A335275 -id:A335275 - cp's OEIS frontend

A368169 The number of divisors of the largest unitary divisor of n that is a cubefull exponentially odd number (A368167).

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 14 2023

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

Cf. A000005, A013661, A098198, A335275, A335988, A368167, A368168.

f[p_, e_] := If[e == 1 || EvenQ[e], 1, e+1]; 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] == 1 || !(f[i, 2]%2), 1, f[i, 2]+1));}

Multiplicative with a(p^e) = e+1 if e is odd that is larger than 1, and 1 otherwise.
a(n) = A000005(A368167(n)).
a(n) >= 1, with equality if and only if n is in A335275.
a(n) <= A000005(n), with equality if and only if n is in A335988.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 3/p^3 - 2/p^4 - 1/p^5 + 1/p^6) = 1.47140789970892803631... .

A370080 The product of the even exponents of the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Feb 09 2024

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

Cf. A005361, A013661, A268335, A335275, A350386, A350388, A370079.

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

a(n) = vecprod(apply(x -> if(x%2, 1, x), factor(n)[, 2]));

a(n) = A005361(A350388(n)).
Multiplicative with a(p^e) = e if e is even, and 1 if e is odd.
a(n) = A005361(n)/A370079(n).
a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
a(n) <= A005361(n), with equality if and only if n is in A335275.
Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 1/p^s - 1/p^(3*s) + 1/p^(4*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)^2 * Product_{p prime} (1 - 1/p^2 - 1/p^3 +  2/p^4 - 1/p^5) = 1.53318063378623623841... .
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + (p^(2*s) + 1)/(p^s*(p^s - 1)*(p^s + 1)^2)). - Vaclav Kotesovec, Feb 11 2024

A377845 Numbers that have more than one odd exponent larger than 1 in their prime factorization.

Original entry on oeis.org

216, 864, 1000, 1080, 1512, 1944, 2376, 2744, 2808, 3000, 3375, 3456, 3672, 4000, 4104, 4320, 4968, 5400, 6048, 6264, 6696, 6750, 7000, 7560, 7776, 7992, 8232, 8856, 9000, 9261, 9288, 9504, 9720, 10152, 10584, 10648, 10976, 11000, 11232, 11448, 11880, 12000, 12744, 13000

Offset: 1

Amiram Eldar, Nov 09 2024

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p^2*(p+1))) * (1 + Sum_{p prime} (1/(p^3+p^2-1))) = 0.0035024748296318122535... .

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

Complement of the union of A335275 and A377844.
Subsequence of A295661.
Subsequences: A162142, A179671, A190011.
Cf. A065465.

q[n_] := Count[FactorInteger[n][[;; , 2]], _?(# > 1 && OddQ[#] &)] > 1; Select[Range[13000], q]

is(k) = #select(x -> x>1 && x%2, factor(k)[, 2]) > 1;

A336223 Numbers k such that the largest square dividing k is a unitary divisor of k and its square root has an even number of distinct prime divisors.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100, 101, 102, 103

Offset: 1

Amiram Eldar, Jul 12 2020

nonn

First differs from A333634 at n = 47.
Terms k of A335275 such that A000188(k) is a term of A030231.
Numbers whose powerful part (A057521) is a square term of A030231.
The squarefree numbers (A005117) are terms of this sequence since if k is squarefree, then the largest square dividing k is 1 which is a unitary divisor, sqrt(1) = 1 has 0 prime divisors, and 0 is even.
The asymptotic density of this sequence is (Product_{p prime} (1 - 1/(p^2*(p+1))) + Product_{p prime} (1 - (2*p+1)/(p^2*(p+1))))/2 = (0.881513... + 0.394391...)/2 = 0.637952807730728551636349961980617856650450613867264... (Cohen, 1964; the first product is A065465).

			36 is a term since the largest square dividing 36 is 36, which is a unitary divisor, sqrt(36) = 6, 6 = 2 * 3 has 2 distinct prime divisors, and 2 is even.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Trans. Amer. Math. Soc., Vol. 112 (1964), pp. 214-227.

Intersection of A333634 and A335275.

Cf. A000188, A001221, A005117, A030231, A057521, A065465.

seqQ[n_] := EvenQ @ Length[(e = Select[FactorInteger[n][[;; , 2]], # > 1 &])] && AllTrue[e, EvenQ[#] &]; Select[Range[100], seqQ]

A370079 The product of the odd exponents of the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Feb 09 2024

First differs from A363329 at n = 32.

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

Cf. A005361, A013661, A268335, A335275, A350387, A350389, A363329, A368472, A370080.

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

a(n) = vecprod(apply(x -> if(x%2, x, 1), factor(n)[, 2]));

a(n) = A005361(A350389(n)).
Multiplicative with a(p^e) = e if e is odd, and 1 if e is even.
a(n) = A005361(n)/A370080(n).
a(n) >= 1, with equality if and only if n is in A335275.
a(n) <= A005361(n), with equality if and only if n is an exponentially odd number (A268335).
Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 1/p^s - 1/p^(2*s) + 1/p^(3*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 1/p^4) = 1.32800597172596287374... .
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 2/((p^s - 1)*(p^s + 1)^2)). - Vaclav Kotesovec, Feb 11 2024

A375031 Numbers whose prime factorization has at least one exponent that equals 2 and no higher even exponent.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207, 212, 220, 225, 228, 234, 236, 242, 244, 245

Offset: 1

Amiram Eldar, Jul 28 2024

Subsequence of A304365 and differs from it by not having the terms 1, 144, 216, 324, 400, ... .
Subsequence of A038109 and differs from it by not having the terms 144, 324, 400, 576, 720, ... .
Numbers whose largest unitary divisor that is a square (A350388) is a square of squarefree number (A062503) that is larger than 1.
Each term is a product of two coprime numbers: an exponentially odd number (A268335) and a square of a squarefree number (A062503) that is larger than 1.
The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^3*(p+1))) - Product_{p prime}(1 - 1/(p*(p+1))) = A065466 - A065463 = 0.2432910611445097832029... .

			4 = 2^2 is a term because it has the exponent 2 in its prime factorization, and no higher even exponent.
144 = 2^4 * 3^2 is not a term because it has the exponent 4 in its prime factorization which is even and larger than 2.

Subsequence of A013929, A038109 and A304365.
A062503 \ {1} is a subsequence.
Cf. A065463, A065466, A268335, A335275, A350388, A375033.

q[n_] := Max[Select[FactorInteger[n][[;; , 2]], EvenQ]] == 2; Select[Range[250], q]

is(k) = {my(e = select(x -> !(x % 2), factor(k)[,2])); #e > 0 && vecmax(e) == 2;}

A375033(a(n)) = 2.

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);}

A336224 Numbers k such that the largest square dividing k is a unitary divisor of k and its square root has an even number of prime divisors (counted with multiplicity).

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, 36, 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, 100

Offset: 1

Amiram Eldar, Jul 12 2020

Terms k of A335275 such that A000188(k) is a term of A028260.
Numbers whose powerful part (A057521) is the square of a term of A028260.
The squarefree numbers (A005117) are terms of this sequence since if k is squarefree, then the largest square dividing k is 1 which is a unitary divisor, sqrt(1) has 0 prime divisors, and 0 is even.
The asymptotic density of this sequence is (5 * Product_{p prime} (1 - 1/(p^2*(p+1))) + 2 * Product_{p prime} (1 + 1/(p^2*(p+1))))/10 = (5 * 0.881513... + 2 * 1.125606...)/10 = 0.665878294481337275662425136416469977597382409701642... (Cohen, 1964; the first product is A065465).

			16 is a term since the largest square dividing 16 is 16, which is a unitary divisor, sqrt(16) = 4, 4 = 2 * 2 has 2 prime divisors, and 2 is even.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Trans. Amer. Math. Soc., Vol. 112 (1964), pp. 214-227.

Intersection of A335275 and A336222.

Cf. A000188, A001222, A005117, A028260, A057521, A065465.

seqQ[n_] := AllTrue[(e = FactorInteger[n][[;; , 2]]), # == 1 || EvenQ[#] &] && EvenQ @ Total[Select[e, # > 1 &]/2]; Select[Range[100], seqQ]

A382787 The product of exponents in the prime factorization of the numbers whose prime factorization contains exponents that are either 1 or even.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Apr 05 2025

First differs from A368473 at n = 57.

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

Cf. A005361, A013661, A065465, A335275, A368473.

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, # == 1 || EvenQ[#] &], Times @@ e, Nothing]]; Array[f, 150]

list(lim) = {my(e, ok); for(k = 1, lim, e = factor(k)[, 2]; ok = 1; for(i = 1, #e, if(e[i] > 1 && e[i]%2, ok = 0; break)); if(ok, print1(vecprod(e), ", ")));}

a(n) = A005361(A335275(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2)^2 / A065465) * Product_{p prime} (1 - 1/p^2 - 2/p^3 + 3/p^4 - 1/p^6) = 1.568148713987289233406... .

cp's OEIS Frontend

A368169 The number of divisors of the largest unitary divisor of n that is a cubefull exponentially odd number (A368167).

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A370080 The product of the even exponents of the prime factorization of n.

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A377845 Numbers that have more than one odd exponent larger than 1 in their prime factorization.

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A336223 Numbers k such that the largest square dividing k is a unitary divisor of k and its square root has an even number of distinct prime divisors.

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A370079 The product of the odd exponents of the prime factorization of n.

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A375031 Numbers whose prime factorization has at least one exponent that equals 2 and no higher even exponent.

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

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A336224 Numbers k such that the largest square dividing k is a unitary divisor of k and its square root has an even number of prime divisors (counted with multiplicity).

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