A241625 -id:A241625 - cp's OEIS frontend

A240667 a(n) is the GCD of the solutions x of sigma(x) = n; sigma(n) = A000203(n) = sum of divisors of n.

1, 0, 2, 3, 0, 5, 4, 7, 0, 0, 0, 1, 9, 13, 8, 0, 0, 1, 0, 19, 0, 0, 0, 1, 0, 0, 0, 12, 0, 29, 1, 1, 0, 0, 0, 22, 0, 37, 18, 27, 0, 1, 0, 43, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 49, 0, 0, 1, 0, 61, 32, 0, 0, 0, 0, 67, 0, 0, 0, 1, 0, 73, 0, 0, 0, 45, 0, 1, 0, 0

Offset: 1

Michel Marcus, Apr 10 2014

nonn

From n = 1 to 5, the least integers such that a(x) = n, depending on if singletons (see A007370 and A211656) are accepted or not, are 1, 3, 4, 7, 6 or 12, 126, 124, 210, 22152.

Is it possible to find an integer n such that a(n) = 6? Answer: n = A241625(6) = 6187272.

			There are no integers such that sigma(x) = 2, so a(2) = 0.
There is a single integer, x = 2, such that sigma(x) = 3, so a(3) = 2.
There are 2 integers, x = 6 and 11, such that sigma(x)=12, their gcd is 1, so a(12) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A007369, A007370, A211656, A241479 (a variant), A241625.

A240667 := n -> igcd(op(select(k->sigma(k)=n, [$1..n]))):
seq(A240667(n), n=1..82); # Peter Luschny, Apr 13 2014

a[n_] := GCD @@ Select[Range[n], DivisorSigma[1, #] == n&];
Array[a, 100] (* Jean-François Alcover, Jul 30 2018 *)

sigv(n) =  select(i->sigma(i) == n, vector(n, i, i));
a(n) = {v = sigv(n); if (#v == 0, 0, gcd(v));}

a(n) = my(s = invsigma(n)); if(#s, gcd(s), 0); \\ Amiram Eldar, Dec 19 2024, using Max Alekseyev's invphi.gp

a(A007369(n)) = 0.

A241646 Numbers m such that the GCD of the x's that satisfy sigma(x) = m is 1.

Original entry on oeis.org

1, 12, 18, 24, 31, 32, 42, 48, 54, 56, 60, 72, 80, 84, 90, 96, 98, 104, 108, 114, 120, 128, 132, 140, 144, 152, 156, 168, 180, 182, 192, 216, 224, 228, 234, 240, 248, 252, 264, 270, 272, 280, 288, 294, 308, 312, 324, 336, 342, 360, 372, 384, 390, 408, 420

Offset: 1

Michel Marcus, Apr 26 2014

nonn

			We have sigma(6) = sigma(11) = 12, and gcd(6, 11) = 1, hence 12 is in the sequence.
For x in [20, 26, 41], sigma(x) = 42, and gcd(20, 26, 41) = 1, hence 42 is here.

Robert Israel, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A240667, A241625, A241646, A241647, A241648, A241649, A241650.

N:= 10^4: # for terms <= N
V:= Vector(N):
for x from 1 to N do
  s:= numtheory:-sigma(x);
  if s <= N then
    if V[s] = 0 then V[s]:= x
    else V[s]:= igcd(V[s], x)
    fi
  fi
od: select(t -> V[t]=1, [$1..N]);  # Robert Israel, Aug 18 2019

is(k) = gcd(invsigma(k)) == 1; \\ Amiram Eldar, Dec 19 2024, using Max Alekseyev's invphi.gp

A241647 Numbers m such that the GCD of the x's that satisfy sigma(x) = m is 2.

Original entry on oeis.org

3, 126, 186, 399, 924, 1350, 1386, 1530, 1806, 2106, 2646, 2652, 2814, 2916, 3066, 3150, 3654, 3870, 4662, 4914, 6162, 6426, 6846, 6882, 6930, 7098, 7566, 7620, 8190, 8910, 9270, 10842, 11076, 12222, 12870, 14586, 14910, 15210, 15246, 15930, 16506, 17010

Offset: 1

Michel Marcus, Apr 26 2014

nonn

			We have sigma(68) = sigma(82) = 126, and gcd(68, 82) = 2, hence 126 is in the sequence.
On the other hand, for x in [20, 26, 41], sigma(x) = 42, and gcd(20, 26, 41) = 1, hence 42 is not here, although gcd(20, 26) is 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1849 from Robert Israel)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A240667, A241625, A241646, A241647, A241648, A241649, A241650.

N:= 10^5: # for terms <= N
V:= Vector(N):
for x from 1 to N do
  s:= numtheory:-sigma(x);
  if s <= N then
    if V[s] = 0 then V[s]:= x
    else V[s]:= igcd(V[s], x)
    fi
  fi
od: select(t -> V[t]=2, [$1..N]); # Robert Israel, Aug 18 2019

is(k) = gcd(invsigma(k)) == 2; \\ Amiram Eldar, Dec 19 2024, using Max Alekseyev's invphi.gp

A241648 Numbers m such that the GCD of the x's that satisfy sigma(x) = m is 3.

Original entry on oeis.org

4, 124, 320, 392, 416, 800, 1352, 1520, 2912, 2960, 3536, 3872, 5720, 5936, 6320, 7112, 8216, 9176, 9912, 10472, 11816, 12152, 12896, 13280, 14960, 15176, 16080, 16400, 16536, 18032, 18392, 18560, 19136, 19880, 20000, 21632, 21680, 21920, 22736, 23120, 23816

Offset: 1

Michel Marcus, Apr 26 2014

nonn

			We have sigma(48) = sigma(75) = 124, and gcd(48, 75) = 3, hence 124 is in the sequence.
Likewise, we have sigma(x) = 2912 for x = [1116, 1236, 1701, 2007, 2181], with gcd 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..850 from Robert Israel)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A240667, A241625, A241646, A241647, A241648, A241649, A241650.

N:= 10^5: # for terms <= N
V:= Vector(N):
for x from 1 to N do
  s:= numtheory:-sigma(x);
  if s <= N then
    if V[s] = 0 then V[s]:= x
    else V[s]:= igcd(V[s], x)
    fi
  fi
od: select(t -> V[t]=3, [$1..N]); # Robert Israel, Aug 18 2019

is(k) = gcd(invsigma(k)) == 3; \\ Amiram Eldar, Dec 19 2024, using Max Alekseyev's invphi.gp

A241649 Numbers m such that the GCD of the x's that satisfy sigma(x) = m is 4.

Original entry on oeis.org

7, 210, 378, 630, 1904, 3570, 6188, 6510, 7154, 9296, 9800, 10220, 12446, 13664, 14378, 17654, 17780, 18536, 19110, 19376, 19530, 20034, 20580, 21266, 23240, 23310, 24150, 24584, 25298, 26754, 27930, 28938, 29106, 29610, 30380, 31640, 34146, 34230, 34664

Offset: 1

Michel Marcus, Apr 26 2014

nonn

			sigma(104) = sigma(116) = 210, and gcd(104, 116) = 4, hence 210 is in the sequence.
Likewise 6510 is obtained with sigma of [2600, 2900, 3464, 3716], with gcd 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..698 from Robert Israel)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A240667, A241625, A241646, A241647, A241648, A241650.

N:= 10^5: # for terms <= N
V:= Vector(N):
for x from 1 to N do
  s:= numtheory:-sigma(x);
  if s <= N then
    if V[s] = 0 then V[s]:= x
    else V[s]:= igcd(V[s],x)
    fi
  fi
od:
select(t -> V[t]=4, [$1..N]); # Robert Israel, Aug 18 2019

is(k) = gcd(invsigma(k)) == 4; \\ Amiram Eldar, Dec 19 2024, using Max Alekseyev's invphi.gp

A241650 Numbers m such that the GCD of the x's that satisfy sigma(x) = m is 5.

Original entry on oeis.org

6, 22152, 35724, 125892, 132444, 146484, 166764, 182052, 192504, 202332, 239304, 242580, 248664, 267852, 291252, 300612, 375492, 375804, 434772, 460044, 494364, 536952, 547992, 550524, 618852, 628212, 646668, 707304, 708984, 752232, 777852, 824304, 828984

Offset: 1

Michel Marcus, Apr 26 2014

nonn

			Only sigma(5) = 6, with gcd(5) = 5, so 6 is in the sequence.
Also, sigma(12735) = sigma(18455) = 22152, and gcd(12735, 18455) = 5, hence 22152 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..5000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A240667, A241625, A241646, A241647, A241648, A241649.

is(k) = gcd(invsigma(k)) == 5; \\ Amiram Eldar, Dec 19 2024, using Max Alekseyev's invphi.gp

A380303 Numbers m such that GCD of the solutions x to sigma(x) = sigma(m) are setwise coprime.

Original entry on oeis.org

1, 6, 10, 11, 14, 15, 16, 17, 20, 21, 23, 24, 25, 26, 28, 30, 31, 33, 34, 35, 38, 39, 40, 41, 42, 44, 46, 47, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 65, 66, 69, 70, 71, 74, 76, 77, 78, 79, 83, 84, 85, 86, 87, 88, 89, 90, 92, 93, 94, 95, 96, 97, 99, 102, 103, 105, 107, 108, 110, 111, 112, 113, 114, 115, 117, 118, 119, 120, 123, 124, 125, 126, 127, 130, 131, 132, 135, 136, 138, 139, 140, 141, 142, 143, 145, 147, 150

Offset: 1

Max Alekseyev, Jan 19 2025

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A206036, A211656, A241480, A241481, A241625, A380304.

isok(k) = if(k == 1, 1, my(v = invsigma(sigma(k))); #v > 1 && gcd(v) == 1); \\ Amiram Eldar, May 28 2025, using Max Alekseyev's invphi.gp (see links).

Union of {1} and the set difference of A206036 and A241481.

A380304 a(n) = sigma(A380303(n)).

Original entry on oeis.org

1, 12, 18, 12, 24, 24, 31, 18, 42, 32, 24, 60, 31, 42, 56, 72, 32, 48, 54, 48, 60, 56, 90, 42, 96, 84, 72, 48, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 96, 104, 84, 144, 96, 144, 72, 114, 140, 96, 168, 80, 84, 224, 108, 132, 120, 180, 90, 234, 168, 128, 144, 120, 252, 98, 156, 216, 104, 192, 108, 280, 216, 152, 248, 114, 240, 144, 182, 180, 144, 360, 168, 224, 156, 312, 128, 252, 132, 336, 240, 270, 288, 140, 336, 192

Offset: 1

Max Alekseyev, Jan 19 2025

nonn

Except for a(1)=1, subsequence of A206421.

Cf. A000203, A206036, A211656, A241480, A241481, A241625, A380303.

a(n) = sigma(A380303(n)) = A000203(A380303(n)).

cp's OEIS Frontend

A240667 a(n) is the GCD of the solutions x of sigma(x) = n; sigma(n) = A000203(n) = sum of divisors of n.

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A241646 Numbers m such that the GCD of the x's that satisfy sigma(x) = m is 1.

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A241647 Numbers m such that the GCD of the x's that satisfy sigma(x) = m is 2.

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A241648 Numbers m such that the GCD of the x's that satisfy sigma(x) = m is 3.

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A241649 Numbers m such that the GCD of the x's that satisfy sigma(x) = m is 4.

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A241650 Numbers m such that the GCD of the x's that satisfy sigma(x) = m is 5.

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A380303 Numbers m such that GCD of the solutions x to sigma(x) = sigma(m) are setwise coprime.

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A380304 a(n) = sigma(A380303(n)).

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