A356192 -id:A356192 - cp's OEIS frontend

A356191 a(n) is the smallest exponentially odd number that is divisible by n.

1, 2, 3, 8, 5, 6, 7, 8, 27, 10, 11, 24, 13, 14, 15, 32, 17, 54, 19, 40, 21, 22, 23, 24, 125, 26, 27, 56, 29, 30, 31, 32, 33, 34, 35, 216, 37, 38, 39, 40, 41, 42, 43, 88, 135, 46, 47, 96, 343, 250, 51, 104, 53, 54, 55, 56, 57, 58, 59, 120, 61, 62, 189, 128, 65

Offset: 1

Amiram Eldar, Jul 29 2022

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

Cf. A064549, A007913, A268335, A336643, A350390.

Similar sequences: A053149, A053143, A053167, A066638, A087320, A087321, A197863, A356192, A356193, A356194.

f[p_, e_] := If[OddQ[e], p^e, p^(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]%2, f[i,1]^f[i,2], f[i,1]^(f[i,2]+1)))};

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - p^2*X^2) * (1 + p*X + p^3*X^2 - p^2*X^2))[n], ", ")) \\ Vaclav Kotesovec, Sep 09 2023

Multiplicative with a(p^e) = p^e if e is odd and p^(e+1) otherwise.
a(n) = n iff n is in A268335.
a(n) = A064549(n)/A007913(n).
a(n) = n*A336643(n).
a(n) = n^2/A350390(n).
From Vaclav Kotesovec, Sep 09 2023: (Start)
Let f(s) = Product_{p prime} (1 - p^(6-5*s) + p^(7-5*s) + 2*p^(5-4*s) - p^(6-4*s) + p^(3-3*s) - p^(4-3*s) - 2*p^(2-2*s)).
Sum_{k=1..n} a(k) ~ Pi^2 * f(2) * n^2 / 24 * (log(n) + 3*gamma - 1/2 + 12*zeta'(2)/Pi^2 + f'(2)/f(2)), where
f(2) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = A256392 = 0.2177787166195363783230075141194468131307977550013559376482764035236264911...,
f'(2) = f(2) * Sum_{p prime} (11*p - 5) * log(p) / (p^3 + p^2 - 3*p + 1) = f(1) * 4.7165968208567630786609552448708126340725121316268495170070986645608062483...
and gamma is the Euler-Mascheroni constant A001620. (End)

A356193 a(n) is the smallest cubefull number (A036966) that is a multiple of n.

Original entry on oeis.org

1, 8, 27, 8, 125, 216, 343, 8, 27, 1000, 1331, 216, 2197, 2744, 3375, 16, 4913, 216, 6859, 1000, 9261, 10648, 12167, 216, 125, 17576, 27, 2744, 24389, 27000, 29791, 32, 35937, 39304, 42875, 216, 50653, 54872, 59319, 1000, 68921, 74088, 79507, 10648, 3375, 97336

Offset: 1

Amiram Eldar, Jul 29 2022

First differs from A053149 and A356192 at n=16.

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

Cf. A002117, A036966, A360541.

Similar sequences: A053143, A053149, A053167, A066638, A087320, A087321, A197863, A356191, A356192, A356194.

f[p_, e_] := p^Max[e, 3]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]

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

Multiplicative with a(p^e) = p^max(e,3).
a(n) = n iff n is in A036966.
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + (3*p-2)/(p^3*(p-1))) = 1.76434793373691907811... . - Amiram Eldar, Jul 29 2022
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(3)/4) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 2/p^5 - 1/p^6 - 1/p^8 + 2/p^9 - 1/p^10) = 0.1559111567... . - Amiram Eldar, Nov 13 2022
a(n) = n * A360541(n). - Amiram Eldar, Sep 01 2023

A356194 a(n) is the smallest multiple of n whose prime factorization exponents are all powers of 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 16, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 48, 25, 26, 81, 28, 29, 30, 31, 256, 33, 34, 35, 36, 37, 38, 39, 80, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 162, 55, 112, 57, 58, 59, 60, 61, 62, 63, 256, 65, 66, 67

Offset: 1

Amiram Eldar, Jul 29 2022

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

Cf. A138302, A353897.

Similar sequences: A053143, A053149, A053167, A066638, A087320, A087321, A197863, A356191, A356192, A356193.

f[p_, e_] := p^(2^Ceiling[Log2[e]]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

Multiplicative with a(p^e) = p^(2^ceiling(log_2(e))).

a(n) = n iff n is in A138302.

A365298 a(n) is the smallest number k such that k*n is a cubefull exponentially odd number (A335988).

Original entry on oeis.org

1, 4, 9, 2, 25, 36, 49, 1, 3, 100, 121, 18, 169, 196, 225, 2, 289, 12, 361, 50, 441, 484, 529, 9, 5, 676, 1, 98, 841, 900, 961, 1, 1089, 1156, 1225, 6, 1369, 1444, 1521, 25, 1681, 1764, 1849, 242, 75, 2116, 2209, 18, 7, 20, 2601, 338, 2809, 4, 3025, 49, 3249, 3364

Offset: 1

Amiram Eldar, Aug 31 2023

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

Cf. A157292, A335988, A356192.

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

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

Multiplicative with a(p) = p^2, a(p^e) = p if e is even, and a(p^e) = 1 is e is odd and > 1.
a(n) = A356192(n)/n.
a(n) = 1 if and only if n is in A335988.
Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 1/p^(3*s) - 1/p^(3*s-2) - 1/p^(2*s) + 1/p^(2*s-1) + 1/p^(s-2)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = (2*Pi^4/315) * Product_{p prime} (1 - p^2 - p^3 + p^4 + p^8 + p^9)/(p^8*(p+1)) = 0.207915752545... .

A367933 a(n) is the smallest multiple of n that is an exponentially odious number (A270428).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 16, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 48, 25, 26, 81, 28, 29, 30, 31, 128, 33, 34, 35, 36, 37, 38, 39, 80, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 162, 55, 112, 57, 58, 59, 60, 61, 62, 63, 128, 65, 66, 67

Offset: 1

Amiram Eldar, Dec 05 2023

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

Cf. A270428, A367931.

Similar sequences: A356192, A365684, A367934.

f[p_, e_] := Module[{k = e}, While[! OddQ[DigitCount[k, 2 ,1]], k++]; p^k]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(e) = {my(k = e); while(!(hammingweight(k)%2), k++); k; };
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2])); }

Multiplicative with a(p^e) = p^s(e), s(e) = min{k >= e, k is odious}.
a(n) = n * A367931(n).
a(n) >= n, with equality if and only if n is an exponentially odious number (A209061).
Sum_{k=1..n} a(k) ~  c * n^2 / 2, where c = Product_{p prime} f(1/p) = 1.30023300... , where f(x) = (1-x) * (1 + Sum_{k>=1} x^(2*k-s(k))), and s(k) is defined above.

A367934 a(n) is the smallest multiple of n that is an exponentially evil number (A262675).

Original entry on oeis.org

1, 8, 27, 8, 125, 216, 343, 8, 27, 1000, 1331, 216, 2197, 2744, 3375, 32, 4913, 216, 6859, 1000, 9261, 10648, 12167, 216, 125, 17576, 27, 2744, 24389, 27000, 29791, 32, 35937, 39304, 42875, 216, 50653, 54872, 59319, 1000, 68921, 74088, 79507, 10648, 3375, 97336

Offset: 1

Amiram Eldar, Dec 05 2023

First differs from A356192 at n = 64.

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

Cf. A262675, A367932.

Similar sequences: A356192, A365684, A367933.

f[p_, e_] := Module[{k = e}, While[! EvenQ[DigitCount[k, 2 ,1]], k++]; p^k]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(e) = {my(k = e); while(hammingweight(k)%2, k++); k; };
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2])); }

Multiplicative with a(p^e) = p^s(e), s(e) = min{k >= e, k is evil}.
a(n) = n * A367932(n).
a(n) >= n, with equality if and only if n is an exponentially evil number (A262675).
Sum_{k=1..n} a(k) ~  c * n^4 / 4, where c = Product_{p prime} f(1/p) = 0.623746285..., where f(x) = (1-x) * (1 + Sum_{k>=1} x^(4*k-s(k))), and s(k) is defined above.
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^s(k)) = 1.70170328791367919805... .

A368170 The largest cubefull exponentially odd divisor of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 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, 8, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 27, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 14 2023

First differs from A008834 at n = 32, and from A366906 at n = 64.

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

Cf. A004709, A335988, A350390, A356192, A368167, A368171.

Cf. A008834, A366906.

f[p_, e_] := If[e <= 2, 1, If[EvenQ[e], p^(e-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, if(!(f[i,2]%2), f[i,1]^(f[i, 2]-1), f[i, 1]^f[i, 2])))};

Multiplicative with a(p^e) = 1 if e <= 2, a(p^e) = p^e if e is odd and e > 1, and p^(e-1) otherwise.
a(n) = n/A368171(n).
a(n) >= 1, with equality if and only if n is cubefree (A004709).
a(n) <= n, with equality if and only if n is in A335988.

A369757 The number of divisors of the smallest cubefull exponentially odd number that is divisible by n.

Original entry on oeis.org

1, 4, 4, 4, 4, 16, 4, 4, 4, 16, 4, 16, 4, 16, 16, 6, 4, 16, 4, 16, 16, 16, 4, 16, 4, 16, 4, 16, 4, 64, 4, 6, 16, 16, 16, 16, 4, 16, 16, 16, 4, 64, 4, 16, 16, 16, 4, 24, 4, 16, 16, 16, 4, 16, 16, 16, 16, 16, 4, 64, 4, 16, 16, 8, 16, 64, 4, 16, 16, 64, 4, 16, 4

Offset: 1

Amiram Eldar, Jan 31 2024

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

Cf. A000005, A335988, A356192, A365348, A369758, A369759.

f[p_, e_] := If[OddQ[e], Max[e, 3] + 1, e + 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x%2, max(x, 3) + 1, x + 2), factor(n)[, 2]));

a(n) = A000005(A356192(n)).
Multiplicative with a(p^e) = max(e,3) + 1 if e is odd, and e+2 if e is even.
a(n) >= A000005(n), with equality if and only if n is cubefull exponentially odd number (A335988).
Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 + 3/p^s - 1/p^(2*s) - 3/p^(3*s) + 2/p^(4*s)).
From Vaclav Kotesovec, Feb 02 2024: (Start)
Dirichlet g.f.: zeta(s)^4 * Product_{p prime} (1 + (7*p^(2*s) + 2*p^(3*s) - 6*p^(4*s) - 7*p^s + 2) / ((p^s+1)*p^(5*s))).
Sum_{k=1..n} a(k) = c * n*log(n)^3/6 + O(n*log(n)^2), where c = Product_{p prime} (1 - (6*p^4 - 2*p^3 - 7*p^2 + 7*p - 2) / ((p+1)*p^5)) = 0.124604542136592401049820049658828040278... (End)

A369758 The sum of divisors of the smallest cubefull exponentially odd number that is divisible by n.

Original entry on oeis.org

1, 15, 40, 15, 156, 600, 400, 15, 40, 2340, 1464, 600, 2380, 6000, 6240, 63, 5220, 600, 7240, 2340, 16000, 21960, 12720, 600, 156, 35700, 40, 6000, 25260, 93600, 30784, 63, 58560, 78300, 62400, 600, 52060, 108600, 95200, 2340, 70644, 240000, 81400, 21960, 6240

Offset: 1

Amiram Eldar, Jan 31 2024

First differs from A369720 at n = 16.

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

Cf. A000203, A335988, A356192, A365349, A369720, A369757, A369759.

Cf. A013662, A013664.

f[p_, e_] := (p^If[OddQ[e], Max[e, 3] + 1, e + 2] - 1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^if(f[i,2]%2, max(f[i,2], 3) + 1, f[i,2] + 2) - 1)/(f[i,1] - 1));}

from math import prod
from sympy import factorint
def A369758(n): return prod((p**((3 if e==1 else e)+1+(e&1^1))-1)//(p-1) for p,e in factorint(n).items()) # Chai Wah Wu, Feb 03 2024

a(n) = A000203(A356192(n)).
Multiplicative with a(p) = p^3 + p^2 + p + 1, a(p^e) = (p^(e+1)-1)/(p-1) for an odd e >= 3, and a(p^e) = (p^(e+2)-1)/(p-1) for an even e.
a(n) >= A000203(n), with equality if and only if n is cubefull exponentially odd number (A335988).
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-3) + 1/p^(s-2) + 1/p^(s-1) - 1/p^(2*s-2) - 1/p^(3*s-5) - 1/p^(3*s-4) - 1/p^(3*s-3) + 1/p^(4*s-5) + 1/p^(4*s-4)).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(4) * zeta(6) * Product_{p prime} (1 - 1/p^4 - 1/p^6 + 1/p^10 + 1/p^11 - 1/p^13) = 1.00040193512214077945... .
Equivalently, c = Product_{p prime} (1 + 1/(p^3*(p^4 - 1)*(p^4 + p^2 + 1))). - Vaclav Kotesovec, Feb 02 2024

A369759 The sum of unitary divisors of the smallest cubefull exponentially odd number that is divisible by n.

Original entry on oeis.org

1, 9, 28, 9, 126, 252, 344, 9, 28, 1134, 1332, 252, 2198, 3096, 3528, 33, 4914, 252, 6860, 1134, 9632, 11988, 12168, 252, 126, 19782, 28, 3096, 24390, 31752, 29792, 33, 37296, 44226, 43344, 252, 50654, 61740, 61544, 1134, 68922, 86688, 79508, 11988, 3528, 109512

Offset: 1

Amiram Eldar, Jan 31 2024

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

Cf. A034448, A335988, A356192, A365480, A369757, A369758.

Cf. A013661, A013662, A013664.

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

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

a(n) = A034448(A356192(n)).
Multiplicative with a(p) = p^3 + 1, a(p^e) = p^e + 1 for an odd e >= 3, and a(p^e) = p^(e+1) + 1 for an even e.
a(n) >= A034448(n), with equality if and only if n is cubefull exponentially odd number (A335988).
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-3) - 1/p^(2*s-2) - 1/p^(3*s-5) + 1/p^(4*s-5) - 1/p^(4*s-3)).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = (zeta(4)*zeta(6)/zeta(2)) * Product_{p prime} (1 - 1/p^6 + 1/p^11 - 1/p^12) = 0.65813930591740259189... .

cp's OEIS Frontend

A356191 a(n) is the smallest exponentially odd number that is divisible by n.

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A356193 a(n) is the smallest cubefull number (A036966) that is a multiple of n.

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A356194 a(n) is the smallest multiple of n whose prime factorization exponents are all powers of 2.

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A365298 a(n) is the smallest number k such that k*n is a cubefull exponentially odd number (A335988).

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A367933 a(n) is the smallest multiple of n that is an exponentially odious number (A270428).

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A367934 a(n) is the smallest multiple of n that is an exponentially evil number (A262675).

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A368170 The largest cubefull exponentially odd divisor of n.

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A369757 The number of divisors of the smallest cubefull exponentially odd number that is divisible by n.

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