A057521 -id:A057521 - cp's OEIS frontend

A326184 a(n) = sigma(n) - A057521(n), where A057521 gives the powerful part of n, and sigma gives the sum of divisors of n.

0, 2, 3, 3, 5, 11, 7, 7, 4, 17, 11, 24, 13, 23, 23, 15, 17, 30, 19, 38, 31, 35, 23, 52, 6, 41, 13, 52, 29, 71, 31, 31, 47, 53, 47, 55, 37, 59, 55, 82, 41, 95, 43, 80, 69, 71, 47, 108, 8, 68, 71, 94, 53, 93, 71, 112, 79, 89, 59, 164, 61, 95, 95, 63, 83, 143, 67, 122, 95, 143, 71, 123, 73, 113, 99, 136, 95, 167, 79, 170, 40, 125, 83

Offset: 1

Antti Karttunen, Jul 11 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A057521, A326185.

Cf. also A326187.

A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521.
A326184(n) = (sigma(n)-A057521(n));

a(n) = A000203(n) - A057521(n).

A349350 Dirichlet inverse of A057521, the powerful part of n.

Original entry on oeis.org

1, -1, -1, -3, -1, 1, -1, -1, -8, 1, -1, 3, -1, 1, 1, 5, -1, 8, -1, 3, 1, 1, -1, 1, -24, 1, -10, 3, -1, -1, -1, 7, 1, 1, 1, 24, -1, 1, 1, 1, -1, -1, -1, 3, 8, 1, -1, -5, -48, 24, 1, 3, -1, 10, 1, 1, 1, 1, -1, -3, -1, 1, 8, -3, 1, -1, -1, 3, 1, -1, -1, 8, -1, 1, 24, 3, 1, -1, -1, -5, 28, 1, -1, -3, 1, 1, 1, 1, -1, -8

Offset: 1

Antti Karttunen, Nov 18 2021

Multiplicative because A057521 is.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A057521.

Cf. also A349340, A349442.

f[p_, e_] := Module[{B = 1 + p - 2*p^2, C = Sqrt[1 + 2*p - 3*p^2]}, FullSimplify[((B - C)*(p - 1 + C)^(e - 1) - (B + C)*(p - 1 - C)^(e - 1))/(2^e*C)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2023 *)

A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
memoA349350 = Map();
A349350(n) = if(1==n,1,my(v); if(mapisdefined(memoA349350,n,&v), v, v = -sumdiv(n,d,if(dA057521(n/d)*A349350(d),0)); mapput(memoA349350,n,v); (v)));

a(1) = 1; a(n) = -Sum_{d|n, d < n} A057521(n/d) * a(d).

Let p be a prime, B = 1 + p - 2*p^2 and C = sqrt(1 + 2*p - 3*p^2). Then the sequence is multiplicative with a(p^e) = ((B-C)*(p-1+C)^(e-1) - (B+C)*(p-1-C)^(e-1))/(2^e*C). - Sebastian Karlsson, Dec 02 2021

A349379 Möbius transform of A057521 (powerful part of n).

Original entry on oeis.org

1, 0, 0, 3, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 24, 0, 18, 0, 0, 0, 0, 16, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 72, 0, 0

Offset: 1

Antti Karttunen, Nov 18 2021

Multiplicative with a(p^e) = 0 if e = 1, p^2 - 1 if e = 2 and p^e - p^(e-1) otherwise. - Amiram Eldar, Nov 18 2021

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000010, A008683, A057521, A349441.

Cf. also A300717.

f[p_, e_] := Which[e > 2, p^e - p^(e - 1), e == 2, p^2 - 1, e == 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
A349379(n) = sumdiv(n,d,moebius(n/d)*A057521(d));

from math import prod
from sympy import factorint
def A349379(n): return prod(0 if e==1 else p**e - (1 if e==2 else p**(e-1)) for p,e in factorint(n).items()) # Chai Wah Wu, Nov 14 2022

a(n) = Sum_{d|n} A008683(n/d) * A057521(d).

a(n) = Sum_{d|n} A000010(n/d) * A349441(d).

A379583 Numerators of the partial sums of the reciprocals of the powerful part function (A057521).

Original entry on oeis.org

1, 2, 3, 13, 17, 21, 25, 51, 467, 539, 611, 629, 701, 773, 845, 1699, 1843, 1859, 2003, 2039, 2183, 2327, 2471, 2489, 62369, 65969, 198307, 201007, 211807, 222607, 233407, 467489, 489089, 510689, 532289, 532889, 554489, 576089, 597689, 600389, 621989, 643589, 665189

Offset: 1

Amiram Eldar, Dec 26 2024

			Fractions begin with 1, 2, 3, 13/4, 17/4, 21/4, 25/4, 51/8, 467/72, 539/72, 611/72, 629/72, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
Maurice-Étienne Cloutier, Les parties k-puissante et k-libre d'un nombre, Thèse de doctorat, Université Laval, Québec (2018).
Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon, On the powerful and squarefree parts of an integer, Journal of Integer Sequences, Vol. 17 (2014), Article 14.6.6.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.12, p. 33.

Cf. A057521, A191622, A370902, A370903, A379584 (denominators), A379585.

f[p_, e_] := If[e > 1, p^e, 1]; powful[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[1/powful[n], {n, 1, 50}]]]

powerful(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] > 1, f[i, 1]^f[i, 2], 1)); }
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / powerful(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} 1/A057521(k)).

a(n)/A379584(n) = c * n + O(n^(1/2)), where c = A191622 (Cloutier et al., 2014). The error term was improved by Tóth (2017) to O(n^(1/2) * exp(-c1 * log(n)^(3/5) / log(log(n))^(1/5))) unconditionally, and O(n^(2/5) * exp(c2 * log(n) / log(log(n)))) assuming the Riemann hypothesis, where c1 and c2 are positive constants.

A379584 Denominators of the partial sums of the reciprocals of the powerful part function (A057521).

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 4, 8, 72, 72, 72, 72, 72, 72, 72, 144, 144, 144, 144, 144, 144, 144, 144, 144, 3600, 3600, 10800, 10800, 10800, 10800, 10800, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 1058400

Offset: 1

Amiram Eldar, Dec 26 2024

Amiram Eldar, Table of n, a(n) for n = 1..1000
Maurice-Étienne Cloutier, Les parties k-puissante et k-libre d'un nombre, Thèse de doctorat, Université Laval, Québec (2018).
Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon, On the powerful and squarefree parts of an integer, Journal of Integer Sequences, Vol. 17 (2014), Article 14.6.6.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.12, p. 33.

Cf. A057521, A370902, A370903, A379583 (numerators), A379586.

f[p_, e_] := If[e > 1, p^e, 1]; powful[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[1/powful[n], {n, 1, 50}]]]

powerful(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] > 1, f[i, 1]^f[i, 2], 1)); }
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / powerful(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} 1/A057521(k)).

A379585 Numerators of the partial alternating sums of the reciprocals of the powerful part function (A057521).

Original entry on oeis.org

1, 0, 1, 3, 7, 3, 7, 13, 125, 53, 125, 107, 179, 107, 179, 349, 493, 53, 69, 65, 81, 65, 81, 79, 1991, 1591, 43357, 40657, 51457, 40657, 51457, 102239, 123839, 102239, 123839, 123239, 144839, 123239, 144839, 142139, 163739, 142139, 163739, 158339, 160739, 139139

Offset: 1

Amiram Eldar, Dec 26 2024

			Fractions begin with 1, 0, 1, 3/4, 7/4, 3/4, 7/4, 13/8, 125/72, 53/72, 125/72, 107/72, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.12, p. 33.

Cf. A057521, A191622, A370902, A370903, A379583, A379586 (denominators).

f[p_, e_] := If[e > 1, p^e, 1]; powful[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[(-1)^(n+1)/powful[n], {n, 1, 50}]]]

powerful(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] > 1, f[i, 1]^f[i, 2], 1)); }
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / powerful(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A057521(k)).

a(n)/A379586(n) = (5/19) * A191622 * n + O(n^(1/2) * exp(-c1 * log(n)^(3/5) / log(log(n))^(1/5))) unconditionally, and with an improved error term O(n^(2/5) * exp(c2 * log(n) / log(log(n)))) assuming the Riemann hypothesis, where c1 and c2 are positive constants.

A379586 Denominators of the partial alternating sums of the reciprocals of the powerful part function (A057521).

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 4, 8, 72, 72, 72, 72, 72, 72, 72, 144, 144, 16, 16, 16, 16, 16, 16, 16, 400, 400, 10800, 10800, 10800, 10800, 10800, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 1058400

Offset: 1

Amiram Eldar, Dec 26 2024

Amiram Eldar, Table of n, a(n) for n = 1..1000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.12, p. 33.

Cf. A057521, A370902, A370903, A379584, A379585 (numerators).

f[p_, e_] := If[e > 1, p^e, 1]; powful[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[(-1)^(n+1)/powful[n], {n, 1, 50}]]]

powerful(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] > 1, f[i, 1]^f[i, 2], 1)); }
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / powerful(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/A057521(k)).

A326186 a(n) = n - A057521(n), where A057521 gives the powerful part of n.

Original entry on oeis.org

0, 1, 2, 0, 4, 5, 6, 0, 0, 9, 10, 8, 12, 13, 14, 0, 16, 9, 18, 16, 20, 21, 22, 16, 0, 25, 0, 24, 28, 29, 30, 0, 32, 33, 34, 0, 36, 37, 38, 32, 40, 41, 42, 40, 36, 45, 46, 32, 0, 25, 50, 48, 52, 27, 54, 48, 56, 57, 58, 56, 60, 61, 54, 0, 64, 65, 66, 64, 68, 69, 70, 0, 72, 73, 50, 72, 76, 77, 78, 64, 0, 81, 82, 80, 84

Offset: 1

Antti Karttunen, Jul 11 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A057521, A326185.

Cf. also A010848.

A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521.
A326186(n) = (n-A057521(n));

from math import prod
from sympy import factorint
def A326186(n): return n-n//prod(p for p, e in factorint(n).items() if e == 1) # Chai Wah Wu, Nov 14 2022

a(n) = n - A057521(n).

A381311 Numbers whose powerful part (A057521) is a power of a prime with an even exponent >= 2.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 112, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 188, 192, 198, 204, 207, 208, 212, 220, 228, 234, 236

Offset: 1

Amiram Eldar, Feb 19 2025

Numbers k whose largest unitary divisor that is a square, A350388(k), is a prime power (A246655), or equivalently, A350388(k) is in A056798 \ {1}.
Numbers having exactly one non-unitary prime factor and its multiplicity is even.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m} with m >= 1, i.e., any number (including zero) of 1's and then a single even number.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} p/((p-1)*(p+1)^2) = 0.24200684327095676029... .

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

Complement of A381312 within A190641.
Cf. A057521, A118914, A246655, A350388.
Subsequence of: A377816.
Subsequences: A001248, A030514, A030516, A030629, A030631, A054753, A056798 \ {1}, A085987, A178739, A179644, A179645, A179668, A179672, A179693, A179747, A189982, A189983, A189985, A189987, A190110, A190292, A190388.

q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;;,2]]]}, EvenQ[e[[1]]] && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000],q]

isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); !(e[1] % 2) && (#e == 1 || e[2] == 1));

A384519 Numbers whose powerful part (A057521) is greater than 1 and is equal to a squarefree number raised to an even power (A384517).

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 36, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 100, 112, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 196, 198, 204, 207, 208, 212, 220

Offset: 1

Amiram Eldar, Jun 01 2025

nonn

Subsequence of A240112 and first differs from it at n = 30: A240112(30) = 108 is not a term of this sequence.
Subsequence of A368714 and differs from it by not having the terms 1, 144, 324, 400, 432, ... .
Numbers whose prime factorization has one distinct exponent that is larger than 1 and it is even.
Numbers that are a product of a squarefree number (A005117) and a coprime nonsquarefree number that is a squarefree number raised to an even power (A384517).
The asymptotic density of this sequence is Sum_{k>=1} (d(2*k)-1)/zeta(2) = 0.265530259454558018819..., where d(k) = zeta(k) * Product_{p prime} (1 + Sum_{i=k+1..2*k-1} (-1)^i/p^i).

Intersection of A335275 and A375142.
Intersection of A368714 and A375142.
Equals A375142 \ A384520.
Subsequence of A013929 and A240112.
Subsequences: A067259, A384517.
Cf. A005117, A013661, A057521.

q[n_] := Module[{u = Union[Select[FactorInteger[n][[;; , 2]], # > 1 &]]}, Length[u] == 1 && EvenQ[u[[1]]]]; Select[Range[250], q]

isok(k) = {my(e = select(x -> (x > 1), Set(factor(k)[, 2]))); #e == 1 && !(e[1] % 2);}

cp's OEIS Frontend

A326184 a(n) = sigma(n) - A057521(n), where A057521 gives the powerful part of n, and sigma gives the sum of divisors of n.

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A349350 Dirichlet inverse of A057521, the powerful part of n.

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A349379 Möbius transform of A057521 (powerful part of n).

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A379583 Numerators of the partial sums of the reciprocals of the powerful part function (A057521).

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A379584 Denominators of the partial sums of the reciprocals of the powerful part function (A057521).

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A379585 Numerators of the partial alternating sums of the reciprocals of the powerful part function (A057521).

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A379586 Denominators of the partial alternating sums of the reciprocals of the powerful part function (A057521).

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A326186 a(n) = n - A057521(n), where A057521 gives the powerful part of n.

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