A351560 -id:A351560 - cp's OEIS frontend

A080398 Largest squarefree number dividing sum of divisors of n.

1, 3, 2, 7, 6, 6, 2, 15, 13, 6, 6, 14, 14, 6, 6, 31, 6, 39, 10, 42, 2, 6, 6, 30, 31, 42, 10, 14, 30, 6, 2, 21, 6, 6, 6, 91, 38, 30, 14, 30, 42, 6, 22, 42, 78, 6, 6, 62, 57, 93, 6, 14, 6, 30, 6, 30, 10, 30, 30, 42, 62, 6, 26, 127, 42, 6, 34, 42, 6, 6, 6, 195, 74, 114, 62, 70, 6, 42, 10

Offset: 1

Labos Elemer, Mar 19 2003

nonn

Not multiplicative, but satisfies a similar condition: For all coprime x, y (with gcd(x,y)=1), a(x*y) = LCM(a(x), a(y)), where LCM is the least common multiply of its arguments, A003990. Compare also with A351560. - Antti Karttunen, Feb 20 2022

			n=12:sigma[12]=1+2+3+4+6+12=28, sqf-kernel=14=a(12)

Cf. A000203, A007947, A019565, A351560.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] cor[x_] := Apply[Times, ba[x]] Table[cor[DivisorSigma[1, w]], {w, 1, 100}]

a(n) = factorback(factor(sigma(n))[,1]); \\ Michel Marcus, Nov 18 2017

a(n) = A007947(A000203(n)).

a(n) = A019565(A351560(n)). - Antti Karttunen, Feb 20 2022

A353783 a(n) = LCM_{p^e||n} sigma(p^e), where n = Product_{p^e||n}, with each p^e the maximal power of prime p that divides n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 6, 12, 28, 14, 24, 12, 31, 18, 39, 20, 42, 8, 12, 24, 60, 31, 42, 40, 56, 30, 12, 32, 63, 12, 18, 24, 91, 38, 60, 28, 30, 42, 24, 44, 84, 78, 24, 48, 124, 57, 93, 36, 14, 54, 120, 12, 120, 20, 30, 60, 84, 62, 96, 104, 127, 42, 12, 68, 126, 24, 24, 72, 195, 74, 114, 124, 140, 24, 84, 80

Offset: 1

Antti Karttunen, May 08 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

Cf. A000203, A080398, A087207, A351560, A353784, A353785.
Cf. also A345044, A345046.
Cf. A336547 (positions where equal to sigma).

Array[LCM @@ DivisorSigma[1, Power @@@ FactorInteger[#]] &, 79] (* Michael De Vlieger, May 08 2022 *)

A353783(n) = { my(f=factor(n)~); lcm(vector(#f, i, sigma(f[1, i]^f[2, i]))); };

a(n) = A000203(n) / A353784(n).
a(n) = A353785(n) * A080398(n).
For all n >= 1, A087207(a(n)) = A351560(n).

A351559 a(n) = A048675(gcd(sigma(n), A019565(n))).

Original entry on oeis.org

0, 2, 1, 0, 1, 2, 1, 0, 0, 2, 3, 8, 9, 2, 3, 0, 1, 2, 1, 0, 1, 2, 3, 0, 0, 10, 1, 8, 5, 2, 1, 0, 1, 2, 3, 32, 1, 6, 1, 0, 9, 2, 1, 8, 33, 2, 3, 0, 0, 2, 3, 0, 1, 6, 3, 0, 1, 2, 3, 8, 1, 2, 33, 0, 1, 2, 65, 0, 1, 2, 3, 0, 1, 2, 1, 12, 1, 10, 5, 0, 16, 2, 3, 0, 1, 18, 7, 0, 1, 2, 9, 8, 1, 2, 7, 0, 1, 2, 35, 0, 65, 2, 33

Offset: 1

Antti Karttunen, Feb 19 2022

Cf. A000203, A004198, A007947, A019565, A080398, A351557, A351560.

Cf. also A324644, A351558.

Table[If[# == 1, 0, Total[#2*2^PrimePi[#1] & @@@ FactorInteger[#]]/2] &@ GCD[DivisorSigma[1, n], Times @@ Prime@ Flatten@ Position[Reverse@ IntegerDigits[n, 2], 1]], {n, 103}] (* Michael De Vlieger, Feb 20 2022 *)

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A351559(n) = A048675(gcd(sigma(n), A019565(n)));

a(n) = A048675(A351557(n)) = A048675(gcd(sigma(n), A019565(n))).

a(n) = n AND A351560(n), where AND is bitwise-and, A004198.

cp's OEIS Frontend

A080398 Largest squarefree number dividing sum of divisors of n.

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A353783 a(n) = LCM_{p^e||n} sigma(p^e), where n = Product_{p^e||n}, with each p^e the maximal power of prime p that divides n.

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A351559 a(n) = A048675(gcd(sigma(n), A019565(n))).

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