]]];
Select[Range[1000],unintQ] (* Gus Wiseman, Dec 26 2018 *)

Sum_{n>=1} 1/a(n) = 1 + Sum_{n>=1} 1/A057588(n) = 2.2397359032... - Amiram Eldar, Oct 20 2020; corrected by Hal M. Switkay and Amiram Eldar, Apr 12 2021

More terms and simpler definition from Ray Chandler, Nov 29 2004

A303606 Powers of composite squarefree numbers that are not squarefree.

Original entry on oeis.org

36, 100, 196, 216, 225, 441, 484, 676, 900, 1000, 1089, 1156, 1225, 1296, 1444, 1521, 1764, 2116, 2601, 2744, 3025, 3249, 3364, 3375, 3844, 4225, 4356, 4761, 4900, 5476, 5929, 6084, 6724, 7225, 7396, 7569, 7776, 8281, 8649, 8836, 9025, 9261, 10000, 10404, 10648, 11025, 11236

Offset: 1

Ilya Gutkovskiy, Apr 26 2018

nonn

			196 is in the sequence because 196 = 2^2*7^2.
4900 is in the sequence because 4900 = 2^2*5^2*7^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Power.
Eric Weisstein's World of Mathematics, Squarefree.

Intersection of A024619 and A072777.
Intersection of A072774 and A126706.
Intersection of A013929 and A182853.
Cf. A000469, A001597, A005117, A120944, A286708.

Select[Range[12000], Length[Union[FactorInteger[#][[All, 2]]]] == 1 && ! SquareFreeQ[#] && ! PrimePowerQ[#] &]
seq[max_] := Module[{sp = Select[Range[Floor@Sqrt[max]], SquareFreeQ[#] && PrimeNu[#] > 1 &], s = {}}, Do[s = Join[s, sp[[k]]^Range[2, Floor@Log[sp[[k]], max]]], {k, 1, Length[sp]}]; Union@s]; seq[10^4] (* Amiram Eldar, Feb 12 2021 *)

from math import isqrt
from sympy import mobius, primepi, integer_nthroot
def A303606(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))-primepi(x))
    def f(x): return n-3+x+(y:=x.bit_length())-sum(g(integer_nthroot(x,k)[0]) for k in range(2,y))
    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 kmax # Chai Wah Wu, Aug 19 2024

Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A120944(n)-1)*A120944(n)) = Sum_{k>=2} (zeta(k)/zeta(2*k) - P(k) - 1) = 0.07547719891508850482..., where P(k) is the prime zeta function. - Amiram Eldar, Feb 12 2021

A340674 Numbers of the form s^(2^e), where s is a squarefree number, and e >= 1.

Original entry on oeis.org

4, 9, 16, 25, 36, 49, 81, 100, 121, 169, 196, 225, 256, 289, 361, 441, 484, 529, 625, 676, 841, 900, 961, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2209, 2401, 2601, 2809, 3025, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489, 4761, 4900

Offset: 1

Antti Karttunen, Jan 31 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000079, A005117, A050376, A209229, A302777.
Positions of terms larger than 2 in A340673 (also in A340675), and of terms larger than 1 in A340676.
Subsequence of A072777 and of A340682.

Select[Range[10^4], Length[(u = Union[FactorInteger[#][[;; , 2]]])] == 1 && u[[1]] > 1 && u[[1]] == 2^IntegerExponent[u[[1]], 2] &] (* Amiram Eldar, Feb 08 2021 *)

A209229(n) = (n && !bitand(n,n-1));
isA340674(n) = { my(b,e); (((e=ispower(n,,&b))>0)&&issquarefree(b)&&A209229(e)); };

from math import isqrt
from sympy import mobius, integer_nthroot
def A340674(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return n+x-sum(g(integer_nthroot(x,1<Chai Wah Wu, Jun 01 2025

Sum_{n>=1} 1/a(n) = Sum_{k>=1} (zeta(2^k)/zeta(2^(k+1))-1) = 0.6018231854... - Amiram Eldar, Feb 08 2021

A368250 Decimal expansion of Sum_{k>=2} (zeta(k)/zeta(2*k) - 1).

Original entry on oeis.org

8, 4, 8, 6, 3, 3, 8, 6, 7, 9, 6, 4, 8, 8, 3, 6, 3, 2, 6, 8, 4, 9, 0, 0, 1, 2, 0, 9, 0, 4, 3, 0, 4, 6, 2, 9, 6, 0, 0, 1, 6, 6, 4, 4, 6, 8, 8, 1, 7, 5, 5, 1, 7, 1, 6, 7, 9, 6, 2, 0, 3, 0, 9, 0, 0, 3, 6, 5, 4, 2, 2, 1, 3, 7, 1, 3, 0, 2, 1, 2, 9, 1, 8, 8, 6, 6, 3, 4, 8, 1, 0, 1, 1, 5, 3, 7, 0, 2, 0, 6, 3, 4, 4, 3, 7

Offset: 0

Amiram Eldar, Dec 19 2023

			0.84863386796488363268490012090430462960016644688175...

Michael I. Shamos, Shamos's catalog of the real numbers, 2011. See p. 637.

Cf. A008683, A072777, A368249, A368251.

evalf(sum(Zeta(k)/Zeta(2*k) - 1, k = 2 .. infinity), 120);

PARI
```
sumpos(k=2, zeta(k)/zeta(2*k) - 1)
```

Equals Sum_{k>=2} mu(k)^2/(k*(k-1)) = Sum_{k>=2} 1/A368249(k).
Equals Sum_{k>=1} 1/A072777(k).
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A368251(k).

A322793 Proper powers of primorial numbers.

Original entry on oeis.org

4, 8, 16, 32, 36, 64, 128, 216, 256, 512, 900, 1024, 1296, 2048, 4096, 7776, 8192, 16384, 27000, 32768, 44100, 46656, 65536, 131072, 262144, 279936, 524288, 810000, 1048576, 1679616, 2097152, 4194304, 5336100, 8388608, 9261000

Offset: 1

Gus Wiseman, Dec 26 2018

nonn

A primorial number is a product of the first n primes, for some n.

Also Heinz numbers of non-strict uniform integer partitions whose union is an initial interval of positive integers. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of all non-strict uniform integer partitions whose Heinz numbers belong to the sequence begins: (11), (111), (1111), (11111), (2211), (111111), (1111111), (222111), (11111111), (111111111), (332211), (1111111111), (22221111).

Michael De Vlieger, Table of n, a(n) for n = 1..8255

Cf. A000961, A001597, A001694, A002110, A025487, A047966, A055932, A056239, A072774, A072777, A100778, A112798, A304250, A322792.

unintpropQ[n_]:=And[SameQ@@Last/@FactorInteger[n],FactorInteger[n][[1,2]]>1,Length[FactorInteger[n]]==PrimePi[FactorInteger[n][[-1,1]]]];
Select[Range[10000],unintpropQ]
(* Second program: *)
nn = 2^24; k = 1; P = 2; Union@ Reap[While[j = 2; While[P^j < nn, Sow[P^j]; j++]; j > 2, k++; P *= Prime[k]]][[-1, 1]] (* Michael De Vlieger, Oct 04 2023 *)

Sum_{n>=1} 1/a(n) = Sum_{k>=1} 1/(A002110(k)*(A002110(k)-1)) = 0.53450573145072369022... . - Amiram Eldar, Mar 10 2024

A303661 Powers of squarefree semiprimes that are not squarefree.

Original entry on oeis.org

36, 100, 196, 216, 225, 441, 484, 676, 1000, 1089, 1156, 1225, 1296, 1444, 1521, 2116, 2601, 2744, 3025, 3249, 3364, 3375, 3844, 4225, 4761, 5476, 5929, 6724, 7225, 7396, 7569, 7776, 8281, 8649, 8836, 9025, 9261, 10000, 10648, 11236, 12321, 13225, 13924, 14161, 14884

Offset: 1

Ilya Gutkovskiy, Apr 28 2018

nonn

			1089 is in the sequence because 1089 = 3^2*11^2.
1296 is in the sequence because 1296 = 2^4*3^4.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Squarefree.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A006881, A013929, A072774, A072777, A085155, A123711, A200511, A246547, A303606.

Select[Range[15000], Length[Union[FactorInteger[#][[All, 2]]]] == 1 && PrimeNu[#] == 2 && ! SquareFreeQ[#] &]
seq[max_] := Module[{sp = Select[Range[Floor@Sqrt[max]], SquareFreeQ[#] && PrimeNu[#] == 2 &], s = {}}, Do[s = Join[s, sp[[k]]^Range[2, Floor@Log[sp[[k]], max]]], {k, 1, Length[sp]}]; Union@s]; seq[10000] (* Amiram Eldar, Feb 12 2021 *)

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A303661(n):
    def g(x): return int(-(t:=primepi(s:=isqrt(x)))-(t*(t-1)>>1)+sum(primepi(x//k) for k in primerange(1, s+1)))
    def f(x): return n-1+x-sum(g(integer_nthroot(x,k)[0]) 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 kmax # Chai Wah Wu, Aug 19 2024

Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A006881(n)-1)*A006881(n)) = Sum_{k>=2} (P(k)^2 - P(2*k))/2 = 0.07160601536406295068..., where P(k) is the prime zeta function. - Amiram Eldar, Feb 12 2021

cp's OEIS Frontend

A375142 Numbers whose powerful part (A057521) is a power of a squarefree number that is larger than 1 (A072777).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A368251 The number of nonsquarefree divisors of n that are powers of squarefree numbers (A072777).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A001597 Perfect powers: m^k where m > 0 and k >= 2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

Python

Python

Python

Sage

Formula

Extensions

A072774 Powers of squarefree numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

A100778 Integer powers of primorial numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A303606 Powers of composite squarefree numbers that are not squarefree.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica