A318060 -id:A318060 - cp's OEIS frontend

A318059 a(n) is the numerator of sigma(sigma(n))/n.

1, 2, 7, 2, 12, 14, 15, 3, 14, 39, 28, 14, 24, 30, 4, 2, 39, 28, 42, 24, 3, 91, 60, 7, 32, 48, 10, 30, 72, 13, 63, 13, 124, 60, 124, 28, 60, 84, 40, 117, 96, 6, 84, 56, 56, 195, 124, 14, 80, 64, 65, 171, 120, 20, 39, 45, 62, 117, 168, 8, 96, 126, 10, 2, 224, 403, 126, 78, 84, 403

Offset: 1

Michel Marcus, Aug 14 2018

			Fractions begin with 1, 2, 7/3, 2, 12/5, 14/3, 15/7, 3, 14/9, 39/10, 28/11, 14/3, ...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma), A051027.

Cf. A098223 (when denominator is 1), A318060 (denominator).

seq(numer((numtheory:-sigma@@2)(n)/n),n=1..100); # Robert Israel, Aug 14 2018

a[n_] := Numerator[DivisorSigma[1, DivisorSigma[1, n]]/n]; Array[a, 100] (* Amiram Eldar, Apr 04 2024 *)

PARI
```
a(n) = numerator(sigma(sigma(n))/n);
```

For n=2, sigma(sigma(n)) = 4, so a(2) = 2.

A318083 Numbers m such that sigma(sigma(m))/sigma(m) <= sigma(m)/m.

Original entry on oeis.org

1, 2, 4, 8, 9, 12, 16, 18, 25, 32, 36, 48, 50, 52, 64, 72, 80, 81, 84, 90, 98, 100, 108, 112, 128, 144, 150, 156, 162, 180, 192, 196, 200, 208, 225, 240, 242, 244, 252, 256, 288, 289, 292, 300, 320, 324, 336, 338, 360, 372, 388, 392, 400, 420, 432, 441, 444, 448, 450, 468, 484

Offset: 1

Michel Marcus, Aug 15 2018

nonn

A318059 and A318060 give f(n) = sigma(sigma(n))/n.
Let g(n) = sigma(n)/n, then f(n) = g(sigma(n))*g(n).
For g(), see A017665 and A017666.
Then this sequence are the integers m for which g(sigma(m)) <= g(m).
Are there other integers than 1, for which sigma(sigma(m))/sigma(m) = sigma(m)/m?

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

Cf. A000203 (sigma), A017665, A017666, A051027, A318059, A318060, A318084.

q[n_] := Module[{s = DivisorSigma[1, n]}, DivisorSigma[1, s]/s <= s/n]; Select[Range[500], q] (* Amiram Eldar, Apr 04 2024 *)

isok(n) = my(sn=sigma(n)); sigma(sn)/sn <= sn/n;

A318084 Numbers m such that sigma(sigma(m))/m is a square.

Original entry on oeis.org

1, 15, 50, 100, 168, 1023, 1444, 1470, 1600, 1944, 3179, 3822, 4000, 5120, 5776, 6174, 9025, 10752, 12348, 14440, 15125, 21970, 26250, 28416, 28665, 29127, 37544, 39200, 45630, 47151, 49392, 52500, 60984, 66125, 67200, 69819, 71485, 77175, 80000, 90250, 100254, 102300, 102400

Offset: 1

Michel Marcus, Aug 15 2018

nonn

This is a necessary condition to have sigma(sigma(m))/sigma(m) = sigma(m)/m.
Are there other integers than 1, for which this is satisfied?
If m is an odd number such that sigma(sigma(m^2))/2 is a square, and p is in A000043 such that 2^p-1 does not divide sigma(m^2), then 2^(p-1)*m^2 is in the sequence. Such m include 5, 19, 161, 543, 1031, 1899, 3035, 6673. Thus if A000043 is infinite, so is this sequence. - Robert Israel, Aug 17 2018

Giovanni Resta, Table of n, a(n) for n = 1..5000 (first 200 terms from Robert Israel)

Cf. A000203 (sigma), A000043, A051027, A006532, A069070.

Cf. A318059, A318060, A318083.

filter:= proc(n) local t; t:= (numtheory:-sigma @@2)(n)/n; issqr(numer(t)) and issqr(denom(t)) end proc:select(filter, [$1..200000]); # Robert Israel, Aug 17 2018

Select[Range[10^5], IntegerQ@ Sqrt[ DivisorSigma[1, DivisorSigma[1, #]] #] &] (* Giovanni Resta, Aug 19 2018 *)

PARI
```
isok(n) = issquare(sigma(sigma(n))/n);
```

A331019 a(n) is the least k such that the denominator(sigma(sigma(kn))/(kn)) equals n.

Original entry on oeis.org

1, 15, 1, 8, 1, 144, 1, 16, 1, 1, 1, 8, 1, 255, 3, 16, 1, 12, 1, 2, 7, 1, 1, 288, 1, 18, 21, 8, 1, 84, 1, 13, 1, 11, 1, 4096, 1, 4, 3, 270, 1, 2448, 1, 2, 3, 1, 1, 16, 1, 420, 3, 1, 1, 124, 3, 16, 3, 128, 1, 616, 1, 85, 3, 16, 1, 1, 1, 8, 3, 1, 1, 32, 1, 64

Offset: 1

Ray Chandler, Jan 09 2020

nonn

a(n) is the least k such that the A318060(k*n) equals n.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma), A051027 (sigma(sigma)), A318060.

Cf. A019278, A330598, A331033.

a(n) = my(k=n); while (denominator(sigma(sigma(k))/k) != n, k+=n); k/n;

a(n) = A331033(n)/n.

A331033 a(n) is the least k such that the denominator(sigma(sigma(k))/k) equals n.

Original entry on oeis.org

1, 30, 3, 32, 5, 864, 7, 128, 9, 10, 11, 96, 13, 3570, 45, 256, 17, 216, 19, 40, 147, 22, 23, 6912, 25, 468, 567, 224, 29, 2520, 31, 416, 33, 374, 35, 147456, 37, 152, 117, 10800, 41, 102816, 43, 88, 135, 46, 47, 768, 49, 21000, 153, 52, 53, 6696, 165, 896, 171, 7424, 59, 36960

Offset: 1

Michel Marcus, Jan 08 2020

nonn

a(n) is the least k such that the A318060(k) equals n.

a(n) is divisible by n.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma), A051027 (sigma(sigma)), A318060.

Cf. A019278, A330598, A331019.

a(n) = my(k=n); while (denominator(sigma(sigma(k))/k) != n, k+=n); k;

A353751 a(n) = gcd(n, sigma(sigma(n))), where sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 1, 8, 1, 1, 1, 4, 1, 2, 15, 16, 1, 2, 1, 4, 21, 1, 1, 24, 1, 2, 9, 4, 1, 15, 1, 8, 1, 2, 1, 4, 1, 2, 3, 2, 1, 42, 1, 4, 3, 1, 1, 16, 1, 2, 3, 1, 1, 18, 5, 8, 3, 2, 1, 60, 1, 2, 21, 64, 1, 1, 1, 4, 3, 1, 1, 24, 1, 2, 1, 4, 7, 6, 1, 16, 1, 2, 1, 84, 5, 2, 3, 2, 1, 6, 1, 4, 3, 1, 5, 8, 1, 2, 1, 4

Offset: 1

Antti Karttunen, May 08 2022

nonn

Index entries for sequences related to sigma(n)

Cf. A000203, A051027, A318059, A318060.
Cf. A019278 (positions of fixed points, where a(n)=n).
Cf. also A009194, A295310.

PARI
```
A353751(n) = gcd(n, sigma(sigma(n)));
```

a(n) = gcd(n, A051027(n)).

a(n) = A051027(n) / A318059(n) = n / A318060(n).

cp's OEIS Frontend

A318059 a(n) is the numerator of sigma(sigma(n))/n.

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A318083 Numbers m such that sigma(sigma(m))/sigma(m) <= sigma(m)/m.

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A318084 Numbers m such that sigma(sigma(m))/m is a square.

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A331019 a(n) is the least k such that the denominator(sigma(sigma(k*n))/(k*n)) equals n.

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A331033 a(n) is the least k such that the denominator(sigma(sigma(k))/k) equals n.

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A353751 a(n) = gcd(n, sigma(sigma(n))), where sigma is the sum of divisors function.

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PARI

Formula

A331019 a(n) is the least k such that the denominator(sigma(sigma(kn))/(kn)) equals n.