A001694 -id:A001694 - cp's OEIS frontend

A085254 Numbers having a unique representation as sum of two powerful numbers (A001694).

2, 5, 8, 9, 10, 12, 13, 16, 18, 20, 24, 25, 26, 28, 29, 31, 32, 34, 35, 37, 43, 44, 45, 48, 53, 54, 58, 59, 61, 63, 64, 74, 82, 88, 90, 91, 96, 98, 99, 100, 101, 106, 112, 121, 122, 124, 126, 127, 128, 134, 135, 140, 141, 146, 149, 150, 155, 161, 162, 169, 171

Offset: 1

Reinhard Zumkeller, Jun 23 2003

nonn

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

Cf. A001694, A076871, A085252, A085253, A085255.

With[{m = 180}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; Position[BinCounts[Select[Plus @@@ Union[Sort /@ Tuples[pow, {2}]], # <= m &], {1, m, 1}], 1] // Flatten] (* Amiram Eldar, Jan 30 2023 *)

A085252(a(n)) = 1.

A085255 Numbers having at least two representations as a sum of two powerful numbers (A001694).

Original entry on oeis.org

17, 33, 36, 40, 41, 50, 52, 57, 65, 68, 72, 73, 76, 80, 81, 85, 89, 97, 104, 108, 109, 113, 116, 117, 125, 129, 130, 132, 133, 136, 137, 144, 145, 148, 152, 153, 157, 160, 164, 170, 172, 177, 180, 185, 189, 193, 197, 200, 201, 204, 205, 208, 209, 216, 221

Offset: 1

Reinhard Zumkeller, Jun 23 2003

nonn

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

Cf. A001694, A076871, A085252, A085253, A085254.

With[{m = 222}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; Position[BinCounts[Select[Plus @@@ Union[Sort /@ Tuples[pow, {2}]], # <= m &], {1, m, 1}], ?(# > 1 &)] // Flatten] (* _Amiram Eldar, Jan 30 2023 *)

A085252(a(n)) > 1.

A115651 Powerful(1) numbers (A001694) which are the sum of distinct double factorials (A006882).

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 49, 64, 72, 108, 121, 125, 128, 169, 392, 400, 432, 441, 500, 512, 961, 968, 972, 1125, 1331, 1352, 1444, 3844, 3888, 3969, 4225, 4232, 4356, 4900, 4913, 5184, 5292, 5324, 10404, 10800, 10952, 11449, 11881, 14283, 14400

Offset: 1

Giovanni Resta, Jan 28 2006

nonn

Double factorials 0!! and 1!! are not considered distinct. Note that double factorial (n!!) is different from (n!)!.

			392 = 2^3*7^2 = 8!! + 4!!.

Index entries for sequences related to powerful numbers

Cf. A001694, A006882, A115652, A115653, A115654.

A115656 Both n and the reverse of n are powerful(1) numbers (A001694).

Original entry on oeis.org

1, 4, 8, 9, 27, 72, 100, 121, 144, 169, 343, 400, 441, 484, 576, 675, 676, 800, 900, 961, 1000, 1089, 1331, 1800, 2700, 3087, 4000, 7200, 7803, 8000, 9000, 9801, 10000, 10201, 10404, 10609, 12100, 12321, 12544, 12769, 14400, 14641, 14884

Offset: 1

Giovanni Resta, Jan 28 2006

			9801=3^4*11^2 and 1089=3^2*11^2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..6228

Cf. A001694, A115655, A085751.

is(n)=ispowerful(n) && ispowerful(subst(Polrev(digits(n)),'x,10)) \\ Charles R Greathouse IV, Sep 16 2014

has(n)=ispowerful(subst(Polrev(digits(n)),'x,10))
list(lim)=my(v=List(),t,t2); for(m=1,lim^(1/3), t=m^3; for(n=1,sqrtint(lim\t), if(has(t2=t*n^2), listput(v,t2)))); Set(v) \\ Charles R Greathouse IV, Sep 16 2014

A323332 The Dedekind psi function values of the powerful numbers, A001615(A001694(n)).

Original entry on oeis.org

1, 6, 12, 12, 24, 30, 36, 48, 72, 56, 96, 144, 108, 180, 216, 132, 150, 192, 288, 182, 336, 360, 432, 360, 324, 384, 576, 306, 648, 392, 380, 672, 720, 864, 672, 792, 900, 768, 552, 1152, 750, 1296, 1080, 1092, 972, 1344, 1440, 870, 1728, 2160, 992, 1584

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

The sum of the reciprocals of all the terms of this sequence is Pi^2/6 (A013661).

The asymptotic density of a sequence S that possesses the property that an integer k is a term if and only if its powerful part, A057521(k) is a term, is (1/zeta(2)) * Sum_{n>=1, A001694(n) is a term of S} 1/a(n). Examples for such sequences are the e-perfect numbers (A054979), the exponential abundant numbers (A129575), and other sequences listed in the Crossrefs section. - Amiram Eldar, May 06 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, A property of Dedekind's psi-function, Proceedings of the American Mathematical Society, Vol. 12, No. 6 (1961), p. 996.

Cf. A001615, A001694, A013661, A082695, A112526.

Sequences whose density can be calculated using this sequence: A054979, A129575, A307958, A308053, A321147, A322858, A323310, A328135, A339936, A340109, A364990, A382061, A383693, A383695, A383697.

psi[1]=1; psi[n_] := n * Times@@(1+1/Transpose[FactorInteger[n]][[1]]); psi /@ Join[{1}, Select[Range@ 1200, Min@ FactorInteger[#][[All, 2]] > 1 &]] (* after T. D. Noe at A001615 and Harvey P. Dale at A001694 *)

from math import isqrt, prod
from sympy import mobius, integer_nthroot, primefactors
def A323332(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x-squarefreepi(integer_nthroot(x,3)[0]), 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    a = primefactors(m:=bisection(f,n,n))
    return m*prod(p+1 for p in a)//prod(a) # Chai Wah Wu, Sep 14 2024

A356871 Primitive coreful abundant numbers (second definition): coreful abundant numbers (A308053) that are powerful numbers (A001694).

Original entry on oeis.org

72, 108, 144, 200, 216, 288, 324, 400, 432, 576, 648, 784, 800, 864, 900, 972, 1000, 1152, 1296, 1568, 1600, 1728, 1764, 1800, 1936, 1944, 2000, 2304, 2592, 2700, 2704, 2744, 2916, 3136, 3200, 3456, 3528, 3600, 3872, 3888, 4000, 4356, 4500, 4608, 4900, 5000, 5184

Offset: 1

Amiram Eldar, Sep 02 2022

nonn

For squarefree numbers k, csigma(k) = k, where csigma(k) is the sum of the coreful divisors of k (A057723). Thus, if m is a term (csigma(m) > 2*m) and k is a squarefree number coprime to k, then csigma(k*m) = csigma(k) * csigma(m) = k * csigma(m) > 2*k*m, so k*m is a coreful abundant number. Therefore, the sequence of coreful abundant numbers (A308053) can be generated from this sequence by multiplying with coprime squarefree numbers. The asymptotic density of the coreful abundant numbers can be calculated from this sequence (see comment in A308053).

			72 is a term since csigma(72) = 168 > 2 * 72, and 72 = 2^3 * 3^2 is powerful.

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

Intersection of A001694 and A308053.
A339940 is a subsequence.
Cf. A057723.
Similar sequences: A307959, A328136.

f[p_, e_] := (p^(e+1)-1)/(p-1)-1; s[1] = 1; s[n_] := If[AllTrue[(fct = FactorInteger[n])[[;;, 2]], #>1 &], Times @@ f @@@ fct, 0]; seq={}; Do[If[s[n] > 2*n, AppendTo[seq, n]], {n, 1, 5000}]; seq

A360721 a(n) is the number of infinitary divisors of n that are powerful (A001694).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Feb 18 2023

Cf. A000120, A001694, A037445, A077609, A360723.

Similar sequences: A005361 (number of powerful divisors), A323308 (number of unitary powerful divisors).

f[p_, e_] := 2^DigitCount[e, 2, 1] - Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 2^hammingweight(f[i, 2]) - f[i, 2]%2);}

Multiplicative with a(p^e) = 2^A000120(e) - (e mod 2).
a(n) <= A037445(n) with equality if and only if n is a square.
a(n) <= A005361(n) with equality if and only if n is not in A360723.
Sum_{k=1..n} a(k) ~ c * n, where c = Product_{p prime} ((1-1/p) * Sum_{k>=1} ((2^A000120(k)- k mod 2)/p^k)) = 1.72717... .

A363176 Primitive abundant numbers (A091191) that are powerful numbers (A001694).

Original entry on oeis.org

196, 15376, 342225, 570375, 1032256, 3172468, 4636684, 63126063, 99198099, 117234117, 171991125, 280495504, 319600125, 327921075, 404529741, 581549787, 635689593, 762155163, 1029447225, 1148667664, 1356949503, 1435045924, 1501500375, 1558495125, 1596961444, 1757705625

Offset: 1

Amiram Eldar, May 19 2023

nonn

The least cubefull (A036966) term is a(158) = 26376098024367 = 3^6 * 7^4 * 13^3 * 19^3.

A363175 is a subsequence. Terms that are not in A363175: 196, 15376, 1032256, 274810802176, 1125882727038976, 72057319160283136, ... .

Amiram Eldar, Table of n, a(n) for n = 1..2151 (terms below 10^18)
Wikipedia, Powerful number.
Wikipedia, Primitive abundant number.

Intersection of A001694 and A091191.
A363175 is a subsequence.
Subsequence of A363169.
Cf. A036966.

f1[p_, e_] := (p^(e + 1) - 1)/(p^(e + 1) - p^e); f2[p_, e_] := (p^(e + 1) - p)/(p^(e + 1) - 1);
primAbQ[n_] := (r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f <= 2;
seq[max_] := Module[{pow = Union[Flatten[Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]]]}, Select[Rest[pow], primAbQ]]; seq[10^10]

isPrimAb(n) = {my(f = factor(n), r, p, e); r = sigma(f, -1); r > 2 && vecmax(vector(#f~, i, p = f[i, 1]; e = f[i, 2]; (p^(e + 1) - p)/(p^(e + 1) - 1))) * r <= 2; }
lista(lim) = {my(pow = List(), t); for(j=1, sqrtnint(lim\1, 3), for(i=1, sqrtint(lim\j^3), listput(pow, i^2*j^3))); select(x->isPrimAb(x), Set(pow)); }

A240590 Number of primes between successive powerful numbers (A001694).

Original entry on oeis.org

2, 2, 0, 2, 3, 0, 2, 0, 4, 3, 2, 2, 3, 3, 2, 0, 1, 3, 5, 5, 2, 1, 1, 5, 1, 7, 0, 5, 2, 4, 5, 1, 5, 2, 7, 3, 2, 2, 6, 9, 4, 4, 0, 7, 8, 2, 7, 4, 4, 8, 1, 1, 4, 4, 9, 7, 2, 1, 9, 10, 6, 1, 0, 2, 0, 9, 12, 7, 4, 12, 6, 5, 4, 5, 12, 0, 8, 3, 3, 10, 8, 0, 2, 13, 2, 13, 10, 10, 1, 15, 0, 7, 9, 9, 3, 13, 7, 4, 0, 7, 5, 4, 13, 2

Offset: 1

Antonio Roldán, Apr 08 2014

nonn

			a(9) = 4 because A001694(9) = 36, A001694(10) = 49, and there are 4 primes between them: 37, 41, 43 and 47.

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

Cf. A001694, A240591.

ispowerful(n)={local(h);if(n==1,h=1,h=(vecmin(factor(n)[, 2])>1));return(h)}
proxpowerful(n)={local(k);k=n+1;while(!ispowerful(k),k+=1);return(k)}
{for(i=1,5000,if(ispowerful(i),m=proxpowerful(i);p=primepi(m)-primepi(i);print1(p, ", ")))}

from math import isqrt
from sympy import mobius, integer_nthroot, primepi
def A240590(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x,3)[0])-l
        return c
    return -primepi(a:=bisection(f,n,n))+primepi(bisection(lambda x:f(x)+1,a,a)) # Chai Wah Wu, Sep 15 2024

A306458 a(n) = A001694(n)/A007947(A001694(n)), the powerful numbers divided by their squarefree kernel.

Original entry on oeis.org

1, 2, 4, 3, 8, 5, 9, 16, 6, 7, 32, 12, 27, 10, 18, 11, 25, 64, 24, 13, 14, 20, 36, 15, 81, 128, 48, 17, 54, 49, 19, 28, 40, 72, 21, 22, 50, 256, 23, 96, 125, 108, 45, 26, 243, 56, 80, 29, 144, 30, 31, 44, 162, 100, 512, 33, 75, 192, 34, 35, 216, 63, 121, 52

Offset: 1

Amiram Eldar, Feb 17 2019

A permutation of the positive integers.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001694, A007947, A064549.

N:= 10^4: # to get terms corresponding to powerful numbers <= N
rad:= n -> convert(numtheory:-factorset(n), `*`):
S:= {1}:
p:= 1:
do
  p:= nextprime(p);
  if p^2 > N then break fi;
  S:= S union map(t -> seq(t*p^i,i=2..floor(log[p](N/t))),select(`<=`,S,N/p^2));
od:
map(t -> t/rad(t), sort(convert(S,list))); # Robert Israel, Mar 20 2019

p=Join[{1}, Select[ Range@ 12500, Min@ FactorInteger[#][[All, 2]] > 1 &]]; rad[n_] := Times @@ (First@# & /@ FactorInteger@ n);  p/(rad/@p) (* after Harvey P. Dale at A001694 and Robert G. Wilson v at A007947 *)

apply(x->(x/factorback(factorint(x)[, 1])), select(x->ispowerful(x), vector(1600, k, k))) \\ Michel Marcus, Feb 17 2019

from math import isqrt, prod
from sympy import mobius, integer_nthroot, primefactors
def A306458(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x-squarefreepi(integer_nthroot(x,3)[0]), 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    return (m:=bisection(f,n,n))//prod(primefactors(m)) # Chai Wah Wu, Sep 14 2024

A064549(a(n)) = A001694(n).

cp's OEIS Frontend

A085254 Numbers having a unique representation as sum of two powerful numbers (A001694).

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A085255 Numbers having at least two representations as a sum of two powerful numbers (A001694).

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A115651 Powerful(1) numbers (A001694) which are the sum of distinct double factorials (A006882).

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A115656 Both n and the reverse of n are powerful(1) numbers (A001694).

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A323332 The Dedekind psi function values of the powerful numbers, A001615(A001694(n)).

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A356871 Primitive coreful abundant numbers (second definition): coreful abundant numbers (A308053) that are powerful numbers (A001694).

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A360721 a(n) is the number of infinitary divisors of n that are powerful (A001694).

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A363176 Primitive abundant numbers (A091191) that are powerful numbers (A001694).

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A240590 Number of primes between successive powerful numbers (A001694).

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