A001694 -id:A001694 - cp's OEIS frontend

A085252 Number of ways to write n as sum of two powerful numbers (A001694).

0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 2, 1, 0, 0, 2, 2, 0, 1, 1, 1, 0, 0, 1, 0, 2, 0, 2, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 1, 1, 2, 0, 0, 2, 0, 0, 0, 2, 2, 1, 0, 2, 0, 0, 0, 2, 2, 1, 0, 0, 2, 0, 0, 1, 2, 1, 1, 0, 0, 0, 0, 1, 2, 1, 1, 1, 1, 0

Offset: 1

Reinhard Zumkeller, Jun 23 2003

nonn

			a(81) = 2: 81 = 9 + 72 = A001694(4) + A001694(12) = 32 + 49 = A001694(8) + A001694(10).

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

Cf. A001694, A076871, A085253, A085254, A085255.

With[{m = 120}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; BinCounts[Select[Plus @@@ Union[Sort /@ Tuples[pow, {2}]], # <= m &], {1, m, 1}]] (* Amiram Eldar, Jan 30 2023 *)

a(A085253(n)) = 0.
a(A076871(n)) > 0.
a(A085254(n)) = 1.
a(A085255(n)) > 1.

A361177 Exponentially powerful numbers: numbers whose exponents in their canonical prime factorization are all powerful numbers (A001694).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102

Offset: 1

Amiram Eldar, Mar 03 2023

nonn

First differs from it subsequence A197680 at n = 167: a(167) = 256 is not a term of A197680.

The asymptotic density of this sequence is Product_{p prime} ((1 - 1/p)*(1 + Sum_{i>=1} 1/p^A001694(i))) = 0.6427901996... .

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

Cf. A001694.

Similar sequences: A197680, A209061, A138302, A268335.

powQ[n_] := n == 1 || Min[FactorInteger[n][[;; , 2]]] > 1; Select[Range[100], AllTrue[FactorInteger[#][[;;, 2]], powQ] &]

ispow(n) = {n == 1 || vecmin(factor(n)[,2]) > 1; }
is(n) = {my(e = factor(n)[, 2]); if(n == 1, return(1)); for(i=1, #e, if(!ispow(e[i]), return(0))); 1;}

A363169 Powerful abundant numbers: numbers that are both powerful (A001694) and abundant (A005101).

Original entry on oeis.org

36, 72, 100, 108, 144, 196, 200, 216, 288, 324, 392, 400, 432, 500, 576, 648, 784, 800, 864, 900, 968, 972, 1000, 1152, 1296, 1352, 1372, 1568, 1600, 1728, 1764, 1800, 1936, 1944, 2000, 2304, 2500, 2592, 2700, 2704, 2744, 2916, 3136, 3200, 3456, 3528, 3600, 3872, 3888, 4000

Offset: 1

Amiram Eldar, May 19 2023

nonn

The least odd term is a(90) = 11025, and the least term that is coprime to 6 is 1382511906801025.

Are there two consecutive integers in this sequence? There are none below 10^22.

			36 = 2^2 * 3^2 is a term since it is powerful, and sigma(36) = 91 > 2*36 = 72.

Amiram Eldar, Table of n, a(n) for n = 1..10534 (terms not exceeding 10^8)

Intersection of A001694 and A005101.
Cf. A180114, A363170, A363171.
Subsequences: A307959, A328136, A356871.

Select[Range[4000], DivisorSigma[-1, #] > 2 && Min[FactorInteger[#][[;;, 2]]] > 1 &]

is(n) = { my(f = factor(n)); n > 1 && vecmin(f[, 2]) > 1 && sigma(f, -1) > 2; }

A363189 Indices of the odd terms in the sequence of powerful numbers (A001694).

Original entry on oeis.org

1, 4, 6, 7, 10, 13, 16, 17, 20, 24, 25, 28, 30, 31, 35, 39, 41, 43, 45, 48, 51, 56, 57, 60, 62, 63, 65, 68, 71, 75, 79, 82, 83, 84, 87, 90, 94, 97, 98, 99, 102, 103, 105, 107, 110, 114, 117, 120, 122, 125, 127, 129, 133, 138, 141, 142, 144, 145, 148, 151, 152

Offset: 1

Amiram Eldar, May 21 2023

The asymptotic density of this sequence is (2-sqrt(2))/(3-sqrt(2)) = 0.369398... .

If A001694(k) is a term of A363190 then k and k+1 are consecutive integers in this sequence.

			The first 6 powerful numbers are 1, 4, 8, 9, 16 and 25. 1, 9 and 25 are odd and their positions in the sequence are 1, 4 and 6, respectively.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Teerapat Srichan, The odd/even dichotomy for the set of square-full numbers, Applied Mathematics E-Notes, Vol. 20 (2020), pp. 528-531.
Wikipedia, Powerful number.

Cf. A001694, A062739, A363190, A363191, A363192.

Position[Select[Range[7000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &], _?(OddQ[#] &)] // Flatten

lista(kmax) = {my(c = 0); for(k = 1, kmax, if(ispowerful(k), c++; if(k%2, print1(c, ", ")))); }

A363194 Number of divisors of the n-th powerful number A001694(n).

Original entry on oeis.org

1, 3, 4, 3, 5, 3, 4, 6, 9, 3, 7, 12, 5, 9, 12, 3, 4, 8, 15, 3, 9, 12, 16, 9, 6, 9, 18, 3, 15, 4, 3, 12, 15, 20, 9, 9, 12, 10, 3, 21, 5, 20, 12, 9, 7, 15, 18, 3, 24, 27, 3, 12, 18, 16, 11, 9, 12, 24, 9, 9, 25, 12, 4, 12, 3, 12, 9, 9, 18, 21, 3, 28, 27, 36, 3, 15

Offset: 1

Amiram Eldar, May 21 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (8).

Cf. A000005, A001694, A180114, A306458, A343443.

Similar sequences: A072048, A076400, A363195.

DivisorSigma[0, Select[Range[3000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]]

apply(numdiv, select(ispowerful, [1..3000]))

from itertools import count, islice
from math import prod
from sympy import factorint
def A363194_gen(): # generator of terms
    for n in count(1):
        f = factorint(n).values()
        if all(e>1 for e in f):
            yield prod(e+1 for e in f)
A363194_list = list(islice(A363194_gen(),20)) # Chai Wah Wu, May 21 2023

a(n) = A000005(A001694(n)).
Sum_{A001694(k) < x} a(k) = c_1 * sqrt(x) * log(x)^2 + c_2 * sqrt(x) * log(x) + c_3 * sqrt(x) + O(x^(5/12 + eps)), where c_1, c_2 and c_3 are constants. c_1 = Product_{p prime} (1 + 4/p^(3/2) - 1/p^2 - 6/p^(5/2) + 2/p^(7/2))/8 = 0.516273682988566836609... . [corrected Sep 21 2024]
a(n) = A343443(A306458(n)). - Amiram Eldar, Sep 01 2023

A258567 a(1) = 1; thereafter a(n) = smallest prime factor of the powerful number A001694(n).

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 3, 2, 2, 7, 2, 2, 3, 2, 2, 11, 5, 2, 2, 13, 2, 2, 2, 3, 3, 2, 2, 17, 2, 7, 19, 2, 2, 2, 3, 2, 2, 2, 23, 2, 5, 2, 3, 2, 3, 2, 2, 29, 2, 2, 31, 2, 2, 2, 2, 3, 3, 2, 2, 5, 2, 3, 11, 2, 37, 2, 2, 3, 2, 2, 41, 2, 2, 2, 43, 2, 2, 2, 3, 2, 2, 3

Offset: 1

Reinhard Zumkeller, Jun 06 2015

nonn

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

Cf. A001694, A020639, A258599, A258568, A258569, A258570, A258571.

Haskell
```
a258567 = a020639 . a001694
```

Table[If[Min[(f = FactorInteger[n])[[;; , 2]]] > 1 || n == 1, f[[1, 1]], Nothing], {n, 1, 3000}] (* Amiram Eldar, Jan 30 2023 *)

from math import isqrt
from sympy import mobius, integer_nthroot, primefactors
def A258567(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 min(primefactors(bisection(f,n,n)),default=1) # Chai Wah Wu, Sep 10 2024

a(n) = A020639(A001694(n)).

a(A258599(n)) = A000040(n) and a(m) != A000040(n) for m < A258599(n).

Definition made more precise by N. J. A. Sloane, Apr 29 2024

A349062 Powerful numbers (A001694) with a record gap to the next powerful number.

Original entry on oeis.org

1, 4, 9, 16, 36, 49, 81, 144, 169, 256, 289, 441, 529, 576, 676, 729, 900, 1024, 1156, 1225, 1372, 1444, 1600, 1849, 2209, 2401, 2916, 3600, 3721, 4096, 4356, 4624, 4761, 5041, 5625, 6400, 6561, 7225, 7396, 8281, 9025, 9409, 9801, 11025, 11236, 11664, 12544, 14400

Offset: 1

Amiram Eldar, Nov 07 2021

nonn

This sequence is infinite since the asymptotic density of the powerful numbers is 0.
The corresponding record gaps are 3, 4, 7, 9, 13, 15, 19, 25, 27, 32, 35, 43, ...
Apparently, most of the terms are squares. The nonsquare terms are 1372, 465125, 4879688, ... (A371191).

			The sequence of powerful numbers begins with 1, 4, 8, 9, 16, 25, 27, 32, 36 and 49. The differences between these terms are 3, 4, 1, 7, 9, 2, 5, 4 and 13. The record values, 3, 4, 7, 9 and 13 occur after the powerful numbers 1, 4, 9, 16 and 36, the first 5 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Cf. A001694, A060355, A076445, A371191.

powQ[n_] := Min[FactorInteger[n][[;; , 2]]] > 1; seq[nmax_] := Module[{s = {}, n1 = 1, gapmax = 0, gap}, Do[If[powQ[n], gap = n - n1; If[gap > gapmax, gapmax = gap; AppendTo[s, n1]]; n1 = n], {n, 2, nmax}]; s]; seq[10^5]

A363191 a(n) is the least start of a run of exactly n consecutive powerful numbers (A001694) that are even, or -1 if no such run exists.

Original entry on oeis.org

16, 4, 196, 968, 8712, 437400, 85730400, 5030690600, 264615012500, 5239012864, 550886816376, 2494017320776852

Offset: 1

Amiram Eldar, May 21 2023

No more terms below 10^18.

At most one of the n even consecutive powerful numbers in the run is a perfect square. - David A. Corneth, May 21 2023

			a(1) = 16, since 16 = 2^4 is an even powerful number, preceded by an odd powerful number, 9 = 3^2, and followed by an odd powerful number, 25 = 5^2.
a(2) = 4, since 4 = 2^2 and 8 = 2^3 are two consecutive even powerful numbers, preceded by an odd powerful number, 1, and followed by an odd powerful number, 9 = 3^2.

Cf. A001694, A062739, A349062, A363189, A363190, A363192.

seq[lim_] := Module[{pow = Union[Flatten[Table[i^2*j^3, {j, 1, lim^(1/3)}, {i, 1, Sqrt[lim/j^3]}]]], s = {}, rem, ind}, rem = Mod[pow, 2]; Do[ind = SequencePosition[rem, Join[{1}, Table[0, {k}], {1}], 1]; If[ind == {}, Break[]]; AppendTo[s, pow[[ind[[1, 1]] + 1]]], {k, 1, Infinity}]; s]; seq[10^10]

A363192 a(n) is the least start of a run of exactly n consecutive powerful numbers (A001694) that are odd, or -1 if no such run exists.

Original entry on oeis.org

1, 25, 2187, 703125, 93096125, 10229709861, 197584409639, 32044275110699, 164029657560618375

Offset: 1

Amiram Eldar, May 21 2023

No more terms below 10^18.

At most one of the n odd consecutive powerful numbers in the run is a perfect square. - David A. Corneth, May 21 2023

			a(1) = 1, since 1 is an odd powerful number, followed by an even powerful number, 4 = 2^2.
a(2) = 25, since 25 = 5^2 and 27 = 3^3 are two consecutive odd powerful numbers, preceded by an even powerful number, 16 = 2^4, and followed by an even powerful number, 32 = 2^5.

Cf. A001694, A062739, A349062, A363189, A363190, A363191.

seq[lim_] := Module[{pow = Union[Flatten[Table[i^2*j^3, {j, 1, lim^(1/3)}, {i, 1, Sqrt[lim/j^3]}]]], s = {}, rem, ind}, rem = Join[{0}, Mod[pow, 2]]; Do[ind = SequencePosition[rem, Join[{0}, Table[1, {k}], {0}], 1]; If[ind == {}, Break[]]; AppendTo[s, pow[[ind[[1, 1]]]]], {k, 1, Infinity}]; s]; seq[1.1*10^10]

A379716 The second Jordan totient function applied to the powerful numbers: a(n) = A007434(A001694(n)).

Original entry on oeis.org

1, 12, 48, 72, 192, 600, 648, 768, 864, 2352, 3072, 3456, 5832, 7200, 7776, 14520, 15000, 12288, 13824, 28392, 28224, 28800, 31104, 43200, 52488, 49152, 55296, 83232, 69984, 115248, 129960, 112896, 115200, 124416, 169344, 174240, 180000, 196608, 279312, 221184, 375000

Offset: 1

Amiram Eldar, Dec 30 2024

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

Cf. A001694, A007434, A323333 (analogous with J_1 = phi), A379715, A379717, A379718.

f[p_, e_] := (p^2-1) * p^(2*e-2); j2[1] = 1; j2[n_] := Times @@ f @@@ FactorInteger[n]; seq[lim_] := j2 /@ Union[Flatten[Table[i^2*j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}]]]; seq[1000]

j2(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^2 - 1) * f[i,1]^(2*f[i,2] - 2));}
list(lim) = apply(j2, select(ispowerful, vector(lim, i, i)));

Sum_{n>=1} 1/a(n) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 2/p^4) = 1.13107206648894940601... .

In general, Sum_{m powerful} 1/J_k(m) = zeta(k)^2 * Product_{p prime} (1 - 2/p^k + 2/p^(2*k)), for k >= 2, where J_k is the k-th Jordan totient function.

cp's OEIS Frontend

A085252 Number of ways to write n as sum of two powerful numbers (A001694).

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A361177 Exponentially powerful numbers: numbers whose exponents in their canonical prime factorization are all powerful numbers (A001694).

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A363169 Powerful abundant numbers: numbers that are both powerful (A001694) and abundant (A005101).

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A363189 Indices of the odd terms in the sequence of powerful numbers (A001694).

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A363194 Number of divisors of the n-th powerful number A001694(n).

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A258567 a(1) = 1; thereafter a(n) = smallest prime factor of the powerful number A001694(n).

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A349062 Powerful numbers (A001694) with a record gap to the next powerful number.

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A363191 a(n) is the least start of a run of exactly n consecutive powerful numbers (A001694) that are even, or -1 if no such run exists.

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