A351568 -id:A351568 - cp's OEIS frontend

A351575 Positions of primes in A351568.

4, 9, 12, 16, 18, 20, 25, 28, 44, 45, 48, 50, 52, 60, 63, 64, 68, 72, 75, 76, 80, 84, 90, 92, 99, 108, 112, 116, 117, 124, 126, 132, 140, 148, 150, 153, 156, 164, 171, 172, 175, 176, 188, 192, 198, 200, 204, 207, 208, 212, 220, 228, 234, 236, 240, 244, 260, 261, 268, 272, 275, 276, 279, 284, 288, 289, 292, 304, 306

Offset: 1

Antti Karttunen, Feb 24 2022

nonn

Numbers k such that A350388(k) is one of the terms of A023194.

Positions of primes in A351568, positions of ones in A351570.

Cf. A000203, A023194, A350388.

f[p_, e_] := If[EvenQ[e], (p^(e + 1) - 1)/(p - 1), 1]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[300], PrimeQ[s[#]] &] (* Amiram Eldar, Feb 25 2022 *)

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));
isA351575(n) = isprime(A351568(n));

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)}.

A342925 a(n) = A003415(sigma(n)), where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

Original entry on oeis.org

0, 1, 4, 1, 5, 16, 12, 8, 1, 21, 16, 32, 9, 44, 44, 1, 21, 16, 24, 41, 80, 60, 44, 92, 1, 41, 68, 92, 31, 156, 80, 51, 112, 81, 112, 20, 21, 92, 92, 123, 41, 272, 48, 124, 71, 156, 112, 128, 22, 34, 156, 77, 81, 244, 156, 244, 176, 123, 92, 332, 33, 272, 164, 1, 124, 384, 72, 165, 272, 384, 156, 119, 39, 101, 128, 188

Offset: 1

Antti Karttunen, Apr 07 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A023194 (positions of ones, which is a subsequence of prime powers, A000961).
Cf. A342922, A342923, A342924, A342926, A351568, A351569, A351570, A351571.
Cf. A342021 (fixed points), A343216 [positions k where a(k) < k], A343217 [a(k) >= k], A343218 [a(k) > k].
Cf. A347870 (parity of terms), A347872, A347873, A347877 (positions of odd terms), A347878 (of even terms), A343218, A343220, A344024.

Array[If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ DivisorSigma[1, #] &, 76] (* Michael De Vlieger, Apr 08 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342925(n) = A003415(sigma(n));

a(A023194(n)) = 1.
If gcd(m,n) = 1, a(m*n) = sigma(m)*A003415(sigma(n)) + sigma(n)*A003415(sigma(m)) = sigma(m)*a(n) + sigma(n)*a(m).
a(n) = (A351568(n)*A351571(n)) + (A351569(n)*A351570(n)). - Antti Karttunen, Feb 23 2022

A358347 a(n) is the sum of the unitary divisors of n that are squares.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 1, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 1, 26, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 50, 1, 1, 1, 1, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 10, 65, 1, 1, 1, 5, 1, 1, 1, 10, 1, 1, 26, 5, 1, 1, 1, 17, 82, 1

Offset: 1

Amiram Eldar, Nov 11 2022

The number of unitary divisors of n that are squares is A056624(n).

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

Cf. A056624, A077610, A078434, A247041, A268335, A350388, A351568.

Similar sequences: A033634, A034448, A035316, A358346.

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

a(n) >= 1 with equality if and only if n is an exponentially odd number (A268335).
Multiplicative with a(p^e) = p^e + 1 if e is even, and 1 otherwise.
a(n) = A034448(n)/A358346(n).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)/(3*zeta(5/2)) = 0.6491241554... .
Dirichlet g.f.: zeta(s)*zeta(2*s-2)/zeta(3*s-2). - Amiram Eldar, Jan 29 2023
a(n) = A034448(A350388(n)). - Amiram Eldar, Sep 09 2023

A351569 Sum of divisors of the largest unitary divisor of n that is an exponentially odd number.

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 15, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 60, 1, 42, 40, 8, 30, 72, 32, 63, 48, 54, 48, 1, 38, 60, 56, 90, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 120, 72, 120, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 15, 74, 114, 4, 20, 96, 168, 80, 6, 1, 126, 84

Offset: 1

Antti Karttunen, Feb 23 2022

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n).

Cf. A000203, A013662, A028982 (positions of odd terms), A268335 (exponentially odd numbers), A350389, A351568, A351571.

Coincides with A001615 on squarefree numbers, A005117.

f[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 23 2022 *)

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));

from math import prod
from sympy import factorint
def A351569(n): return prod((p**(e+1)-1)//(p-1) if e % 2 else 1 for p, e in factorint(n).items()) # Chai Wah Wu, Feb 24 2022

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if e is odd and 1 otherwise.
a(n) = A000203(A350389(n)).
a(n) = A000203(n) / A351568(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(4)/2 = Pi^4/180 = 0.541161... . - Amiram Eldar, Nov 20 2022
Dirichlet g.f.: zeta(2*s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^s - 1/p^(2*s-2)). - Amiram Eldar, Sep 03 2023

A351570 Arithmetic derivative of the sum of the divisors of the largest unitary divisor of n that is a square.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 22, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 22, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 22, 1, 38

Offset: 1

Antti Karttunen, Feb 23 2022

nonn

Observation: There seems to be no terms in range 2..19 in this sequence.

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

Cf. A000203, A003415, A342925, A350388, A351568, A351571, A351572, A351575 (positions of ones).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
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); };
A351570(n) = A003415(sigma(A350388(n)));

a(n) = A003415(A351568(n)) = A342925(A350388(n)) = A003415(A000203(A350388(n))).

A365401 The number of divisors of the largest unitary divisor of n that is a square.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 5, 3, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 5, 5, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Sep 03 2023

First differs from A212181 at n = 32.
The sum of these divisors is A351568(n).
All the terms are odd.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000005, A000290, A212181, A268335 A350388, A351568, A365402.

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

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

a(n) = A000005(A350388(n)).
a(n) = A000005(n) / A365402(n).
a(n) <= A000005(n) with equality if and only if n is a square (A000290).
a(n) >= 1 with equality if and only if n is an exponentially odd number (A268335).
Multiplicative with a(p^e) = 1 if e is odd, and e+1 if e is even.
Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 1/p^s + 1/p^(2*s) - 1/p^(3*s)).
From Vaclav Kotesovec, Sep 05 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s)^2 * Product_{p prime} (1 - 2/p^(3*s) + 1/p^(4*s)).
Let f(s) = Product_{p prime} (1 - 2/p^(3*s) + 1/p^(4*s)).
Sum_{k=1..n} a(k) ~ f(1) * Pi^4 * n / 36 + sqrt(n) * zeta(1/2) * f(1/2)/2 * (log(n) + 4*gamma - 2 + zeta'(1/2)/zeta(1/2) + f'(1/2)/f(1/2)), where
f(1) = Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.7446954979060674204391238715944543281179691329049241118630718137015097502...,
f(1/2) = Product_{p prime} (1 - 2/p^(3/2) + 1/p^2) = 0.2312522106782016049013780988087017618011735848676872392115785564006277675...,
f'(1/2) = f(1/2) * Sum_{p prime} 2*(3*sqrt(p) - 2) * log(p) / (1 - 2*sqrt(p) + p^2) = f(1/2) * 6.937179176924511608542644054340717439502789953858512457656... and gamma is the Euler-Mascheroni constant A001620. (End)

A367171 The sum of divisors of the largest unitary divisor of n that is a term of A138302.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 1, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 4, 31, 42, 1, 56, 30, 72, 32, 1, 48, 54, 48, 91, 38, 60, 56, 6, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 3, 72, 8, 80, 90, 60, 168, 62, 96, 104, 1, 84, 144, 68, 126

Offset: 1

Amiram Eldar, Nov 07 2023

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

Cf. A000203, A048298, A138302, A306633, A367168, A367169, A367170.

Similar sequences: A351568, A351569.

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

Multiplicative with a(p^e) = (p^(A048298(e)+1)-1)/(p-1).
a(n) = A000203(A367168(n)).
a(n) <= A000203(n), with equality if and only if n is in A138302.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2)/zeta(3) = 1.368432... (A306633).

A367988 The sum of the divisors of the square root of the largest unitary divisor of n that is a square.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 1, 3, 1, 1, 1, 7, 1, 4, 1, 3, 1, 1, 1, 1, 6, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 7, 8, 6, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 15, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 6, 3, 1, 1, 1, 7, 13, 1, 1, 3, 1, 1

Offset: 1

Amiram Eldar, Dec 07 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000203, A071974, A268335, A351568, A367987.

Cf. A065463.

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

Multiplicative with a(p^e) = (p^(e/2+1)-1)/(p-1) if e is even and 1 otherwise.
a(n) = A000203(A071974(n)).
a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
Dirichlet g.f.: zeta(2*s) * zeta(2*s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s-1)).
From Vaclav Kotesovec, Apr 20 2025: (Start)
Let f(s) = Product_{p prime} (1 - 1/((p^s + 1)*p^(2*s - 1))).
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * f(s).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where
f(1) = A065463 = Product_{p prime} (1 - 1/(p*(1+p))) = 0.704442200999165592736603350326637210188586431417098049414226842591097056682...
f'(1) = f(1) * Sum_{p prime} (3*p+2)*log(p)/((p+1)*(p^2+p-1)) = f(1) * 1.167129912223800181472507785468113632129480568043855995406075158923507536957...
and gamma is the Euler-Mascheroni constant A001620. (End)

A377990 a(n) = sigma(sigma(A350388(n))) * sigma(sigma(A350389(n))), where A350388 and A350389 are the largest unitary divisor of n that is a square, and the largest unitary divisor of n that is an exponentially odd number, respectively.

Original entry on oeis.org

1, 4, 7, 8, 12, 28, 15, 24, 14, 39, 28, 56, 24, 60, 60, 32, 39, 56, 42, 96, 63, 91, 60, 168, 32, 96, 90, 120, 72, 195, 63, 104, 124, 120, 124, 112, 60, 168, 120, 234, 96, 252, 84, 224, 168, 195, 124, 224, 80, 128, 195, 192, 120, 360, 195, 360, 186, 234, 168, 480, 96, 252, 210, 128, 224, 403, 126, 312, 252, 403, 195

Offset: 1

Antti Karttunen, Nov 15 2024

nonn

Differs from A051027 at 52, 98, 156, 164, 245, ..., = A377991.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A051027, A350388, A350389, A351568, A351569, A377991.

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); };
A377990(n) = (sigma(sigma(A350388(n))) * sigma(sigma(A350389(n))));

a(n) = A051027(A350388(n)) * A051027(A350389(n)).

a(n) = sigma(A351568(n)) * sigma(A351569(n)).

cp's OEIS Frontend

A351575 Positions of primes in A351568.

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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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A342925 a(n) = A003415(sigma(n)), where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

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A358347 a(n) is the sum of the unitary divisors of n that are squares.

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A351569 Sum of divisors of the largest unitary divisor of n that is an exponentially odd number.

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A351570 Arithmetic derivative of the sum of the divisors of the largest unitary divisor of n that is a square.

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A365401 The number of divisors of the largest unitary divisor of n that is a square.

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A367171 The sum of divisors of the largest unitary divisor of n that is a term of A138302.

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