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 1-6 of 6 results.

A339873 a(n) = 1 + n - A143771(n).

Original entry on oeis.org

0, 0, 4, 0, 6, 0, 6, 8, 10, 0, 12, 0, 12, 8, 16, 0, 18, 0, 18, 20, 22, 0, 24, 24, 24, 24, 28, 0, 30, 0, 30, 32, 34, 24, 36, 0, 36, 32, 40, 0, 42, 0, 42, 44, 46, 0, 48, 48, 48, 48, 52, 0, 54, 48, 54, 56, 58, 0, 60, 0, 60, 56, 64, 60, 66, 0, 66, 68, 70, 0, 72, 0, 72, 72, 76, 72, 78, 0, 78, 80, 82, 0, 84, 84, 84, 80
Offset: 2

Views

Author

Antti Karttunen, Dec 25 2020

Keywords

Links

Crossrefs

Cf. A143771, A339914.

Programs

A339914 Ordinal transform of sequence b(1) = 1, b(n) = A143771(n) for n > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 2, 1, 4, 1, 5, 1, 3, 2, 6, 1, 7, 1, 4, 2, 8, 1, 9, 3, 5, 2, 10, 1, 11, 1, 6, 4, 12, 2, 13, 1, 7, 3, 14, 1, 15, 1, 8, 5, 16, 1, 17, 6, 9, 3, 18, 1, 19, 4, 10, 7, 20, 1, 21, 1, 11, 5, 22, 2, 23, 1, 12, 8, 24, 1, 25, 1, 13, 4, 26, 3, 27, 1, 14, 9, 28, 1, 29, 10, 15, 6, 30, 1, 31, 5, 16, 11, 32, 2
Offset: 1

Views

Author

Antti Karttunen, Dec 28 2020

Keywords

Links

Crossrefs

Cf. A143771, A339873.

Programs

  • Mathematica
    f[n_] := If[n == 1, 1, GCD @@ Table[d + n/d, {d, Divisors[n]}]];
b[_] = 1;
a[n_] := a[n] = With[{t = f[n]}, b[t]++];
Array[a, 105] (* Jean-François Alcover, Dec 19 2021 *)
  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A143771b(n) = if(1==n,1, my(d = divisors(n)); gcd(vector(#d, k, d[k]+n/d[k])));
v339914 = ordinal_transform(vector(up_to,n,A143771b(n)));
A339914(n) = v339914[n];

A342918 a(n) = (1+n) / A143771(n).

Original entry on oeis.org

1, 1, 1, 5, 1, 7, 1, 3, 5, 11, 1, 13, 1, 5, 2, 17, 1, 19, 1, 7, 11, 23, 1, 25, 13, 9, 7, 29, 1, 31, 1, 11, 17, 35, 3, 37, 1, 13, 5, 41, 1, 43, 1, 15, 23, 47, 1, 49, 25, 17, 13, 53, 1, 55, 7, 19, 29, 59, 1, 61, 1, 21, 8, 65, 11, 67, 1, 23, 35, 71, 1, 73, 1, 25, 19, 77, 13, 79, 1, 27, 41, 83, 1, 85, 43, 29, 11, 89, 1, 91, 23
Offset: 1

Views

Author

Antti Karttunen, Mar 29 2021

Keywords

Links

Crossrefs

Cf. A143771, A339873.
After n=1 differs from A342916 for the first time at n=44, where a(44) = 15, while A342916(44) = 5.

Programs

Formula

a(n) = (1+n) / A143771(n).

A342915 a(n) = gcd(1+n, A001615(n)), where A001615 is Dedekind psi, n * Product_{p|n, p prime} (1 + 1/p).

Original entry on oeis.org

1, 3, 4, 1, 6, 1, 8, 3, 2, 1, 12, 1, 14, 3, 8, 1, 18, 1, 20, 3, 2, 1, 24, 1, 2, 3, 4, 1, 30, 1, 32, 3, 2, 1, 12, 1, 38, 3, 8, 1, 42, 1, 44, 9, 2, 1, 48, 1, 2, 3, 4, 1, 54, 1, 8, 3, 2, 1, 60, 1, 62, 3, 32, 1, 6, 1, 68, 3, 2, 1, 72, 1, 74, 3, 4, 1, 6, 1, 80, 9, 2, 1, 84, 1, 2, 3, 8, 1, 90, 1, 4, 3, 2, 1, 24, 1, 98, 3, 4, 1, 102
Offset: 1

Views

Author

Antti Karttunen, Mar 29 2021

Keywords

Links

Crossrefs

Cf. A001615, A342916, A342917.
Cf. also A049559, A342458.
After n=1 differs from A143771 for the first time at n=44, where a(44) = 9, while A143771(44) = 3.

Programs

  • Mathematica
    psi[n_] := If[n==1, 1, Times @@ ((#1+1)*#1^(#2-1)& @@@ FactorInteger[n])];
a[n_] := GCD[n+1, psi[n]];
Array[a, 105] (* Jean-François Alcover, Dec 22 2021 *)
  • PARI
    A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A342915(n) = gcd(1+n,A001615(n));

Formula

a(n) = gcd(1+n, A001615(n)).
a(n) = (1+n) / A342916(n) = A001615(n) / A342917(n).

A104076 If k(m) is the m-th divisor (when the divisors are ordered by size) of n, then a(n) = gcd(k(1)+k(2), k(2)+k(3), k(3)+k(4), ..., k(j-1)+k(j)), where j is the number of divisors of n.

Original entry on oeis.org

3, 4, 3, 6, 1, 8, 3, 4, 1, 12, 1, 14, 3, 4, 3, 18, 1, 20, 3, 2, 1, 24, 1, 6, 3, 4, 1, 30, 1, 32, 3, 2, 1, 6, 1, 38, 3, 4, 1, 42, 1, 44, 3, 2, 1, 48, 1, 8, 1, 4, 1, 54, 1, 2, 1, 2, 1, 60, 1, 62, 3, 2, 3, 6, 1, 68, 3, 2, 1, 72, 1, 74, 3, 4, 1, 2, 1, 80, 1, 4, 1, 84, 1, 2, 3, 4, 1, 90, 1, 4, 3, 2, 1, 6, 1, 98
Offset: 2

Views

Author

Leroy Quet, Aug 31 2008

Keywords

Examples

			The divisors of 14 are 1,2,7,14. So a(14) = gcd(1+2, 2+7, 7+14) = 3.

Links

Crossrefs

Cf. A143771.

Programs

  • Maple
    A104076 := proc(n) local dvs ; dvs := sort(convert(numtheory[divisors](n),list)) ; igcd(seq( op(i,dvs)+op(i+1,dvs), i=1..nops(dvs)-1)) ; end: for n from 2 to 140 do printf("%d,",A104076(n)) ; od: # R. J. Mathar, Sep 05 2008
  • Mathematica
    Table[GCD@@(Total/@Partition[Divisors[n],2,1]),{n,2,100}] (* Harvey P. Dale, Dec 18 2018 *)

Extensions

Extended by R. J. Mathar, Sep 05 2008
Definition corrected by Leroy Quet, Sep 21 2008

A143772 If m is the n-th composite, then a(n) = gcd(k + m/k), where k is over all divisors of m.

Original entry on oeis.org

1, 1, 3, 2, 1, 1, 3, 8, 1, 1, 3, 2, 1, 1, 2, 3, 4, 1, 1, 3, 2, 1, 12, 1, 3, 8, 1, 1, 3, 2, 1, 1, 2, 3, 4, 1, 1, 8, 3, 2, 1, 1, 3, 8, 1, 6, 1, 3, 2, 1, 1, 3, 4, 1, 6, 1, 3, 2, 1, 1, 2, 3, 8, 1, 1, 4, 3, 2, 1, 24, 1, 3, 4, 1, 1, 3, 2, 1, 1, 3, 8, 1, 1, 4, 3, 2, 1, 24, 1, 2, 3, 4, 1, 6, 1, 3, 2, 1, 1, 2, 3, 8, 1, 1, 3
Offset: 1

Views

Author

Leroy Quet, Aug 31 2008

Keywords

Comments

Conjecture: All even numbers are terms and the only odd numbers which are terms are 1 and 3. - Robert G. Wilson v, Sep 08 2008

Examples

			For n=11, 20 is the 11th composite. So we have a(11) = gcd(1+20, 2+10, 4+5, 5+4, 10+2, 20+1) = 3.

Crossrefs

Cf. A143771.

Programs

  • Mathematica
    Composite[n_Integer] := FixedPoint[n + PrimePi@# + 1 &, n + PrimePi@n + 1]; f[n_] := Block[{m = Composite@n}, Last@ FoldList[ GCD, m!, # + m/# & /@ Divisors@m]]; Array[f, 105] (* Robert G. Wilson v, Sep 08 2008 *)

Extensions

More terms from Robert G. Wilson v, Sep 08 2008
Showing 1-6 of 6 results.

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