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.

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A133875 n modulo 5 repeated 5 times.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1
Offset: 0

Views

Author

Hieronymus Fischer, Oct 10 2007

Keywords

Comments

Periodic with length 5^2 = 25.

Links

Crossrefs

Cf. A133620-A133625, A133630, A038509, A133634-A133636.
Cf. A133885, A133880, A133890, A133900, A133910.

Programs

  • Magma
    [(1 + Floor(n/5)) mod 5 : n in [0..50]]; // Wesley Ivan Hurt, Jun 06 2014
  • Maple
    A133875:=n->((1+floor(n/5)) mod 5); seq(A133875(n), n=0..100); # Wesley Ivan Hurt, Jun 06 2014
  • Mathematica
    Table[Mod[1 + Floor[n/5], 5], {n, 0, 100}] (* Wesley Ivan Hurt, Jun 06 2014 *)
LinearRecurrence[{1,0,0,0,-1,1,0,0,0,-1,1,0,0,0,-1,1,0,0,0,-1,1},{1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,0},120] (* Harvey P. Dale, Dec 14 2017 *)

Formula

a(n) = (1 + floor(n/5)) mod 5.
a(n) = A010874(A002266(n+5)).
a(n) = 1 + floor(n/5) - 5*floor((n+5)/25).
a(n) = (((n+5) mod 25) - (n mod 5)) / 5.
a(n) = ((n + 5 - (n mod 5)) / 5) mod 5.
a(n) = A010874((n + 5 - A010874(n))/5).
a(n) = binomial(n+5, n) mod 5 = binomial(n+5, 5) mod 5.
a(n) = +a(n-1) -a(n-5) +a(n-6) -a(n-10) +a(n-11) -a(n-15) +a(n-16) -a(n-20) +a(n-21). - R. J. Mathar, Sep 03 2011
G.f.: ( 1+2*x^5+3*x^10+4*x^15 ) / ( (1-x)*(x^20+x^15+x^10+x^5+1) ). - R. J. Mathar, Sep 03 2011

A135120 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 10 all are equal.

Original entry on oeis.org

1, 21, 222, 223, 1230, 1231, 1502, 2200, 2201, 3012, 3013, 10431, 12214, 12215, 12250, 12251, 14102, 15003, 15021, 16011, 20040, 20041, 22130, 23211, 23230, 23231, 24003, 30070, 30071, 30105, 30231, 30321, 31005, 31150, 31151, 31420
Offset: 1

Views

Author

Hieronymus Fischer, Dec 24 2007

Keywords

Examples

			a(2)=21, since ds_2(21)=ds_3(21)=ds_10(21)=3.

Links

Crossrefs

Cf. A000040, A007953, A054899, A131451, A133620, A133900, A134599, A135110.

Programs

  • Mathematica
    Select[Range[5000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 3]] ==  Total[IntegerDigits[#, 10]] &] (* G. C. Greubel, Sep 26 2016 *)
  • PARI
    is(n)=my(t=sumdigits(n)); t==hammingweight(n) && t==sumdigits(n,3) \\ Charles R Greathouse IV, Sep 26 2016

A133885 Binomial(n+5,n) mod 5^2.

Original entry on oeis.org

1, 6, 21, 6, 1, 2, 12, 17, 12, 2, 3, 18, 13, 18, 3, 4, 24, 9, 24, 4, 5, 5, 5, 5, 5, 6, 11, 1, 11, 6, 7, 17, 22, 17, 7, 8, 23, 18, 23, 8, 9, 4, 14, 4, 9, 10, 10, 10, 10, 10, 11, 16, 6, 16, 11, 12, 22, 2, 22, 12, 13, 3, 23, 3, 13, 14, 9, 19, 9, 14, 15, 15, 15, 15, 15, 16, 21, 11, 21, 16, 17
Offset: 0

Views

Author

Hieronymus Fischer, Oct 10 2007

Keywords

Comments

Periodic with length 5^3=125.

Links

  • Ray Chandler, Table of n, a(n) for n = 0..1000
  • Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1).

Crossrefs

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.
Cf. A133875, A133880, A133890, A133900, A133910.
For the sequence regarding binomial(n+5, n) mod 5 see A133875.

Programs

  • Mathematica
    Table[Mod[Binomial[n+5,n],25],{n,0,90}] (* Harvey P. Dale, Jan 12 2023 *)

Formula

a(n)=binomial(n+5,5) mod 5^2.
G.f. g(x)=sum{0<=k<125, a(k)*x^k}/(1-x^125).

A133911 Number of prime factors (counted with multiplicity) of the period numbers defined by A133900.

Original entry on oeis.org

0, 2, 2, 4, 2, 5, 2, 6, 4, 6, 2, 8, 2, 6, 5, 8, 2, 9, 2, 8, 5, 7, 2, 10, 4, 7, 6, 8, 2, 12, 2, 10, 6, 8, 5, 12, 2, 8, 6, 11, 2, 12, 2, 9, 8, 8, 2, 13, 4, 10, 6, 9, 2, 12, 5, 11, 6, 8, 2, 14, 2, 8, 8, 12, 5, 13, 2, 10, 6, 13, 2, 14, 2, 9, 8, 10, 5, 13, 2, 13, 8, 10, 2, 17, 5, 9, 7, 11, 2, 16, 5, 10, 7, 9
Offset: 1

Views

Author

Hieronymus Fischer, Oct 20 2007

Keywords

Examples

			a(6)=5, since A133900(6)=72=2*2*2*3*3.
a(12)=8, since A133900(12)=864=2*2*2*2*2*3*3*3.

Crossrefs

Cf. A000040, A001222, A100118, A046363, A133620, A133621, A133623, A133630, A133635.
Cf. A133872, A133880, A133890, A133900, A133910, A134331.

Formula

a(n)=A001222(A133900(n)).

A135100 Numbers which divide their digital sumorial (see A131383).

Original entry on oeis.org

1, 3, 4, 15, 26, 2573, 17226, 19786, 22083, 58133, 67693, 223657, 376460, 464713, 497068, 2621204, 4553376, 6000136, 7671158, 13975944, 14074903, 52731198, 82594577
Offset: 1

Views

Author

Hieronymus Fischer, Dec 24 2007

Keywords

Examples

			a(5)=26, since 26 divides its digital sumorial, which is A131383(26)=182.

Crossrefs

Cf. A007953, A054899, A131383, A131451, A133620, A133900, A134599.

Programs

  • PARI
    a=1;for(n=2,10^6,if(a%(n-1)==0,print1(n-1","));x=divisors(n);L=numdiv(n);a+=n; for(i=2,L-1,d=x[i];k=n;while(k%d==0,a-=d-1;k\=d))) \\ Robert Gerbicz, May 09 2008

Extensions

a(12)-a(15) from Robert Gerbicz, May 09 2008
a(16)-a(23) from Hieronymus Fischer, Jul 31 2008

A134331 Sum of prime factors (counted with multiplicity) of the period numbers defined by A133900.

Original entry on oeis.org

0, 4, 6, 8, 10, 12, 14, 12, 12, 18, 22, 19, 26, 22, 19, 16, 34, 22, 38, 22, 23, 32, 46, 23, 20, 36, 18, 26, 58, 37, 62, 20, 34, 46, 29, 29, 74, 50, 38, 31, 82, 38, 86, 36, 30, 58, 94, 30, 28, 32, 46, 40, 106, 30, 37, 37, 50, 70, 118, 41, 122, 74, 36, 24, 41, 48, 134, 50, 58, 50
Offset: 1

Views

Author

Hieronymus Fischer, Oct 20 2007

Keywords

Examples

			a(6)=12, since A133900(6)=72=2*2*2*3*3 and 2+2+2+3+3=12.
a(12)=19, since A133900(12)=864=2*2*2*2*2*3*3*3 and 2+2+2+2+2+3+3+3=19.

Crossrefs

Cf. A000040, A100118, A046363, A133620, A133621, A133623, A133630, A133635.
Cf. A133872, A133880, A133890, A133900, A133910, A133911.

A362686 Binomial(n+p, n) mod n where p=6.

Original entry on oeis.org

0, 0, 0, 2, 2, 0, 1, 3, 1, 8, 1, 0, 1, 8, 9, 5, 1, 10, 1, 10, 15, 12, 1, 15, 6, 14, 1, 8, 1, 12, 1, 9, 12, 18, 8, 10, 1, 20, 27, 19, 1, 36, 1, 12, 10, 24, 1, 45, 1, 36, 18, 14, 1, 28, 12, 15, 39, 30, 1, 48, 1, 32, 1, 17, 14, 12, 1, 18, 24, 50, 1, 19, 1, 38
Offset: 1

Views

Author

Ray Chandler, Apr 29 2023

Keywords

Links

Crossrefs

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.
Cf. A133875, A133885, A133880, A133890, A133900, A133910, A362687, A362688, A362689.

Programs

  • Mathematica
    Table[Mod[Binomial[n+6, n], n], {n, 90}]

Formula

a(n)=binomial(n+6,n) mod n.
For n > 1452, a(n) = 2*a(n-720) - a(n-1440).

A362687 Binomial(n+p, n) mod n where p=7.

Original entry on oeis.org

0, 0, 0, 2, 2, 0, 2, 3, 1, 8, 1, 0, 1, 10, 9, 5, 1, 10, 1, 10, 18, 12, 1, 15, 6, 14, 1, 12, 1, 12, 1, 9, 12, 18, 13, 10, 1, 20, 27, 19, 1, 0, 1, 12, 10, 24, 1, 45, 8, 36, 18, 14, 1, 28, 12, 23, 39, 30, 1, 48, 1, 32, 10, 17, 14, 12, 1, 18, 24, 60, 1, 19, 1
Offset: 1

Views

Author

Ray Chandler, Apr 29 2023

Keywords

Links

Crossrefs

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.
Cf. A133875, A133885, A133880, A133890, A133900, A133910, A362686, A362688, A362689.

Programs

  • Mathematica
    Table[Mod[Binomial[n+7,n],n],{n,90}]

Formula

a(n)=binomial(n+7,n) mod n.
For n > 10122, a(n) = 2*a(n-5040) - a(n-10080).

A362688 Binomial(n+p, n) mod n where p=8.

Original entry on oeis.org

0, 1, 0, 3, 2, 3, 2, 6, 1, 8, 1, 6, 1, 10, 9, 15, 1, 1, 1, 5, 18, 1, 1, 12, 6, 14, 1, 12, 1, 12, 1, 13, 12, 1, 13, 19, 1, 1, 27, 34, 1, 0, 1, 34, 10, 24, 1, 27, 8, 11, 18, 1, 1, 1, 12, 16, 39, 30, 1, 48, 1, 32, 10, 25, 14, 45, 1, 35, 24, 25, 1, 46, 1, 38, 66
Offset: 1

Views

Author

Ray Chandler, Apr 29 2023

Keywords

Links

Crossrefs

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.
Cf. A133875, A133885, A133880, A133890, A133900, A133910, A362686, A362687, A362689.

Programs

  • Mathematica
    Table[Mod[Binomial[n+8,n],n],{n,90}]

Formula

a(n)=binomial(n+8,n) mod n.
For n > 645240, a(n) = 2*a(n-322560) - a(n-645120).

A362689 Binomial(n+p, n) mod n where p=9.

Original entry on oeis.org

0, 1, 1, 3, 2, 1, 2, 6, 2, 8, 1, 2, 1, 10, 14, 15, 1, 3, 1, 5, 18, 1, 1, 12, 6, 14, 4, 12, 1, 22, 1, 13, 1, 1, 13, 23, 1, 1, 14, 34, 1, 14, 1, 34, 15, 24, 1, 27, 8, 11, 18, 1, 1, 7, 12, 16, 1, 30, 1, 28, 1, 32, 17, 25, 14, 23, 1, 35, 47, 25, 1, 54, 1, 38, 66
Offset: 1

Views

Author

Ray Chandler, Apr 29 2023

Keywords

Links

Crossrefs

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.
Cf. A133875, A133885, A133880, A133890, A133900, A133910, A362686, A362687, A362688.

Programs

  • Mathematica
    Table[Mod[Binomial[n+9,n],n],{n,90}]

Formula

a(n)=binomial(n+9,n) mod n.
For n > 5806081, a(n) = 2*a(n-2903040) - a(n-5806080).
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