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 1-5 of 5 results.

A077104 Duplicate of A073815.

Original entry on oeis.org

1, 3, 18, 12, 200, 14, 3364, 15, 722, 328, 9801, 42, 25281, 116, 1800, 165, 36992, 810
Offset: 1

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Author

Keywords

A074389 a(n) = gcd(n, sigma(n), phi(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 4, 1, 1, 1, 2, 1, 6, 1, 8, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 6, 1, 4, 1, 2, 1, 4, 1, 1, 3, 1, 1, 2, 1, 2, 3
Offset: 1

Views

Author

Labos Elemer, Aug 23 2002

Keywords

Links

Crossrefs

Cf. A000010, A000203, A073815, A050399, A009194, A009195, A009223, A074465, A074466.
In the old definition the erroneously given formula gcd(n, A000005(n), A000010(n)) is now sequence A318459. - Antti Karttunen, Sep 07 2018

Programs

  • Mathematica
    Table[Apply[GCD, {w, DivisorSigma[1, w], EulerPhi[w]}], {w, 1, 128}]
  • PARI
    A074389(n) = gcd([n, sigma(n), eulerphi(n)]); \\ Antti Karttunen, Sep 07 2018

Formula

a(n) = gcd(n, A000010(n), A000203(n)).
a(n) = gcd(n, A009223(n)). - Antti Karttunen, Sep 07 2018

Extensions

Name corrected by Antti Karttunen, Sep 07 2018

A074391 a(n) is the smallest number such that gcd(a(n), sigma(a(n))) = n.

Original entry on oeis.org

1, 10, 15, 12, 95, 6, 91, 56, 153, 40, 473, 24, 117, 182, 135, 336, 1139, 90, 703, 380, 861, 946, 3151, 168, 3725, 468, 1431, 28, 5017, 570, 775, 992, 891, 2176, 4865, 792, 2701, 1406, 585, 280, 6683, 546, 11051, 1892, 1305, 6302, 13207, 528, 4753, 5800
Offset: 1

Views

Author

Labos Elemer, Aug 23 2002

Keywords

Comments

a(n) is the smallest number k such that A017666(k), the denominator of sigma(k)/k, is equal to k/n. - Jaroslav Krizek, Sep 23 2014
Each term a(n) is divisible by its index n. - Michel Marcus, Jan 13 2015

Examples

			n=6: a(6)=6 because gcd(6, sigma(6))=6 and a(6)=6 is the smallest.

Links

Crossrefs

Cf. A000203, A017666, A073815, A050399, A009195, A009194.

Programs

  • Magma
    A074391:=func; [A074391(n): n in[1..100]] // Jaroslav Krizek, Sep 23 2014
  • Maple
    f:= proc(n) local k;
  for k from n by n do
    if igcd(k, numtheory:-sigma(k))=n then return k fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Feb 11 2020
  • Mathematica
    f[x_] := GCD[DivisorSigma[1, x], x] t=Table[0, {100}]; Do[s=f[n]; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 1000000}];
  • PARI
    a(n) = my(k=1); while (gcd(sigma(k), k) != n, k++); k; \\ Michel Marcus, Jan 13 2015

Formula

a(n) = Min{x; gcd(x, sigma(x))} = Min{x; gcd(x, A000203(x))} = n. - corrected by Michel Marcus, Jan 13 2015

A074390 a(n) is the least number k that A074389(k) = n.

Original entry on oeis.org

1, 6, 18, 12, 200, 42, 196, 56, 459, 950, 5203, 396, 9243, 980, 1800, 336, 19363, 270, 13357, 600, 1764, 10406, 72473, 168, 18625, 34814, 4293, 812, 145493, 1350, 15376, 992, 19602, 38726, 41615, 1836, 99937, 26714, 1521, 440, 274003, 3822, 475193
Offset: 1

Views

Author

Labos Elemer, Aug 23 2002

Keywords

Examples

			For n = 79: a(79) = 979837 because GCD(979837,998718,961272) = 79 and 979837 is the smallest.

Links

Crossrefs

Cf. A000005, A000010, A073815, A050399, A009195, A009194.

Programs

  • Mathematica
    f[x_] := GCD[DivisorSigma[1, x], EulerPhi[x], x]; t=Table[0, {100}]; Do[s=f[n]; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 1000000}]; t
  • PARI
    lista(len) = {my(v = vector(len), c = 0, k = 1, f, i); while(c < len, f = factor(k); i = gcd([k, sigma(k), eulerphi(k)]); if(i <= len && v[i] == 0, c++; v[i] = k); k++); v;} \\ Amiram Eldar, Nov 14 2024

Formula

a(n) = Min{x; GCD(x, sigma(x), phi(x)) = n} = Min{x; GCD(x, A000005(x), A000010(x)) = n}.

A082057 Least x=a(n) such that product of common prime-divisors [without multiplicity] of sigma(x) and phi(x) equals n; or 0 if n is not a squarefree number or if no such x exists. Among indices n only squarefree numbers arise because multiplicity of prime factors is ignored.

Original entry on oeis.org

1, 3, 18, 0, 200, 14, 3364, 0, 0, 88, 9801, 0, 25281, 116, 1800, 0, 36992, 0, 4414201, 0, 196, 2881, 541696, 0, 0, 711, 0, 0, 98942809, 209, 1547536, 0, 19602, 6901, 814088, 0, 49042009, 8473, 1521, 0, 3150464641, 377, 245178368, 0, 0, 6439, 9265217536, 0, 0
Offset: 1

Views

Author

Labos Elemer, Apr 03 2003

Keywords

Examples

			For n = 85: a(85) = 924800 = 128*5*5*17*17; sigma(924800) = 2426835 = 3*5*17*31*307; phi(924800) = 348160 = 4096*5*17; common prime factor 5.17 = 85.

Crossrefs

Cf. A000203, A000010, A082054, A082055, A082056.
Cf. A073815.

Programs

  • Mathematica
    ffi[x_] := Flatten[FactorInteger[x]]
lf[x_] := Length[FactorInteger[x]]
ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]
t=Table[0, {100}]; Do[s=Apply[Times, Intersection
[ba[EulerPhi[n]], ba[DivisorSigma[1, n]]]];
If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 1000000}]; t

Formula

a(n) = Min{x; A082055(x)=n}; 0 if n is not squarefree.

Extensions

Corrected and extended by David Wasserman, Aug 27 2004
Showing 1-5 of 5 results.

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