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.

A086358 Digital root of n!.

Original entry on oeis.org

1, 1, 2, 6, 6, 3, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9
Offset: 0

Views

Author

Labos Elemer, Jul 21 2003

Keywords

Comments

a(n) = 9 for n >= 6.

Examples

			n = 5, 5 != 120, iteration list = {120,3}, a(5) = 3.

Links

Crossrefs

Cf. A000142, A086353-A086361, A007953, A010888, A038194, A029898, A004152.

Programs

  • Mathematica
    sud[x_] := Apply[Plus, DeleteCases[IntegerDigits[x], 0]]; Table[FixedPoint[sud, w!], {w, 1, 87}]

Formula

a(n) = A010888(n!) = fixed-point of A007953(n!). It equals n! modulo(9); at r = 0 use 9.
G.f.: (1 + x^2 + 4*x^3 - 3*x^5 + 6*x^6)/(1 - x). - Stefano Spezia, Jan 26 2023

Extensions

a(0) = 1 prepended by Alois P. Heinz, Dec 05 2018

A086360 The n-th primorial number reduced modulo 9.

Original entry on oeis.org

1, 2, 6, 3, 3, 6, 6, 3, 3, 6, 3, 3, 3, 6, 6, 3, 6, 3, 3, 3, 6, 6, 6, 3, 6, 6, 3, 3, 6, 6, 3, 3, 6, 3, 3, 6, 6, 6, 6, 3, 6, 3, 3, 6, 6, 3, 3, 3, 3, 6, 6, 3, 6, 6, 3, 6, 3, 6, 6, 6, 3, 3, 6, 6, 3, 3, 6, 6, 6, 3, 3, 6, 3, 3, 3, 3, 6, 3, 3, 6, 6, 3, 3, 6, 6, 6, 3, 6
Offset: 0

Views

Author

Labos Elemer, Jul 21 2003

Keywords

Comments

a(n) is the fixed point reached by decimal-digit-sum-function (A007953), when starting the iteration from the value of the n-th primorial, A002110(n). - The (edited) original definition of the sequence, which is equal to a simple definition a(n) = A002110(n) mod 9, because taking the decimal digit sum preserves congruence modulo 9. - Antti Karttunen, Nov 14 2024
Only a(0)=1 and a(1)=2; each subsequent term is either a 3 or a 6.

Examples

			For n=7, 7th primorial = 510510, list of iterated digit sums is {510510,12,3}, thus a(7)=3.

Links

Crossrefs

Cf. A002110, A007953, A010878, A010888, A029898, A038194, A086353-A086361.
Cf. also A377876, A377877.

Programs

  • Maple
    A086360 := proc(n) option remember: if(n=1)then return 2:fi: return ithprime(n)*procname(n-1) mod 9: end: seq(A086360(n), n=1..100); # Nathaniel Johnston, May 04 2011
  • Mathematica
    sud[x_] := Apply[Plus, DeleteCases[IntegerDigits[x], 0]] q[x_] := Apply[Times, Table[Prime[w], {w, 1, x}]] Table[FixedPoint[sud, q[w]], {w, 1, 128}]
  • PARI
    up_to = 19683;
A086360list(up_to_n) = { my(m=9, v=vector(1+up_to_n), pr=1); v[1] = 1; for(n=1, up_to_n, pr = (pr*prime(n))%m; v[1+n] = pr); (v); };
v086360 = A086360list(up_to);
A086360(n) = v086360[1+n]; \\ Antti Karttunen, Nov 14 2024

Formula

a(n) = A010878(A002110(n)) = A002110(n) mod 9.
a(n) = A010888(A002110(n)).

Extensions

Term a(0)=1 prepended, old definition moved to comments and replaced with one of the formulas, keyword:base removed because not really base-dependent - Antti Karttunen, Nov 14 2024

A086354 Fixed point if (nonzero-digit product)-function at initial value 2^n is iterated.

Original entry on oeis.org

2, 4, 8, 6, 6, 8, 6, 6, 1, 8, 8, 2, 6, 2, 2, 4, 8, 2, 1, 6, 2, 2, 6, 8, 2, 8, 2, 8, 2, 2, 8, 6, 6, 2, 2, 6, 2, 2, 6, 8, 8, 6, 3, 4, 2, 2, 6, 6, 2, 8, 6, 2, 2, 9, 8, 6, 6, 5, 8, 2, 8, 8, 2, 6, 2, 8, 8, 8, 5, 8, 8, 8, 2, 8, 6, 4, 8, 6, 2, 7, 1, 8, 8, 4, 2, 8, 8
Offset: 1

Views

Author

Labos Elemer, Jul 21 2003

Keywords

Examples

			n=20, 2^20=1048576, iteration list={1048576,6720,84,32,6}, so a(20)=6.

Links

Crossrefs

Cf. A051801, A051802, A086353-A086361, A029898, A010888.

Programs

  • Maple
    A051801 := proc(n) local d,j: d:=convert(n,base,10): return mul(`if`(d[j]=0,1,d[j]), j=1..nops(d)): end: A086354 := proc(n) local m: m:=2^n: while(length(m)>1)do m:=A051801(m): od: return m: end: seq(A086354(n),n=1..100); # Nathaniel Johnston, May 04 2011
  • Mathematica
    prd[x_] := Apply[Times, DeleteCases[IntegerDigits[x], 0]] Table[FixedPoint[prd, 2^w], {w, 1, 128}]

Formula

a(n) = A051802(2^n) = fixed point of A051801(2^n).

A086355 Fixed point if [nonzero-digit product]-function at initial-value=prime(n) is iterated.

Original entry on oeis.org

2, 3, 5, 7, 1, 3, 7, 9, 6, 8, 3, 2, 4, 2, 6, 5, 2, 6, 8, 7, 2, 8, 8, 4, 8, 1, 3, 7, 9, 3, 4, 3, 2, 4, 8, 5, 5, 8, 8, 2, 8, 8, 9, 4, 8, 8, 2, 2, 6, 8, 8, 2, 8, 1, 7, 8, 8, 4, 4, 6, 6, 2, 2, 3, 9, 2, 9, 8, 6, 8, 2, 5, 2, 8, 4, 4, 2, 4, 4, 8, 8, 8, 2, 8, 8, 6, 6, 4, 8, 4, 6, 2, 6, 8, 8, 5, 2, 1, 3, 2, 4, 5, 9, 4, 5
Offset: 1

Views

Author

Labos Elemer, Jul 21 2003

Keywords

Examples

			n=100, prime(100)=541, iteration list={541,20,2}, a(100)=2.

Links

Crossrefs

Cf. A051801, A051802, A086353-A086361, A029898, A010888, A038194.

Programs

  • Mathematica
    prd[x_] := Apply[Times, DeleteCases[IntegerDigits[x], 0]]; Table[FixedPoint[prd, Prime[w]], {w, 1, 128}]

Formula

a(n) = A051802(A000040(n)) = fixed-point of A051801(n-th prime).

A086356 Fixed point if [nonzero-digit product]-function at initial-value=C[2n,n]=central binomial coefficient is iterated.

Original entry on oeis.org

2, 6, 2, 7, 2, 4, 4, 2, 2, 6, 6, 6, 8, 6, 5, 8, 8, 4, 8, 2, 9, 8, 6, 8, 6, 2, 8, 8, 2, 8, 6, 2, 6, 6, 8, 2, 6, 6, 6, 8, 9, 2, 2, 8, 2, 8, 2, 8, 6, 4, 2, 2, 8, 8, 2, 8, 6, 8, 2, 8, 6, 8, 9, 6, 6, 2, 6, 2, 2, 2, 8, 6, 8, 6, 8, 2, 8, 8, 8, 8, 8, 8, 6, 2, 6, 2, 8, 6, 8, 8, 8, 8, 8, 6, 8, 8, 6, 8, 2, 8, 2, 8, 6, 8, 8
Offset: 1

Views

Author

Labos Elemer, Jul 21 2003

Keywords

Examples

			n=10, C[20,10]=184756, iteration list={184756,7560,210,2},
a(100)=2.

Crossrefs

Cf. A051801, A051802, A086353-A086361, A029898, A010888, A038194, A000984.

Programs

  • Mathematica
    prd[x_] := Apply[Times, DeleteCases[IntegerDigits[x], 0]] Table[FixedPoint[prd, Binomial[2*n, n]], {w, 1, 128}]

Formula

a(n)=A051802[A000984(n)]=fixed-point of A051801[C(2n, n)]

A086357 Fixed point if [nonzero-digit-product]-function at initial-value=A002110(n)=n-th primorial is iterated.

Original entry on oeis.org

2, 6, 3, 2, 6, 9, 1, 6, 8, 2, 8, 8, 8, 8, 6, 8, 6, 7, 8, 2, 8, 8, 8, 6, 2, 8, 1, 6, 8, 2, 2, 2, 8, 6, 8, 8, 8, 6, 4, 6, 2, 8, 2, 6, 6, 2, 2, 2, 6, 6, 8, 6, 2, 8, 8, 8, 5, 8, 6, 8, 2, 8, 8, 1, 2, 2, 8, 6, 8, 2, 2, 8, 2, 8, 8, 2, 2, 8, 8, 9, 6, 8, 4, 6, 8, 8, 8, 2, 8, 6, 8, 1, 6, 2, 2, 8, 1, 6, 6, 8, 6, 9, 2, 6, 2
Offset: 1

Views

Author

Labos Elemer, Jul 21 2003

Keywords

Examples

			n=7, 7th-primorial=510510, iteration list={510510,25,10,1},
a(100)=2.

Crossrefs

Cf. A051801, A051802, A086353-A086361, A029898, A010888, A038194, A000984.

Programs

  • Mathematica
    prd[x_] := Apply[Times, DeleteCases[IntegerDigits[x], 0]] q[x_] := Apply[Times, Table[Prime[w], {w, 1, x}]] Table[FixedPoint[prd, q[w]], {w, 1, 128}]

Formula

a(n)=A051802[A002110(n)]=fixed-point of A051801[A002110(n)]

A086359 Fixed point if [decimal-digit-sum]-function at initial-value=A000984(n)=C[2n,n] is iterated.

Original entry on oeis.org

2, 6, 2, 7, 9, 6, 3, 9, 2, 4, 3, 7, 2, 9, 9, 9, 9, 6, 3, 9, 3, 6, 9, 9, 9, 9, 2, 4, 3, 4, 5, 9, 3, 6, 9, 7, 5, 6, 2, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 6, 3, 9, 3, 6, 9, 9, 9, 9, 3, 6, 9, 6, 3, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 2, 4, 3, 4, 5, 9, 3, 6, 9, 4, 8, 6, 5, 4, 9, 9, 9, 9, 3, 6, 9, 6, 3, 9, 9
Offset: 1

Views

Author

Labos Elemer, Jul 21 2003

Keywords

Examples

			n=10, C[20,10]=184756, iteration list={184756,31,4},a(10)=4.

Crossrefs

Cf. A086353-A086361, A007953, A010888, A038194, A029898.

Programs

  • Mathematica
    sud[x_] := Apply[Plus, DeleteCases[IntegerDigits[x], 0]] Table[FixedPoint[sud, Binomial[2*w, w]], {w, 1, 128}]

Formula

a(n)=A010888[C[2n, n]]=fixed-point of A007953[C[2n, n]]; It equals C[2n, n] modulo(9); at r=0 use 9.
Showing 1-7 of 7 results.

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