A001694 -id:A001694 - cp's OEIS frontend

A361936 Indices of the squares in the sequence of powerful numbers (A001694).

1, 2, 4, 5, 6, 9, 10, 11, 13, 14, 16, 19, 20, 21, 24, 26, 28, 29, 31, 33, 35, 36, 39, 40, 41, 44, 45, 46, 48, 50, 51, 55, 56, 59, 60, 61, 65, 67, 68, 70, 71, 73, 75, 76, 79, 81, 84, 85, 87, 88, 90, 92, 94, 96, 97, 100, 102, 104, 107, 109, 110, 111, 114, 116, 117, 119, 120

Offset: 1

Amiram Eldar, Mar 31 2023

nonn

Equivalently, the number of powerful numbers that do not exceed n^2.
The asymptotic density of this sequence is zeta(3)/zeta(3/2) = 1/A090699 = 0.460139... .
If k is a term of A336175 then a(k) and a(k+1) are consecutive integers, i.e., a(k+1) = a(k) + 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős and George Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), Vol. 7, No. 2 (1935), pp. 95-102.
Solomon W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77, No. 8 (1970), pp. 848-852.

Cf. A000290, A001694, A090699, A119241, A217038, A336175.

Position[Select[Range[5000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &], _?(IntegerQ[Sqrt[#]] &)] // Flatten

lista(kmax) = {my(c = 0); for(k = 1, kmax, if(ispowerful(k), c++); if(issquare(k), print1(c, ", "))); }

from math import isqrt
from sympy import integer_nthroot, factorint
def A361936(n):
    m = n**2
    return int(sum(isqrt(m//k**3) for k in range(1, integer_nthroot(m, 3)[0]+1) if all(d<=1 for d in factorint(k).values()))) # Chai Wah Wu, Sep 10 2024

a(n) = A217038(n^2).
a(n+1) - a(n) = A119241(n) + 1.
a(n) = (zeta(3/2)/zeta(3)) * n + O(n^(2/3)).

A368105 The number of bi-unitary 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, 3, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 12 2023

First differs from A095691 and A365552 at n = 32.

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

Cf. A013661, A001694, A005117, A062503, A286324,
Similar sequences: A005361, A323308, A360721.
Cf. A095691, A365552.

f[p_, e_] := If[e == 2 || OddQ[e], e, e -1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x%2 || x == 2, x, x-1), factor(n)[, 2]));

Multiplicative with a(p^e) = e if e = 2 or e is odd, and e-1 otherwise.
a(n) >= 1, with equality if and only if n is squarefree (A005117).
a(n) <= A286324(n), with equality if and only if n equals the square of a squarefree number (A062503).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 + 1/p^3 - 1/p^4 + 1/p^5) = 1.87133814920590891161... .

A084371 Squarefree kernels of powerful numbers (A001694).

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 3, 2, 6, 7, 2, 6, 3, 10, 6, 11, 5, 2, 6, 13, 14, 10, 6, 15, 3, 2, 6, 17, 6, 7, 19, 14, 10, 6, 21, 22, 10, 2, 23, 6, 5, 6, 15, 26, 3, 14, 10, 29, 6, 30, 31, 22, 6, 10, 2, 33, 15, 6, 34, 35, 6, 21, 11, 26, 37, 14, 38, 39, 14, 10, 41, 6, 42, 30, 43, 22, 6, 10

Offset: 1

Reinhard Zumkeller, Jun 23 2003

nonn

			A001694(11) = 64 = 2^6 -> a(11) = 2,
A001694(12) = 72 = 2^3 * 3^2 -> a(12) = 2*3 = 6,
A001694(13) = 81 = 3^4 -> a(13) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, The kernel of powerful numbers, International Mathematical Forum, Vol. 12, No. 15 (2017), pp. 721-730, Remark 2.5, p. 729.

Cf. A001694, A007947, A090699.

s = {1}; Do[f = FactorInteger[n]; If[Min @ f[[;;, 2]] > 1, AppendTo[s, Times @@ f[[;;, 1]]]], {n, 2, 10^4}]; s (* Amiram Eldar, Aug 22 2019 *)

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
lista(nn) = apply(x->rad(x), select(x->ispowerful(x), [1..nn])); \\ Michel Marcus, Aug 22 2019

from math import prod, isqrt
from sympy import mobius, integer_nthroot, primefactors
def A084371(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, j = n+x-squarefreepi(integer_nthroot(x,3)[0]), 0, 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 prod(primefactors(bisection(f,n,n))) # Chai Wah Wu, Sep 13 2024

a(n) = A007947(A001694(n)).
From Amiram Eldar, May 13 2023: (Start)
Sum_{A001694(k) < x} a(k) = (1/2) * x + o(x) (Jakimczuk, 2017). [corrected Sep 21 2024]
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(3)/zeta(3/2))^2/2 = 0.1058641473... . (End)

A115645 Powerful(1) numbers (A001694) that are sums of distinct factorials.

Original entry on oeis.org

1, 8, 9, 25, 27, 32, 121, 128, 144, 729, 841, 864, 5041, 5184, 40328, 41067, 45369, 45387, 46208, 46225, 363609, 403225, 3674889, 43954688, 6230694987, 1401602635449

Offset: 1

Giovanni Resta, Jan 27 2006

Factorials 0! and 1! are not considered distinct.

a(27) > 10^18, if it exists. - Amiram Eldar, Feb 24 2024

			6230694987 = 13!+10!+8!+7!+4!+2!+1! = 3^3*11^2*1381^2.

Index entries for sequences related to powerful numbers.

Intersection of A001694 and A059590.

Cf. A025494, A115644, A115646, A089359.

pwfQ[n_] := n==1 || Min[Last /@ FactorInteger@n] > 1; fac=Range[20]!;lst={}; Do[ n = Plus@@(fac*IntegerDigits[k, 2, 20]); If[pwfQ[n], AppendTo[lst, n]], {k, 2^20-1}]; lst
q[n_] := Module[{k = n, m = 2, r, ans = True}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r > 1, ans = False; Break[]]; m++]; ans]; With[{max = 2^20-1}, Select[Union[Flatten[Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]]], q]] (* Amiram Eldar, Feb 24 2024 *)

A115689 Powerful(1) numbers (A001694) whose digit reversal is a square.

Original entry on oeis.org

1, 4, 9, 100, 121, 144, 169, 400, 441, 484, 675, 676, 900, 961, 1000, 1089, 1800, 4000, 9000, 9801, 10000, 10201, 10404, 10609, 12100, 12321, 12544, 12769, 14400, 14641, 14884, 16900, 18000, 18252, 40000, 40401, 40804, 44100, 44521, 44944

Offset: 1

Giovanni Resta, Jan 31 2006

			18252=2^2*3^3*13^2 is powerful and 25281=159^2.

Harvey P. Dale, Table of n, a(n) for n = 1..160
Index entries for sequences related to powerful numbers

Cf. A001694, A115690.

Join[{1},Select[Range[50000],Min[FactorInteger[#][[All,2]]]>1&& IntegerQ[ Sqrt[ IntegerReverse[#]]]&]] (* Harvey P. Dale, Aug 08 2020 *)

A115690 Squares whose digit reversal is a powerful(1) number (A001694).

Original entry on oeis.org

1, 4, 9, 100, 121, 144, 169, 400, 441, 484, 576, 676, 900, 961, 1089, 9801, 10000, 10201, 10404, 10609, 12100, 12321, 12544, 12769, 14400, 14641, 14884, 16900, 25281, 27225, 40000, 40401, 40804, 44100, 44521, 44944, 48400, 48841, 57600

Offset: 1

Giovanni Resta, Jan 31 2006

If x is a member, then so is 100*x. - Robert Israel, Mar 16 2020

			25281=159^2 and 18252=2^2*3^3*13^2 is powerful.

Robert Israel, Table of n, a(n) for n = 1..1500

Subsequence of A115656.
A033294 is a subsequence, and the main entry for this sequence.
Cf. A001694, A115689.

filter:= proc(n) local L,i,x;
  L:= convert(n,base,10);
  x:=add(L[-i]*10^(i-1),i=1..nops(L));
  andmap(t -> t[2]>=2, ifactors(x)[2]):
end proc:select(filter, [seq(i^2,i=1..10^4)]); # Robert Israel, Mar 16 2020

is(k) = ispowerful(fromdigits(Vecrev(digits(k))));
select(is, vector(300, n, n^2)) \\ Michel Marcus, Nov 01 2022

Trivially, n^2 <= a(n) <= 100^(n-1). - Charles R Greathouse IV, Nov 01 2022

A115693 Powerful(1) numbers (A001694) whose digit reversal is a cube.

Original entry on oeis.org

1, 8, 72, 100, 343, 800, 1000, 1331, 7200, 8000, 10000, 34300, 72000, 80000, 100000, 133100, 343000, 720000, 800000, 1000000, 1030301, 1331000, 1367631, 3430000, 7200000, 8000000, 10000000, 13310000, 34300000, 72000000, 80000000, 100000000, 103030100, 133100000

Offset: 1

Giovanni Resta, Jan 31 2006

			72 = 2^3*3^2 is powerful and 27 = 3^3.

Amiram Eldar, Table of n, a(n) for n = 1..180
Index entries for sequences related to powerful numbers.

Cf. A001694, A115694.

Select[Join[{1},Select[Range[10000000],Min[Transpose[ FactorInteger[#]] [[2]]]>1&]],IntegerQ[Power[FromDigits[Reverse[IntegerDigits[#]]], (3)^-1]]&] (* Harvey P. Dale, Sep 18 2011 *)

a(28)-a(34) from Amiram Eldar, Feb 24 2024

A174172 Partials sums of A001694.

Original entry on oeis.org

1, 5, 13, 22, 38, 63, 90, 122, 158, 207, 271, 343, 424, 524, 632, 753, 878, 1006, 1150, 1319, 1515, 1715, 1931, 2156, 2399, 2655, 2943, 3232, 3556, 3899, 4260, 4652, 5052, 5484, 5925, 6409, 6909, 7421, 7950, 8526, 9151, 9799, 10474, 11150, 11879, 12663

Offset: 1

Jonathan Vos Post, Mar 10 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, The kernel of powerful numbers, International Mathematical Forum, Vol. 12, No. 15 (2017), pp. 721-730, Theorem 2.7, p. 729.

Cf. A001694, A090699.

Accumulate @ Select[Range[1000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &] (* Amiram Eldar, Jan 30 2023 *)

lista(kmax) = {my(s = 0); for(k = 1, kmax, if(k==1 || vecmin(factor(k)[, 2]) > 1, s += k; print1(s, ", ")));} \\ Amiram Eldar, May 13 2023

a(n) = Sum_{i=1..n} A001694(i).
a(n) ~ (zeta(3)^2/(3*zeta(3/2)^2)) * n^3. - Amiram Eldar, Jan 30 2023
a(n) = c * A001694(n)^(3/2) + o(A001694(n)^(3/2)), where c = zeta(3/2)/(3*zeta(3)) = 0.7244181041... (Jakimczuk, 2017). - Amiram Eldar, May 13 2023

A240591 The smaller of a pair of successive powerful numbers (A001694) without any prime number between them.

Original entry on oeis.org

8, 25, 32, 121, 288, 675, 1331, 1369, 1936, 2187, 2700, 3125, 5324, 6724, 9800, 10800, 12167, 15125, 32761, 39200, 48668, 70225, 79507, 88200, 97336, 107648, 143641, 156800, 212521, 228484, 235224, 280900, 312481, 332928, 456968, 465124, 574564, 674028, 744769, 829921, 830297, 857476, 877952, 940896

Offset: 1

Antonio Roldán, Apr 08 2014

nonn

			25 is in the sequence because A001694(6)=25, A001694(7)=27, without primes between them.

Amiram Eldar, Table of n, a(n) for n = 1..373 (terms below 10^15; terms 1..103 from Amiram Eldar, terms 104..235 from Chai Wah Wu)

Supersequence of A060355.

Cf. A001694, A240590.

Select[Partition[Join[{1},Select[Range[10^6],Min@FactorInteger[#][[All, 2]]> 1&]],2,1],PrimePi[#[[1]]]==PrimePi[#[[2]]]&][[All,1]] (* Harvey P. Dale, Mar 28 2018 *)

ispowerful(n)={local(h);if(n==1,h=1,h=(vecmin(factor(n)[, 2])>1));return(h)}
nextpowerful(n)={local(k);k=n+1;while(!ispowerful(k),k+=1);return(k)}
{for(i=1,10^6,if(ispowerful(i),if(nextprime(i)>=nextpowerful(i),print1(i, ", "))))}

from itertools import count, islice
from math import isqrt
from sympy import mobius, integer_nthroot, nextprime
def A240591_gen(): # generator of terms
    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, j = x-squarefreepi(integer_nthroot(x,3)[0]), 0, 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
    m = 1
    for n in count(2):
        k = bisection(lambda x:f(x)+n,m,m)
        if nextprime(m) > k:
            yield m
        m = k
A240591_list = list(islice(A240591_gen(),30)) # Chai Wah Wu, Sep 14 2024

A337044 Numbers k such that both k and sigma(k)=A000203(k) are powerful, i.e., are terms of A001694.

Original entry on oeis.org

1, 81, 343, 400, 9261, 27783, 32400, 137200, 189728, 224939, 972000, 1705636, 2205472, 3087000, 3591200, 3648100, 3704400, 7968032, 11113200, 13645088, 15350724, 15367968, 18220059, 21161304, 24240600, 25992000, 26680500, 29184800, 32832900, 48586824, 51595489

Offset: 1

Andrew Howroyd and Hugo Pfoertner, Aug 12 2020

nonn

From David A. Corneth, Aug 14 2020: (Start)
If coprime numbers k and m are in the sequence then k*m is in the sequence.
Up to 10^15, the largest prime divisor of a term is 178987 for which the product of the primes with multiplicity 1 of sigma(178987^2) is 16653 = 3 * 7 * 13 * 61. The second largest prime divisor is 25073 (for which sigma(25073^2) has a product of primes with multiplicity 1 of 341 = 11 * 31), which is quite a bit smaller than 178987. Can we somehow constrain the list of possible prime divisors to ease computation? (End)

David A. Corneth, Table of n, a(n) for n = 1..10324 (terms <= 10^18)
David A. Corneth, PARI program

Cf. A000203, A001694, A180090, A337045.

for(k=1, 60000000, if(ispowerful(k) && ispowerful(sigma(k)), print1(k, ", ")))

\\ See Corneth link \\ David A. Corneth, Aug 14 2020

cp's OEIS Frontend

A361936 Indices of the squares in the sequence of powerful numbers (A001694).

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A368105 The number of bi-unitary divisors of n that are powerful (A001694).

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A084371 Squarefree kernels of powerful numbers (A001694).

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A115645 Powerful(1) numbers (A001694) that are sums of distinct factorials.

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A115689 Powerful(1) numbers (A001694) whose digit reversal is a square.

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A115690 Squares whose digit reversal is a powerful(1) number (A001694).

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A115693 Powerful(1) numbers (A001694) whose digit reversal is a cube.

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A174172 Partials sums of A001694.

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