A337050 -id:A337050 - cp's OEIS frontend

A368886 The largest unitary divisor of n without an exponent 2 in its prime factorization (A337050).

1, 2, 3, 1, 5, 6, 7, 8, 1, 10, 11, 3, 13, 14, 15, 16, 17, 2, 19, 5, 21, 22, 23, 24, 1, 26, 27, 7, 29, 30, 31, 32, 33, 34, 35, 1, 37, 38, 39, 40, 41, 42, 43, 11, 5, 46, 47, 48, 1, 2, 51, 13, 53, 54, 55, 56, 57, 58, 59, 15, 61, 62, 7, 64, 65, 66, 67, 17, 69, 70

Offset: 1

Amiram Eldar, Jan 09 2024

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

Cf. A062503, A337050, A368884.

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

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 2, 1, f[i,1]^f[i,2]));}

from math import prod
from sympy import factorint
def A368886(n): return prod(p**e for p, e in factorint(n).items() if e!=2) # Chai Wah Wu, Jan 09 2024

Multiplicative with a(p^2) = 1, and a(p^e) = p^e if e != 2.
a(n) = n / A368884(n).
a(n) >= 1, with equality if and only if n is in A062503.
a(n) <= n, with equality if and only if n is in A337050.
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^(2*s-2) + 1/p^(2*s) + 1/p^(3*s-3) - 1/p^(3*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^3 + 1/p^4 - 1/p^5) = 0.78357388280736936739... .

A330596 Decimal expansion of Product_{primes p} (1 - 1/p^2 + 1/p^3).

Original entry on oeis.org

7, 4, 8, 5, 3, 5, 2, 5, 9, 6, 8, 2, 3, 6, 3, 5, 6, 4, 6, 4, 4, 2, 1, 5, 0, 4, 8, 6, 3, 7, 9, 1, 0, 6, 0, 1, 6, 4, 1, 6, 4, 0, 3, 4, 3, 0, 0, 5, 3, 2, 4, 4, 0, 4, 5, 1, 5, 8, 5, 2, 7, 9, 3, 9, 2, 5, 9, 2, 5, 5, 8, 6, 8, 9, 5, 4, 9, 5, 8, 8, 3, 4, 2, 1, 2, 6, 2, 0, 6, 8, 1, 4, 6, 4, 7, 0, 9, 8, 1, 3, 1, 4, 3, 3, 5, 4

Offset: 0

Vaclav Kotesovec, Dec 19 2019

The asymptotic density of A337050. - Amiram Eldar, Aug 13 2020

			0.748535259682363564644215048637910601641640343005324404515852793925925...

Cf. A057723, A065464, A065473, A065476, A065487, A328017, A330594, A330595, A337050, A372636.

Do[Print[N[Exp[-Sum[q = Expand[(p^2 - p^3)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 110]], {t, 20, 200, 20}]

prodeulerrat(1 - 1/p^2 + 1/p^3) \\ Amiram Eldar, Mar 17 2021

Equals (6/Pi^2) * A065487. - Amiram Eldar, Jun 10 2020

A360540 a(n) is the cubefull part of n: the largest divisor of n that is a cubefull number (A036966).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 16, 81, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Feb 11 2023

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

Cf. A004709, A036966, A057521, A360539.

f[p_, e_] := If[e > 2, p^e, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] > 2, f[i, 1]^f[i, 2], 1));}

a(n) = 1 if and only if n is a cubefree number (A004709).
a(n) = n if and only if n is a cubefull number (A036966).
a(n) <= A057521(n) with equality if and only if n is in A337050.
a(n) = n/A360539(n).
Multiplicative with a(p^e) = p^e if e >= 3, and 1 otherwise.
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - p^(1-s) + p^(-s) - p^(1-3*s) - p^(1-2*s) + p^(-2*s) + p^(3-3*s)).

A360539 a(n) is the cubefree part of n: the largest unitary divisor of n that is a cubefree number (A004709).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 1, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 3, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69, 70

Offset: 1

Amiram Eldar, Feb 11 2023

Equivalently, a(n) is the least divisor d of n such that n/d is a cubefull number (A036966).

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

Cf. A004709, A007948, A036966, A055231, A360540.

f[p_, e_] := If[e < 3, p^e, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] < 3, f[i, 1]^f[i, 2], 1));}

from math import prod
from sympy import factorint
def A360539(n): return prod(p**e for p,e in factorint(n).items() if e<=2) # Chai Wah Wu, Aug 06 2024

a(n) = 1 if and only if n is a cubefull number (A036966).
a(n) = n if and only if n is a cubefree number (A004709).
a(n) >= A055231(n) with equality if and only if n is in A337050.
a(n) = n/A360540(n).
Multiplicative with a(p^e) = p^e if e <= 2, and 1 otherwise.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(1-s) - p^(-s) + p^(2-2*s) - p^(1-2*s) - p^(2-3*s) + p^(-3*s)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^5 + 1/p^6 - 1/p^7) = 0.4213813264... .

A386796 Numbers that have exactly one exponent in their prime factorization that is equal to 2.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 108, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 164, 169, 171, 172, 175, 188, 198, 200, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 260, 261, 268

Offset: 1

Amiram Eldar, Aug 02 2025

First differs from its subsequence A060687 at n = 16: a(16) = 72 is not a term of A060687.
Differs from A286228 by having the terms 60, 72, 84, 90, ..., and not having the term 1.
Numbers k such that A369427(k) = 1.
The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^2 + 1/p^3) * Sum_{p prime} (p-1)/(p^3 - p + 1) = 0.22661832022705616779... (the product is A330596) (Elma and Martin, 2024).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ertan Elma and Greg Martin, Distribution of the number of prime factors with a given multiplicity, Canadian Mathematical Bulletin, Vol. 67, No. 4 (2024), pp. 1107-1122; arXiv preprint, arXiv:2406.04574 [math.NT], 2024.
Index entries for sequences computed from indices in prime factorization.

A060687 is a subsequence.
Cf. A286228, A330596, A369427.
Numbers that have exactly one exponent in their prime factorization that is equal to k: A119251 (k=1), this sequence (k=2), A386800 (k=3), A386804 (k=4), A386808 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 2: A337050 (m=0), this sequence (m=1), A386797 (m=2), A386798 (m=3).

f[p_, e_] := If[e == 2, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[300], s[#] == 1 &]

isok(k) = vecsum(apply(x -> if(x == 2, 1, 0), factor(k)[, 2])) == 1;

A386797 Numbers that have exactly two exponents in their prime factorization that are equal to 2.

Original entry on oeis.org

36, 100, 180, 196, 225, 252, 300, 396, 441, 450, 468, 484, 588, 612, 676, 684, 700, 828, 882, 980, 1044, 1089, 1100, 1116, 1156, 1225, 1260, 1300, 1332, 1444, 1452, 1476, 1521, 1548, 1575, 1692, 1700, 1800, 1900, 1908, 1980, 2028, 2100, 2116, 2124, 2156, 2178, 2196

Offset: 1

Amiram Eldar, Aug 02 2025

First differs from its subsequence A375144 at n = 38: a(38) = 1800 = 2^3 * 3^2 * 5^2 is not a term of A375144.
Numbers k such that A369427(k) = 2.
The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^2 + 1/p^3) * ((Sum_{p prime} (p-1)/(p^3 - p + 1))^2 - Sum_{p prime} ((p-1)^2/(p^3 - p + 1)^2)) / 2 = 0.023701044250873975412... (the product is A330596) (Elma and Martin, 2024).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ertan Elma and Greg Martin, Distribution of the number of prime factors with a given multiplicity, Canadian Mathematical Bulletin, Vol. 67, No. 4 (2024), pp. 1107-1122; arXiv preprint, arXiv:2406.04574 [math.NT], 2024.
Index entries for sequences computed from indices in prime factorization.

A375144 is a subsequence.	
Cf. A330596, A369427.
Numbers that have exactly two exponents in their prime factorization that are equal to k: this sequence (k=2), A386801 (k=3), A386805 (k=4), A386809 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 2: A337050 (m=0), A386796 (m=1), this sequence (m=2), A386798 (m=3).

f[p_, e_] := If[e == 2, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[2200], s[#] == 2 &]

isok(k) = vecsum(apply(x -> if(x == 2, 1, 0), factor(k)[, 2])) == 2;

A386798 Numbers that have exactly three exponents in their prime factorization that are equal to 2.

Original entry on oeis.org

900, 1764, 4356, 4900, 6084, 6300, 8820, 9900, 10404, 11025, 11700, 12100, 12996, 14700, 15300, 16900, 17100, 19044, 19404, 20700, 21780, 22050, 22932, 23716, 26100, 27225, 27900, 28900, 29988, 30276, 30420, 30492, 33124, 33300, 33516, 34596, 36100, 36300, 36900, 38025, 38700

Offset: 1

Amiram Eldar, Aug 02 2025

Numbers k such that A369427(k) = 2.

The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^2 + 1/p^3) * (s(1)^3 + 3*s(1)*s(2) + 2*s(3)) / 6 = 0.0011175284878980531468... (the product is A330596), where s(m) = (-1)^(m-1) * Sum_{p prime} (1/(p^3/(p-1)-1))^m (Elma and Martin, 2024).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ertan Elma and Greg Martin, Distribution of the number of prime factors with a given multiplicity, Canadian Mathematical Bulletin, Vol. 67, No. 4 (2024), pp. 1107-1122; arXiv preprint, arXiv:2406.04574 [math.NT], 2024.
Index entries for sequences computed from indices in prime factorization.

Cf. A330596, A369427.
Numbers that have exactly three exponents in their prime factorization that are equal to k: this sequence (k=2), A386802 (k=3), A386806 (k=4), A386810 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 2: A337050 (m=0), A386796 (m=1), A386797 (m=2), this sequence (m=3).

f[p_, e_] := If[e == 2, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[40000], s[#] == 3 &]

isok(k) = vecsum(apply(x -> if(x == 2, 1, 0), factor(k)[, 2])) == 3;

A386799 Numbers without an exponent 3 in their prime factorization.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75

Offset: 1

Amiram Eldar, Aug 02 2025

First differs from its subsequence A336592 at n = 116: a(116) = 128 = 2^7 is not a term of A336592.
Numbers k such that A295883(k) = 0.
These numbers were named semi-3-free integers by Suryanarayana (1971).
The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^3 + 1/p^4) = 0.90470892696874750603... (Suryanarayana, 1971).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ertan Elma and Greg Martin, Distribution of the number of prime factors with a given multiplicity, Canadian Mathematical Bulletin, Vol. 67, No. 4 (2024), pp. 1107-1122; arXiv preprint, arXiv:2406.04574 [math.NT], 2024.
D. Suryanarayana, Semi-k-free integers, Elemente der Mathematik, Vol. 26 (1971), pp. 39-40.
D. Suryanarayana and R. Sitaramachandra Rao, Distribution of semi-k-free integers, Proceedings of the American Mathematical Society, Vol. 37, No. 2 (1973), pp. 340-346.
Index entries for sequences computed from indices in prime factorization.

Complement of A176297.	
A336592 is a subsequence.
Cf. A295883.
Numbers without an exponent k in their prime factorization: A001694 (k=1), A337050 (k=2), this sequence (k=3), A386803 (k=4), A386807 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 3: this sequence (m=0), A386800 (m=1), A386801 (m=2), A386802 (m=3).

Select[Range[100], !MemberQ[FactorInteger[#][[;; , 2]], 3] &]

isok(k) = vecsum(apply(x -> if(x == 3, 1, 0), factor(k)[, 2])) == 0;

A386803 Numbers without an exponent 4 in their prime factorization.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Amiram Eldar, Aug 03 2025

First differs from its subsequence A209061 at n = 246: a(246) = 256 = 2^8 is not a term of A209061.
First differs from its subsequences A115063 and A369939 at n = 62: a(62) = 64 = 2^6 is not a term of A115063.
The complement of this sequence is a subsequence of A336595.
These numbers were named semi-4-free integers by Suryanarayana (1971).
The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^4 + 1/p^5) = 0.95908865419555719109... (Suryanarayana, 1971).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ertan Elma and Greg Martin, Distribution of the number of prime factors with a given multiplicity, Canadian Mathematical Bulletin, Vol. 67, No. 4 (2024), pp. 1107-1122; arXiv preprint, arXiv:2406.04574 [math.NT], 2024.
D. Suryanarayana, Semi-k-free integers, Elemente der Mathematik, Vol. 26 (1971), pp. 39-40.
D. Suryanarayana and R. Sitaramachandra Rao, Distribution of semi-k-free integers, Proceedings of the American Mathematical Society, Vol. 37, No. 2 (1973), pp. 340-346.
Index entries for sequences computed from indices in prime factorization.

Subsequences: A115063, A209061, A369939.
Numbers without an exponent k in their prime factorization: A001694 (k=1), A337050 (k=2), A386799 (k=3), this sequence (k=4), A386807 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 4: this sequence (m=0), A386804 (m=1), A386805 (m=2), A386806 (m=3).

Select[Range[100], !MemberQ[FactorInteger[#][[;; , 2]], 4] &]

isok(k) = vecsum(apply(x -> if(x == 4, 1, 0), factor(k)[, 2])) == 0;

A386807 Numbers without an exponent 5 in their prime factorization.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Amiram Eldar, Aug 03 2025

First differs from its subsequence A166718 at n = 47: a(47) = 48 = 2^4 * 3 is not a term of A166718.
Differs from A373868 by having the terms 1, 1024, 32768, 59049, ..., and not having the terms 96, 160, 224, ... .
These numbers were named semi-5-free integers by Suryanarayana (1971).
The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^5 + 1/p^6) = 0.98136375107187963656... (Suryanarayana, 1971).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ertan Elma and Greg Martin, Distribution of the number of prime factors with a given multiplicity, Canadian Mathematical Bulletin, Vol. 67, No. 4 (2024), pp. 1107-1122; arXiv preprint, arXiv:2406.04574 [math.NT], 2024.
D. Suryanarayana, Semi-k-free integers, Elemente der Mathematik, Vol. 26 (1971), pp. 39-40.
D. Suryanarayana and R. Sitaramachandra Rao, Distribution of semi-k-free integers, Proceedings of the American Mathematical Society, Vol. 37, No. 2 (1973), pp. 340-346.
Index entries for sequences computed from indices in prime factorization.

A166718 is a subsequence.
Cf. A373868.
Numbers without an exponent k in their prime factorization: A001694 (k=1), A337050 (k=2), A386799 (k=3), A386803 (k=4), this sequence (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 5: this sequence (m=0), A386808 (m=1), A386809 (m=2), A386810 (m=3).

Select[Range[100], !MemberQ[FactorInteger[#][[;; , 2]], 5] &]

isok(k) = vecsum(apply(x -> if(x == 5, 1, 0), factor(k)[, 2])) == 0;

cp's OEIS Frontend

A368886 The largest unitary divisor of n without an exponent 2 in its prime factorization (A337050).

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A330596 Decimal expansion of Product_{primes p} (1 - 1/p^2 + 1/p^3).

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A360540 a(n) is the cubefull part of n: the largest divisor of n that is a cubefull number (A036966).

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A360539 a(n) is the cubefree part of n: the largest unitary divisor of n that is a cubefree number (A004709).

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A386796 Numbers that have exactly one exponent in their prime factorization that is equal to 2.

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A386797 Numbers that have exactly two exponents in their prime factorization that are equal to 2.

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A386798 Numbers that have exactly three exponents in their prime factorization that are equal to 2.

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A386799 Numbers without an exponent 3 in their prime factorization.

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