cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A241625 Smallest number m such that the GCD of the x's that satisfy sigma(x)=m is n.

Original entry on oeis.org

1, 3, 4, 7, 6, 6187272, 8, 15, 13, 196602, 8105688, 28, 14
Offset: 1

Views

Author

Michel Marcus, Apr 26 2014

Keywords

Comments

This sequence is a sequel to A240667.
Some large known terms: a(16)=2031554, a(25)=1355816, a(31)=8880128, a(80)=11532, a(97)=5488.
a(14) > 10^9. - Michel Marcus, May 09 2014
a(n) is a multiple of A353783(n). Some further terms: a(15) = 497943732, a(17) = 962949708, a(20) = 612372264, a(48) = 12692888, a(53) = 39887316. - Max Alekseyev, Jan 19 2025

Examples

			a(2) = 3, because the only x such that sigma(x)=3 is 2.
a(6) = 6187272, because the x's that satisfy sigma(x)=6187272 are [2651676, 2855646] and their GCD is 6.

Links

Crossrefs

Cf. A000203, A211656, A240667, A353783, A380303, A380304.

Programs

  • PARI
    lista() = {lim = 12000000; nn = 100; out = "a241625.txt"; v = vector(lim, i, sigma(i)); w = vector(lim); for (i=1, lim, vi = v[i]; if (vi <= lim, if (w[vi] == 0, w[vi] = i, w[vi] = concat(w[vi], i)););); for (i=1, nn, got = 0; write1(out, i, " "); for (j=1, #w, wj = w[j]; if (gcd(wj) == i, got = 1; write(out, j);break;);); if (! got, write(out, );););}
  • PARI
    a(n) = my(m=1); while(gcd(invsigma(m)) != n, m++); m; \\ Michel Marcus, Jan 16 2025; using Max Alekseyev's invphi.gp

Formula

For n in A211656, a(n) = sigma(n).

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

Views

Author

Michel Marcus, Apr 26 2014

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Maple
    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
  • PARI
    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

Views

Author

Michel Marcus, Apr 26 2014

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Maple
    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
  • PARI
    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

Views

Author

Michel Marcus, Apr 26 2014

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Maple
    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
  • PARI
    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

Views

Author

Michel Marcus, Apr 26 2014

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Maple
    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
  • PARI
    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

Views

Author

Michel Marcus, Apr 26 2014

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

A241479 GCD of the solutions x of sigma(x) = sigma(n), where sigma(n) = A000203(n) = sum of divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 7, 8, 9, 1, 1, 12, 13, 1, 1, 1, 1, 18, 19, 1, 1, 22, 1, 1, 1, 1, 27, 1, 29, 1, 1, 32, 1, 1, 1, 36, 37, 1, 1, 1, 1, 1, 43, 1, 45, 1, 1, 3, 49, 50, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 61, 1, 1, 64, 1, 1, 67, 2, 1, 1, 1, 72, 73, 1, 3, 1, 1, 1, 1, 2
Offset: 1

Views

Author

Michel Marcus, Apr 23 2014

Keywords

Comments

A variant of A240667 without zeros.

Examples

			a(6) = 1 since sigma(6) = 12 and sigma(11) = 12 and gcd(6, 11) = 1.

Links

Crossrefs

Cf. A000203, A240667.

Programs

  • PARI
    sigv(n) =  select(i->sigma(i) == n, vector(n, i, i));
a(n) = gcd(sigv(sigma(n)));
  • PARI
    a(n) = gcd(invsigma(sigma(n))); \\ Amiram Eldar, Dec 19 2024, using Max Alekseyev's invphi.gp

Formula

a(n) = A240667(A000203(n)).
Showing 1-7 of 7 results.

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