A350389 -id:A350389 - cp's OEIS frontend

A368471 a(n) is the sum of exponentially odd divisors of the largest unitary divisor of n that is an exponentially odd number (A268335).

1, 3, 4, 1, 6, 12, 8, 11, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 44, 1, 42, 31, 8, 30, 72, 32, 43, 48, 54, 48, 1, 38, 60, 56, 66, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 93, 72, 88, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144

Offset: 1

Amiram Eldar, Dec 26 2023

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

Cf. A000203, A000290, A005117, A033634, A268335, A350389, A368470.

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

a(n) = A033634(A350389(n)).
Multiplicative with a(p^e) = (p^(e+2) - p)/(p^2 - 1) + 1 if e is odd and 1 otherwise.
a(n) >= 1, with equality if and only if n is a square (A000290).
a(n) <= A000203(n), with equality if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^6/1080) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.51287686448947428073... .

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

A374538 a(n) is the sum of the squares of the unitary divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 5, 10, 1, 26, 50, 50, 65, 1, 130, 122, 10, 170, 250, 260, 1, 290, 5, 362, 26, 500, 610, 530, 650, 1, 850, 730, 50, 842, 1300, 962, 1025, 1220, 1450, 1300, 1, 1370, 1810, 1700, 1690, 1682, 2500, 1850, 122, 26, 2650, 2210, 10, 1, 5, 2900, 170, 2810, 3650, 3172

Offset: 1

Amiram Eldar, Jul 11 2024

The number of unitary divisors of n that are exponentially odd is A055076(n) and their sum is A358346(n).

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

Cf. A000290, A005117, A034676, A055076, A268335, A350389, A358346, A374537.

Cf. A002117, A013661, A183699.

f[p_, e_] := 1 + If[OddQ[e], p^(2*e), 0]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + if(f[i, 2]%2,  f[i, 1]^(2*f[i, 2]), 0));}

a(n) = A034676(A350389(n)).
a(n) >= 1 with equality if and only if n is a square (A000290).
a(n) <= A374537(n) with equality if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = p^(2*e) + 1 if e is odd, and 1 otherwise.
Dirichlet g.f.: zeta(s) * zeta(2*s-4) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-4) - 1/p^(2*s-2)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 2/p^2 + 1/p^3 - 1/p^4 + 1/p^5) = 0.79482441214759383925... .

A377820 Powerful numbers that have a single odd exponent in their prime factorization.

Original entry on oeis.org

8, 27, 32, 72, 108, 125, 128, 200, 243, 288, 343, 392, 432, 500, 512, 648, 675, 800, 968, 972, 1125, 1152, 1323, 1331, 1352, 1372, 1568, 1728, 1800, 2000, 2048, 2187, 2197, 2312, 2592, 2700, 2888, 3087, 3125, 3200, 3267, 3528, 3872, 3888, 4232, 4500, 4563, 4608, 4913, 5000

Offset: 1

Amiram Eldar, Nov 09 2024

First differs from A370786 at n = 124: A370786(124) = 27000 = 2^3 * 3^3 * 5*3 is not a term of this sequence.

Powerful numbers k such that A350389(k) is a prime power with an odd exponent (A246551).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Intersection of A001694 and A229125.
Intersection of A000037 and A377821.
Cf. A013661, A085541, A246551, A350389, A370786.

With[{max = 5000}, Select[Union@ Flatten@Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], # > 1 && Count[FactorInteger[#][[;; , 2]], _?OddQ] == 1 &]]

is(k) = if(k == 1, 0, my(e = factor(k)[, 2]); vecmin(e) > 1 && #select(x -> (x%2), e) == 1);

Sum_{n>=1} 1/a(n) = zeta(2) * P(3) = A013661 * A085541 = 0.28747301899596333866... .

A377991 Numbers k such that A351568(k) and A351569(k) are not coprime, where A351568 and A351569 are the sum of divisors of the largest unitary divisor of n that is a square, and of the largest unitary divisor of n that is an exponentially odd number, respectively.

Original entry on oeis.org

52, 98, 156, 164, 245, 260, 294, 332, 338, 364, 388, 392, 468, 490, 492, 539, 556, 572, 668, 722, 724, 735, 780, 820, 833, 845, 882, 884, 892, 927, 972, 976, 980, 988, 996, 1004, 1014, 1078, 1092, 1125, 1127, 1148, 1164, 1172, 1176, 1196, 1228, 1274, 1300, 1352, 1396, 1404, 1421, 1470, 1476, 1508, 1525, 1568, 1573

Offset: 1

Antti Karttunen, Nov 15 2024

nonn

			A351568(52) = 7 and A351569(52) = 14, so they share a factor (7), and therefore 52 is included as a term.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

Positions k where A377990(k) is larger than A051027(k).
Cf. A350388, A350389, A351568, A351569.
Subsequence of A336548.

A350388(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(0==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A351568(n) = sigma(A350388(n));
isA377991(n) = (1A351568(n), sigma(n)/A351568(n)));

{k such that gcd(A351568(n),A351569(n)) > 1}.

{k such that A377990(k) > A051027(k)}.

A380165 a(n) is the value of the Euler totient function when applied to the largest unitary divisor of n that is an exponentially odd number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jan 14 2025

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

Cf. A000010, A000290, A013662, A077591, A268335, A350389, A351569, A365402, A374456, A380164.

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

a(n) = A000010(A350389(n)).
a(n) >= 1, with equality if and only if n is either a square (A000290) or twice and odd square (A077591 \ {1}).
a(n) <= A000010(n), with equality if and only if n is an exponentially odd number (A268335).
Multiplicative with a(p^e) = (p-1)*p^(e-1) if e is odd, and 1 otherwise.
Dirichlet g.f.: zeta(2*s-2) * zeta(2*s) * Product_{p prime} (1 - 1/p^s + 1/p^(s-1) - 1/p^(2*s-2) - 1/p^(3*s-1) + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 2/p^4 + 1/p^5) = 0.50115112192510092436... .

A385007 The largest unitary divisor of n that is a biquadratefree number (A046100).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 15 2025

First differs from A053165 at n = 32 = 2^5: a(32) = 1 while A053165(32) = 2.
First differs from A383764 at n = 32 = 2^5: a(32) = 1 while A383764(32) = 32.
Equivalently, a(n) is the least divisor d of n such that n/d is a 4-full number (A036967).

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

Cf. A013661, A036967, A046100, A053165, A058035, A077610, A383764.

The largest unitary divisor of n that is: A000265 (odd), A006519 (power of 2), A055231 (squarefree), A057521 (powerful), A065330 (5-rough), A065331 (3-smooth), A350388 (square), A350389 (exponentially odd), A360539 (cubefree), A360540 (cubefull), A366126 (cube), A367168 (exponentially 2^n), this sequence (biquadratefree).

f[p_, e_] := If[e < 4, 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] < 4, f[i, 1]^f[i, 2], 1)); }

a(n) = 1 if and only if n is a 4-full number (A036967).
a(n) = n if and only if n is a biquadratefree number (A046100).
Multiplicative with a(p^e) = p^e if e <= 3, 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^(3-3*s) - p^(3-4*s) + p^(-4*s)).
Sum_{k=1..n} a(k) ~ c * zeta(2) * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^6 + 1/p^8 - 1/p^9) = 0.56331392082909224894... .

A374485 Lexicographically earliest infinite sequence such that a(i) = a(j) => A350388(i) = A350388(j) and A351569(i) = A351569(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 6, 11, 12, 13, 13, 14, 10, 15, 16, 17, 18, 19, 13, 20, 21, 22, 23, 24, 25, 26, 18, 27, 28, 29, 28, 30, 31, 20, 32, 33, 22, 34, 35, 36, 37, 26, 28, 38, 39, 40, 26, 41, 29, 42, 26, 42, 43, 33, 20, 44, 45, 34, 46, 47, 48, 49, 50, 51, 34, 49, 26, 52, 53, 54, 55, 56, 34, 57, 43, 58, 59, 60, 48, 61, 62, 63, 42, 64, 33, 65, 66, 44, 67, 49, 42, 68

Offset: 1

Antti Karttunen, Aug 06 2024

nonn

Restricted growth sequence transform of the ordered pair [A350388(n), A351569(n)].

For all i, j >= 1: a(i) = a(j) => A000203(i) = A000203(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A350388, A350389, A351569.

Cf. also A286603, A291751.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A350388(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(0==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A350389(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(1==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A351569(n) = sigma(A350389(n));
Aux374485(n) = [A350388(n), A351569(n)];
v374485 = rgs_transform(vector(up_to, n, Aux374485(n)));
A374485(n) = v374485[n];

cp's OEIS Frontend

A368471 a(n) is the sum of exponentially odd divisors of the largest unitary divisor of n that is an exponentially odd number (A268335).

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

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A374538 a(n) is the sum of the squares of the unitary divisors of n that are exponentially odd numbers (A268335).

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A377820 Powerful numbers that have a single odd exponent in their prime factorization.

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A377991 Numbers k such that A351568(k) and A351569(k) are not coprime, where A351568 and A351569 are the sum of divisors of the largest unitary divisor of n that is a square, and of the largest unitary divisor of n that is an exponentially odd number, respectively.

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A380165 a(n) is the value of the Euler totient function when applied to the largest unitary divisor of n that is an exponentially odd number.

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A385007 The largest unitary divisor of n that is a biquadratefree number (A046100).

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A374485 Lexicographically earliest infinite sequence such that a(i) = a(j) => A350388(i) = A350388(j) and A351569(i) = A351569(j), for all i, j >= 1.

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