A082695 -id:A082695 - cp's OEIS frontend

A363217 Odd powerful numbers that are not powers of primes.

225, 441, 675, 1089, 1125, 1225, 1323, 1521, 2025, 2601, 3025, 3087, 3249, 3267, 3375, 3969, 4225, 4563, 4761, 5625, 5929, 6075, 6125, 7225, 7569, 7803, 8281, 8575, 8649, 9025, 9261, 9747, 9801, 10125, 11025, 11907, 11979, 12321, 13225, 13689, 14161, 14283, 15125, 15129, 16641, 16875, 17689, 18225, 19773

Offset: 1

Michael De Vlieger, May 21 2023

nonn

This sequence is { A286708 INTERSECT A005408 } = { A001694 INTERSECT A360769 }.

Subset of A001694, A062739, A126706, and A360769.

			a(1) = 225 = 3^2 * 5^2, the smallest odd number with multiple distinct prime factors, each of which have multiplicity exceeding 1.
a(2) = 441 = 3^2 * 7^2,
a(3) = 675 = 3^3 * 5^2, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000961, A001694, A062739, A126706, A286708, A363216.

Cf. A002117, A013661, A013664, A082695, A136141.

With[{nn = 20000}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], And[OddQ[#], ! PrimePowerQ[#]] &] ]

isok(k) = (k>1) && (k%2) && ispowerful(k) && !isprimepower(k); \\ Michel Marcus, May 28 2023

This sequence is { k = a^2*b^3 : a >= 1, b >= 1, omega(k) > 1, k mod 2 = 1 }.

Sum_{n>=1} 1/a(n) = 2*zeta(2)*zeta(3)/(3*zeta(6)) - 1/2 - Sum_{p prime} 1/(p*(p-1)) = (2/3) * A082695 - 1/2 - A136141 = 0.0225742... . - Amiram Eldar, May 28 2023

A370834 a(n) is the greatest prime dividing the n-th powerful number, for n >= 2; a(1)=1.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 03 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Rafael Jakimczuk, Summing the largest prime factor over integer sequences, Revista de la Unión Matemática Argentina, Vol. 67, No. 1 (2024), pp. 27-35.
Michael De Vlieger, Log log scatterplot of pi(a(n)), n = 2..2^20, highlighting in red pi(a(n)) such that b(n) is a prime power, where b(n) = A001694(n). The remaining terms are such that b(n) is in A286708. The topmost red line corresponds with b(n) in A001248 (prime squares), the row of red dots at the bottom corresponds with b(n) in A000079 (powers of 2), the topmost blue line corresponds with b(n) > 16 in A069262 (4*p^2, with prime p).

Cf. A001694, A006530, A073482, A082695, A370833, A370835.

s[n_] := Module[{f = FactorInteger[n]}, If[n == 1 || AllTrue[f[[;; , 2]], # > 1 &], f[[-1, 1]], Nothing]]; Array[s, 4000]

lista(kmax) = {my(f); print1(1, ", "); for(k = 2, kmax, f = factor(k); if(vecmin(f[, 2]) > 1, print1(f[#f~, 1], ", ")));}

a(n) = A006530(A001694(n)).

Sum_{A001694(n) <= x} a(n) = Sum_{i=1..k} e_i * x/log(x)^i + O(x/log(x)^(k+1)), for any given positive integer k, where e_i are constants, e_1 = zeta(2)*zeta(3)/zeta(6) = 1.943596... (A082695) (De Koninck and Jakimczuk, 2024).

A370835 a(n) is the greatest prime dividing the n-th cubefull number, for n >= 2; a(1)=1.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 03 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Rafael Jakimczuk, Summing the largest prime factor over integer sequences, Revista de la Unión Matemática Argentina, Vol. 67, No. 1 (2024), pp. 27-35.

Cf. A036966, A006530, A073482, A082695, A370833, A370834.

s[n_] := Module[{f = FactorInteger[n]}, If[n == 1 || AllTrue[f[[;; , 2]], # > 2 &], f[[-1, 1]], Nothing]]; Array[s, 32000]

lista(kmax) = {my(f); print1(1, ", "); for(k = 2, kmax, f = factor(k); if(vecmin(f[, 2]) > 2, print1(f[#f~, 1], ", ")));}

a(n) = A006530(A036966(n)).

Sum_{A036966(n) <= x} a(n) = Sum_{i=1..k} e_i * x^(2/3)/log(x)^i + O(x^(2/3)/log(x)^(k+1)), for any given positive integer k, where e_i are constants, e_1 = (3/2) * Product_{p prime} (1 + Sum_{i>=3} 1/p^(2*i/3)) = 3.44968588450293915243... (De Koninck and Jakimczuk, 2024).

A376217 Powerful numbers whose sum of powerful divisors (including 1) is even.

Original entry on oeis.org

9, 25, 36, 49, 72, 81, 100, 121, 144, 169, 196, 200, 225, 288, 289, 324, 361, 392, 400, 441, 484, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 900, 961, 968, 1089, 1125, 1152, 1156, 1225, 1296, 1323, 1352, 1369, 1444, 1521, 1568, 1600, 1681, 1764, 1800, 1849, 1936

Offset: 1

Amiram Eldar, Sep 16 2024

The primitive terms of A376216: all the terms of A376216 are of the form k*m, where m is a term of this sequence and k is a squarefree number that is coprime to m.
Powerful numbers that have at least one odd prime factor in their prime factorization that has an even exponent.
Equivalently, powerful numbers whose odd part (A000265) is not an exponentially odd number (A268335).

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

Intersection of A001694 and A376216.

Cf. A000265, A005117, A065487, A082695, A183097, A268335.

q[n_] := Module[{f = FactorInteger[n], i = 2 - Mod[n, 2]}, AllTrue[f[[;;, 2]], # > 1 &] && AnyTrue[f[[i;;-1, 2]], EvenQ]]; Select[Range[2000], q]

is(k) = {my(f = factor(k), i = 1 + !(k % 2)); ispowerful(f) && #select(x -> !(x%2), f[i..#f~,2]) > 0;}

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - (9/7) * Product_{p prime} (1 + 1/(p*(p^2-1))) = A082695 - (9/7) * A065487 = 0.36050781682112605291... .

A378769 Intersection of A375055 and A376936.

Original entry on oeis.org

5400, 9000, 10584, 10800, 13500, 16200, 18000, 21168, 21600, 24696, 26136, 27000, 31752, 32400, 36000, 36504, 37044, 40500, 42336, 43200, 45000, 48600, 49000, 49392, 52272, 54000, 62424, 63504, 64800, 67500, 68600, 72000, 73008, 74088, 77976, 78408, 81000, 84672

Offset: 1

Michael De Vlieger, Dec 13 2024

Let omega = A001221, bigomega = A001222, rad = A007947.
Powerful numbers k with bigomega(k) > omega(k) > 2 that are divisible by two distinct prime cubes p^3 and q^3.
Numbers k such that there exists (d, k/d), d | k, such that d neither divides nor is coprime to k/d and vice versa in the following 3 ways:
Type A: rad(d) does not divide d/k and rad(d/k) does not divide d
Type B: rad(d) divides d/k but rad(d/k) does not divide d
Type C: rad(d) | d/k and rad(d/k) | d, hence rad(d) = rad(d/k) = rad(k), a kind of coreful divisor pair.
Since (d, d/k) are noncoprime and do not divide one another, both must be composite, thus k is also composite.
In addition the following kinds of divisor pairs are also seen:
Type D: (d, k/d) such that d | k/d but there exists a factor Q | k/d that does not divide d. Then omega(d) < omega(k/d) = omega(k).
Type E: Nontrivial unitary divisor pairs (d, k/d) such that gcd(d, k/d) = 1, d > 1, k/d > 1. Let prime power factor p^m | k be such that m is maximized. Then set d = p^m and it is clear that for any k in A024619, there exists at least 1 nontrivial unitary divisor pair.
A378767 = { k : omega(k) > 1, p^3 | k for some prime p }, and
A376936 = { k : rad(k)^2 | k, p^3 | k and q^3 | k for distinct primes p, q }.
Therefore, we need only take intersection of A375055 and A376936.

			Table of the first 12 terms of this sequence, showing examples of types A, B, and C described in Comments.
   n     a(n)  Factors of a(n)    Type A      Type B      Type C
  ----------------------------------------------------------------
   1    5400   2^3 * 3^3 * 5^2    24 * 225    4 * 1350    60 * 90
   2    9000   2^3 * 3^2 * 5^3    18 * 500    4 * 2250    60 * 150
   3   10584   2^3 * 3^3 * 7^2    24 * 441    4 * 2646    84 * 126
   4   10800   2^4 * 3^3 * 5^2    48 * 225    8 * 1350    90 * 120
   5   13500   2^2 * 3^3 * 5^3    12 * 1125   9 * 1500    90 * 150
   6   16200   2^3 * 3^4 * 5^2    24 * 675    4 * 4050    60 * 270
   7   18000   2^4 * 3^2 * 5^3    18 * 1000   8 * 2250   120 * 150
   8   21168   2^4 * 3^3 * 7^2    48 * 441    8 * 2646   126 * 168
   9   21600   2^5 * 3^3 * 5^2    50 * 432    8 * 2700    90 * 240
  10   24696   2^3 * 3^2 * 7^3    18 * 1372   4 * 6174    84 * 294
  11   26136   2^3 * 3^3 * 11^2   24 * 1089   4 * 6534   132 * 198
  12   27000   2^3 * 3^3 * 5^3    24 * 1125   4 * 6750    60 * 450

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Listing of select divisor pairs of a(n), n = 1..16, showing divisor pairs of type A in light gray, type B in blue and gold, and type C in black.

Cf. A001694, A024619, A126706, A375055, A376936, A378767, A378900.

Cf. A082020, A082695.

s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[2^16],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
Select[s, PrimeOmega[#] > PrimeNu[#] > 2 &]

Intersection of A375055, A376936, and A378767.
This sequence is { k : rad(k)^2 | k, bigomega(k) > omega(k) > 2, p^3 | k and q^3 | k for distinct primes p, q }.
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - (15/Pi^2) * (1 + Sum_{prime} 1/((p-1)*(p^2+1))) - ((Sum_{p prime} (1/(p^2*(p-1))))^2 - Sum_{p prime} (1/(p^4*(p-1)^2)))/2 = 0.0025524144364532126894... . - Amiram Eldar, Dec 21 2024

A363216 Even powerful numbers that are not prime powers.

Original entry on oeis.org

36, 72, 100, 108, 144, 196, 200, 216, 288, 324, 392, 400, 432, 484, 500, 576, 648, 676, 784, 800, 864, 900, 968, 972, 1000, 1152, 1156, 1296, 1352, 1372, 1444, 1568, 1600, 1728, 1764, 1800, 1936, 1944, 2000, 2116, 2304, 2312, 2500, 2592, 2700, 2704, 2744, 2888, 2916, 3136, 3200, 3364, 3456, 3528, 3600

Offset: 1

Michael De Vlieger, May 21 2023

nonn

This sequence is { A286708 INTERSECT A005843 } = { A001694 INTERSECT A363101 }.

Subset of A001694, A126706, and A363101.

			a(1) = 36 = 2^2 * 3^2, the smallest even number with multiple distinct prime factors, all of which have multiplicity exceeding 1, so it is the first term.
a(2) = 72 = 2^3 * 3^2,
a(3) = 100 = 2^2 * 5^2, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001694, A062739, A126706, A286708, A363101, A363217.

Cf. A002117, A013661, A013664, A082695.

With[{nn = 3600}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], And[EvenQ[#], ! PrimePowerQ[#]] &] ]

isok(k) = !(k%2) && ispowerful(k) && !isprimepower(k); \\ Michel Marcus, May 27 2023

This sequence is { k = a^2*b^3 : a >= 1, b >= 1, omega(k) > 1, k mod 2 = 0 }.

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/(3*zeta(6)) - 1/2 = A082695 / 3 - 1/2 = 0.147865... . - Amiram Eldar, May 28 2023

A374289 5-rough powerful numbers: numbers k coprime to 6 such that if a prime p divides k then p^2 also divides k.

Original entry on oeis.org

1, 25, 49, 121, 125, 169, 289, 343, 361, 529, 625, 841, 961, 1225, 1331, 1369, 1681, 1849, 2197, 2209, 2401, 2809, 3025, 3125, 3481, 3721, 4225, 4489, 4913, 5041, 5329, 5929, 6125, 6241, 6859, 6889, 7225, 7921, 8281, 8575, 9025, 9409, 10201, 10609, 11449, 11881, 12167

Offset: 1

Amiram Eldar, Jul 02 2024

This sequence is closed under multiplication.

The least term that is not a power of a prime (A000961) is a(14) = 5^2*7^2 = 1225.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Rough Number.
Index entries for sequences related to powerful numbers.

Intersection of A007310 and A001694.
Intersection of A001651 and A062739.
A374290 is a subsequence.
Cf. A000961, A082695.

powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Flatten @ Table[6*n + {-1, 1}, {n, 0, 2000}], powQ]

is(k) = gcd(k, 6) == 1 && ispowerful(k);

Sum_{n>=1} 1/a(n) = 4*zeta(2)*zeta(3)/(7*zeta(6)) = (4/7) * A082695 = 1.1106265353... .

A374290 7-rough powerful numbers: numbers k coprime to 30 such that if a prime p divides k then p^2 also divides k.

Original entry on oeis.org

1, 49, 121, 169, 289, 343, 361, 529, 841, 961, 1331, 1369, 1681, 1849, 2197, 2209, 2401, 2809, 3481, 3721, 4489, 4913, 5041, 5329, 5929, 6241, 6859, 6889, 7921, 8281, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 14161, 14641, 16129, 16807, 17161, 17689, 18769

Offset: 1

Amiram Eldar, Jul 02 2024

This sequence is closed under multiplication.

The least term that is not a power of a prime (A000961) is a(25) = 7^2*11^2 = 5929.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Rough Number.
Index entries for sequences related to powerful numbers.

Intersection of A007775 and A001694.
Intersection of A229829 and A062739.
Intersection of A047201 and A374289.
Cf. A000961, A082695.

powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Range[20000], CoprimeQ[#, 30] && powQ[#] &]

is(k) = gcd(k, 30) == 1 && ispowerful(k);

Sum_{n>=1} 1/a(n) = 80*zeta(2)*zeta(3)/(147*zeta(6)) = (80/147) * A082695 = 1.05773955745... .

In general, the sum of reciprocals of the p-rough powerful numbers is (zeta(2)*zeta(3)/zeta(6)) * Product_{prime q < p} ((q-1)*q/(q^2-q+1)).

A376171 Powerful numbers whose prime factorization has an odd maximum exponent.

Original entry on oeis.org

8, 27, 32, 72, 108, 125, 128, 200, 216, 243, 288, 343, 392, 500, 512, 675, 800, 864, 968, 972, 1000, 1125, 1152, 1323, 1331, 1352, 1372, 1568, 1800, 1944, 2048, 2187, 2197, 2312, 2592, 2700, 2744, 2888, 3087, 3125, 3200, 3267, 3375, 3456, 3528, 3872, 3888, 4000

Offset: 1

Amiram Eldar, Sep 13 2024

Subsequence of A102834 and first differs from it at n = 14: A102834(14) = 432 = 2^4 * 3^3 is not a term of this sequence.
Powerful numbers k such that A051903(k) is odd.
Equivalently, numbers whose prime factorization exponents are all larger than 1 and their maximum is odd. The maximum exponent in the prime factorization of 1 is considered to be A051903(1) = 0, and therefore 1 is not a term of this sequence.
The numbers of terms that do not exceed the 10^k-powerful number (A376092(k)), for k = 1, 2, ..., are 3, 40, 416, 4255, 42829, 429393, 4299797, 43022803, ... . Apparently, the asymptotic density of this sequence within the powerful numbers (A001694) exists and approximately equals 0.43.

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

Complement of A376170 within A001694.
Intersection of A001694 and A376142.
Subsequence of A102834.
Subsequences: A030078, A050997, A079395, A092759, A138031, A179665, A335988 \ {1}.
Cf. A051903, A082695, A376092.

seq[lim_] := Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}], # > 1 && OddQ[Max[FactorInteger[#][[;; , 2]]]] &]; seq[10^4]

is(k) = {my(f = factor(k), e = f[,2]); #e && ispowerful(f) && vecmax(e) % 2;}

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{k>=2} (-1)^k * s(k) = 0.29116340833243888282..., where s(k) = Product_{p prime} (1 + Sum_{i=2..k} 1/p^i).

A382421 The product of exponents in the prime factorization of the noncubefree numbers.

Original entry on oeis.org

3, 4, 3, 3, 5, 3, 4, 3, 3, 6, 6, 4, 4, 3, 5, 3, 6, 4, 3, 3, 7, 3, 3, 8, 3, 5, 4, 3, 4, 3, 3, 6, 6, 4, 9, 5, 3, 4, 5, 3, 3, 8, 3, 3, 4, 3, 10, 3, 3, 4, 3, 6, 8, 3, 4, 3, 3, 3, 5, 6, 4, 3, 3, 3, 7, 6, 8, 4, 3, 5, 3, 12, 3, 6, 3, 3, 4, 3, 5, 5, 3, 4, 6, 6, 9, 3, 3

Offset: 1

Amiram Eldar, Mar 25 2025

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

Cf. A002117, A013661, A013664, A005361, A046099, A082695, A330594, A368039, A382419.

s[n_] := Times @@ FactorInteger[n][[;; , 2]]; noncubeFreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] > 2; s /@ Select[Range[600], noncubeFreeQ]

list(kmax) = {my(e); for(k = 2, kmax, e = factor(k)[, 2]; if(vecmax(e) > 2, print1(vecprod(e), ", "))); }

a(n) = A005361(A046099(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2)*zeta(3)^2/zeta(6) - zeta(3) * Product_{p prime} (1 + 1/p^2 - 2/p^3))/(zeta(3) - 1) = (A082695 - A330594) * A002117 / (A002117 - 1) = 4.97723390794900554553... .

cp's OEIS Frontend

A363217 Odd powerful numbers that are not powers of primes.

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A370834 a(n) is the greatest prime dividing the n-th powerful number, for n >= 2; a(1)=1.

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A370835 a(n) is the greatest prime dividing the n-th cubefull number, for n >= 2; a(1)=1.

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A376217 Powerful numbers whose sum of powerful divisors (including 1) is even.

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A378769 Intersection of A375055 and A376936.

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A363216 Even powerful numbers that are not prime powers.

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A374289 5-rough powerful numbers: numbers k coprime to 6 such that if a prime p divides k then p^2 also divides k.

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A374290 7-rough powerful numbers: numbers k coprime to 30 such that if a prime p divides k then p^2 also divides k.

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