A064549 -id:A064549 - cp's OEIS frontend

A361264 Multiplicative with a(p^e) = p^(e + 2), e > 0.

1, 8, 27, 16, 125, 216, 343, 32, 81, 1000, 1331, 432, 2197, 2744, 3375, 64, 4913, 648, 6859, 2000, 9261, 10648, 12167, 864, 625, 17576, 243, 5488, 24389, 27000, 29791, 128, 35937, 39304, 42875, 1296, 50653, 54872, 59319, 4000, 68921, 74088, 79507, 21296, 10125

Offset: 1

Vaclav Kotesovec, Mar 06 2023

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

Cf. A000005, A003557, A064549, A065483, A007947, A330523, A360997, A361266.

g[p_, e_] := p^(e+2); a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

for(n=1, 100, print1(direuler(p=2, n, 1 + p^3*X/(1 - p*X))[n], ", "))

Dirichlet g.f.: Product_{primes p} (1 + p^3/(p^s - p)).
Dirichlet g.f.: zeta(s-3) * zeta(s-1) * Product_{primes p} (1 + p^(4-2*s) - p^(6-2*s) - p^(1-s)).
Sum_{k=1..n} a(k) ~ c * zeta(3) * n^4 / 4, where c = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523 = 0.53589615382833799980850263131854595064822237...
From Amiram Eldar, Sep 01 2023: (Start)
a(n) = n * A007947(n)^2 = A064549(n) * A007947(n) = A064549(A064549(n)).
A000005(a(n)) = A360997(n).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^2*(p-1))) = A065483. (End)

A367406 The exponentially odd numbers (A268335) multiplied by their squarefree kernels (A007947).

Original entry on oeis.org

1, 4, 9, 25, 36, 49, 16, 100, 121, 169, 196, 225, 289, 361, 441, 484, 529, 144, 676, 81, 841, 900, 961, 64, 1089, 1156, 1225, 1369, 1444, 1521, 400, 1681, 1764, 1849, 2116, 2209, 2601, 2809, 324, 3025, 784, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489, 4761

Offset: 1

Amiram Eldar, Nov 17 2023

Analogous to A355038, with the exponentially odd numbers instead of the square numbers (A000290).

This sequence is a permutation of the square numbers.

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

Cf. A000290, A007947, A053143, A064549, A065463, A268335, A355038, A367407, A367417, A367418.

s[n_] := n * Times @@ FactorInteger[n][[;;, 1]]; s /@ Select[Range[100], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]

b(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, f[i,1]^(f[i,2]+1), 0));}
lista(kmax) = {my(b1); for(k = 1, kmax, b1 = b(k); if(b1 > 0, print1(b1, ", ")));}

a(n) = A064549(A268335(n)).
a(n) = A268335(n)*A367417(n).
a(n) = A367407(n)^2.
a(n) = A268335(n)^2/A367418(n).
Sum_{k=1..n} a(k) = c * n^3 / 3, where c = (Pi^2/(15*d^3)) * Product_{p prime} (1 - 1/(p^3*(p+1))) = 1.78385074227198915372..., and d = A065463 is the asymptotic density of the exponentially odd numbers.
a(n) = A053143(A268335(n)). - Amiram Eldar, Nov 30 2023

A078326 Numbers n such that n-1 and n are a pair of consecutive powerful numbers.

Original entry on oeis.org

9, 289, 676, 9801, 12168, 235225, 332929, 465125, 1825201, 11309769, 384199201, 592192225, 4931691076, 5425069448, 13051463049, 221322261601, 443365544449, 865363202001, 8192480787001, 11968683934832, 13325427460801, 15061377048201, 28821995554248

Offset: 1

Reinhard Zumkeller, Nov 23 2002

nonn

a(n) = u*rad(u) = v*rad(v)+1 for appropriate u, v, where rad(n) = A007947(n) is the squarefree kernel.

Also numbers n such that n(n-1) is a powerful number. - Charles R Greathouse IV, Aug 08 2013

Donovan Johnson, Table of n, a(n) for n = 1..39 (terms < 10^22)
Jérôme Germoni, Nombres puissants au bac S, Images des Mathématiques, CNRS, 2021 (in French).
Eric Weisstein's World of Mathematics, Powerful Number

Cf. A078310, A064549, A078315, A078325, A060355, A078326.

is(n)=ispowerful(n-1)&&ispowerful(n) \\ Charles R Greathouse IV, Aug 08 2013

a(n) = A060355(n)+1.

a(22)-a(23) from Donovan Johnson, Jul 29 2011

A350996 a(n) = Sum_{k=1..n} k * rad(k).

Original entry on oeis.org

1, 5, 14, 22, 47, 83, 132, 148, 175, 275, 396, 468, 637, 833, 1058, 1090, 1379, 1487, 1848, 2048, 2489, 2973, 3502, 3646, 3771, 4447, 4528, 4920, 5761, 6661, 7622, 7686, 8775, 9931, 11156, 11372, 12741, 14185, 15706, 16106, 17787, 19551, 21400, 22368, 23043, 25159

Offset: 1

Wesley Ivan Hurt, Jan 28 2022

nonn

			a(4) = 22; a(4) = Sum_{k=1..4} k * rad(k) = 1*rad(1) + 2*rad(2) + 3*rad(3) + 4*rad(4) = 1*1 + 2*2 + 3*3 + 4*2 = 22.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Radical of an integer.

Cf. A007947 (rad), A073355, A064549, A065463.

f[n_] := n * Times @@ FactorInteger[n][[;; , 1]]; Accumulate @ Array[f, 50] (* Amiram Eldar, Jan 29 2022 *)

a(n) = sum(k=1, n, k*factorback(factorint(k)[, 1])); \\ Michel Marcus, Jan 30 2022

a(n) = Sum_{k=1..n} A064549(k).

a(n) ~ c * n^3 / 3, where c = A065463. - Amiram Eldar, Dec 09 2023

A367407 a(n) = sqrt(A367406(n)).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 4, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 12, 26, 9, 29, 30, 31, 8, 33, 34, 35, 37, 38, 39, 20, 41, 42, 43, 46, 47, 51, 53, 18, 55, 28, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 44, 89, 91, 93, 94, 95, 24, 97

Offset: 1

Amiram Eldar, Nov 17 2023

A permutation of the positive integers.

Cf. A064549, A268335, A367406, A367417, A367419.

Cf. A065463, A065465, A253905.

s[n_] := Sqrt[n * Times @@ FactorInteger[n][[;;, 1]]]; s /@ Select[Range[100], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]

b(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, f[i,1]^(f[i,2]+1), 0));}
lista(kmax) = {my(b1); for(k = 1, kmax, b1 = b(k); if(b1 > 0, print1(sqrtint(b1), ", ")));}

a(n) = sqrt(A064549(A268335(n))).
a(n) = sqrt(A268335(n)*A367417(n)).
a(n) = A268335(n)/A367419(n).
Sum_{k=1..n} a(k) = c * n^2 / 2, where c = (zeta(3)/(zeta(2)*d^2)) * Product_{p prime} (1 - 1/(p^2*(p+1))) = A253905 * A065465 / d^3 = 1.29812028442810841122..., and d = A065463 is the asymptotic density of the exponentially odd numbers (A268335).

A373318 Numerator of the asymptotic density of numbers that are unitarily divided by n.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 3, 8, 1, 16, 1, 18, 1, 4, 5, 22, 1, 4, 3, 2, 3, 28, 2, 30, 1, 20, 4, 24, 1, 36, 9, 8, 1, 40, 1, 42, 5, 8, 11, 46, 1, 6, 1, 32, 3, 52, 1, 8, 3, 4, 7, 58, 1, 60, 15, 4, 1, 48, 5, 66, 2, 44, 6, 70, 1, 72, 9, 8, 9, 60, 2, 78

Offset: 1

Amiram Eldar, Jun 01 2024

Numbers that are unitarily divided by n are numbers k such that n is a unitary divisor of k, or equivalently, numbers of the form m*n, with gcd(m, n) = 1.

			Fractions begin with: 1, 1/4, 2/9, 1/8, 4/25, 1/18, 6/49, 1/16, 2/27, 1/25, 10/121, 1/36, ...
For n = 2, the numbers that are unitarily divided by 2 are the numbers of the form 4*k+2 whose asymptotic density is 1/4. Therefore a(2) = numerator(1/4) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor.
Wikipedia, Unitary divisor.

Cf. A000010, A003277, A034444, A064549, A076512, A077610, A109395, A173557, A373319(denominators), A373320.
Cf. A001620, A013661, A306016.
Numbers that are unitarily divided by k: A000027 (k=1), A016825 (k=2), A016051 (k=3), A017113 (k=4), A051062 (k=8), A051063 (k=9).

a[n_] := Numerator[EulerPhi[n]/n^2]; Array[a, 100]

PARI
```
a(n) = numerator(eulerphi(n)/n^2);
```

a(n) = 1 if and only if n is in A090778.
a(n) = A000010(n) if and only if n is a cyclic number (A003277).
Let f(n) = a(n)/A373319(n). Then:
f(n) = A000010(n)/n^2 = A076512(n)/(n*A109395(n)).
f(n) = A173557(n)/A064549(n).
f(n) is multiplicative with f(p^e) = (1 - 1/p)/p^e.
Sum_{k=1..n} f(k) = (log(n) + gamma - zeta'(2)/zeta(2)) / zeta(2), where gamma is Euler's constant (A001620).

A073354 Binomial coefficient ( n, squarefree kernel(n) ).

Original entry on oeis.org

1, 1, 1, 6, 1, 1, 1, 28, 84, 1, 1, 924, 1, 1, 1, 120, 1, 18564, 1, 184756, 1, 1, 1, 134596, 53130, 1, 2925, 40116600, 1, 1, 1, 496, 1, 1, 1, 1947792, 1, 1, 1, 847660528, 1, 1, 1, 2104098963720, 344867425584, 1, 1, 12271512, 85900584, 10272278170, 1

Offset: 1

Reinhard Zumkeller, Jul 29 2002

a(n)=1 iff n is squarefree.

Robert Israel, Table of n, a(n) for n = 1..3315

Cf. A007947, A005117, A073353, A066503, A064549, A003557.

f:= proc(n) local k;
  k:= convert(numtheory:-factorset(n),`*`);
  binomial(n,k)
end proc:
map(f, [$1..60]); # Robert Israel, May 07 2021

a[n_] := Binomial[n, Times @@ FactorInteger[n][[All, 1]]];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, May 11 2023 *)

a(n) = binomial(n, A007947(n)).

A308360 Product of positive divisors d of n that are divisible by every prime that divides n.

Original entry on oeis.org

1, 2, 3, 8, 5, 6, 7, 64, 27, 10, 11, 72, 13, 14, 15, 1024, 17, 108, 19, 200, 21, 22, 23, 1728, 125, 26, 729, 392, 29, 30, 31, 32768, 33, 34, 35, 46656, 37, 38, 39, 8000, 41, 42, 43, 968, 675, 46, 47, 82944, 343, 500, 51, 1352, 53, 5832, 55, 21952, 57, 58, 59

Offset: 1

Jaroslav Krizek, May 22 2019

nonn

			The divisors of 12 that are divisible by both 2 and 3 are 6 and 12. So a(12) = 6 * 12 = 72.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

See A005361 and A057723 for number and sum of such divisors.

Cf. A007955, A064549.

[1] cat [&*[d: d in Divisors(n) |  d gt 1  and #[c: c in Divisors(d) | IsPrime(c)] eq #[d: d in Divisors(n) | IsPrime(d)]]: n in [2..100]]

Table[Sqrt[(n*Product[If[PrimeQ[d], d, 1], {d, Divisors[n]}])^Product[ FactorInteger[n][[k, 2]], {k, 1, Length[FactorInteger[n]]}]], {n, 1, 100}] (* Vaclav Kotesovec, Jun 15 2019 *)

a(n) = n for squarefree numbers (A005117).

a(n) = A064549(n)^(A005361(n)/2). - Charlie Neder, Jun 03 2019

A355261 a(n) = largest-nth-power(n, 2) * radical(n) = A000188(n) * A007947(n), where largest-nth-power(n, e) is the largest positive integer b such that b^e divides n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 14, 15, 8, 17, 18, 19, 20, 21, 22, 23, 12, 25, 26, 9, 28, 29, 30, 31, 8, 33, 34, 35, 36, 37, 38, 39, 20, 41, 42, 43, 44, 45, 46, 47, 24, 49, 50, 51, 52, 53, 18, 55, 28, 57, 58, 59, 60, 61, 62, 63, 16, 65, 66, 67, 68

Offset: 1

Peter Luschny, Jul 12 2022

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

Cf. A000188, A002117, A007947, A064549, A355263.

with(NumberTheory): seq(LargestNthPower(n, 2)*Radical(n), n = 1..68);

Array[Apply[Times, #[[All, 1]]]*Apply[Times, #1^Floor[#2/2] & @@ Transpose@ #] &@ FactorInteger[#] &, 68] (* Michael De Vlieger, Jul 12 2022 *)

from math import prod
from sympy import factorint
def A355261(n): return prod(p**((e>>1)+1) for p, e in factorint(n).items()) # Chai Wah Wu, Jul 13 2022

Multiplicative with a(p^e) = p^(1+floor(e/2)). - Amiram Eldar, Jul 13 2022

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(3)/2) * Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.447583182004... . - Amiram Eldar, Nov 13 2022

A355263 a(n) = largest-nth-power(n, 3) * radical(n) = A053150(n) * A007947(n), where the largest-nth-power(n, e) is the largest positive integer b such that b^e divides n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 4, 3, 10, 11, 6, 13, 14, 15, 4, 17, 6, 19, 10, 21, 22, 23, 12, 5, 26, 9, 14, 29, 30, 31, 4, 33, 34, 35, 6, 37, 38, 39, 20, 41, 42, 43, 22, 15, 46, 47, 12, 7, 10, 51, 26, 53, 18, 55, 28, 57, 58, 59, 30, 61, 62, 21, 8, 65, 66, 67, 34, 69

Offset: 1

Peter Luschny, Jul 12 2022

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

Cf. A000188, A013663, A053150, A064549, A355261.

with(NumberTheory): seq(LargestNthPower(n, 3)*Radical(n), n=1..69);

f[p_, e_] := p^(1 + Floor[e/3]); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 13 2022 *)

from math import prod
from sympy import factorint
def A355263(n): return prod(p**(e//3+1) for p, e in factorint(n).items()) # Chai Wah Wu, Jul 13 2022

Multiplicative with a(p^e) = p^(1 + floor(e/3)). - Amiram Eldar, Jul 13 2022

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(5)/2) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 2/p^5 + 1/p^6) = 0.3643121583... . - Amiram Eldar, Nov 13 2022

cp's OEIS Frontend

A361264 Multiplicative with a(p^e) = p^(e + 2), e > 0.

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A367406 The exponentially odd numbers (A268335) multiplied by their squarefree kernels (A007947).

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A078326 Numbers n such that n-1 and n are a pair of consecutive powerful numbers.

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A350996 a(n) = Sum_{k=1..n} k * rad(k).

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A367407 a(n) = sqrt(A367406(n)).

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A373318 Numerator of the asymptotic density of numbers that are unitarily divided by n.

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A073354 Binomial coefficient ( n, squarefree kernel(n) ).

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A308360 Product of positive divisors d of n that are divisible by every prime that divides n.

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