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.

A110760 a(n) = number of divisors of the concatenation of n,n-1,...3,2,1,2,3,...,n-1,n.

Original entry on oeis.org

1, 6, 8, 4, 32, 28, 8, 16, 8, 8, 8, 16, 2, 16, 64, 6, 4, 8, 16, 384, 16, 256, 4, 12, 24, 64, 32, 256, 32, 64, 64, 12, 32, 128, 16, 80, 256, 32, 96, 96, 8, 8, 128, 256, 128, 128, 1024, 192, 64, 96, 128, 64, 32, 32, 128
Offset: 1

Views

Author

Amarnath Murthy, Aug 11 2005

Keywords

Examples

			a(3) = tau(32123) = 8.

Crossrefs

Cf. A000005, A007942, A110756, A110757, A110758, A110759.

Formula

a(n) = A000005(A007942(n)). - Michel Marcus, Mar 22 2023

Extensions

More terms from Ryan Propper, Jul 21 2006
Corrected and extended by Tyler Busby, Feb 12 2023
a(45)-a(55) from Tyler Busby, Feb 26 2023

A110757 a(n) = number of divisors of N, where N = reverse concatenation of 1,2,3,...,n.

Original entry on oeis.org

1, 4, 4, 4, 8, 4, 4, 12, 18, 8, 4, 8, 8, 16, 48, 16, 96, 576, 16, 32, 16, 32, 16, 32, 64, 256, 96, 32, 128, 256, 8, 64, 32, 128, 384, 144, 16, 8, 64, 32, 256, 64, 8, 192, 96, 32, 128, 128, 8, 64, 8, 128, 1280, 2560, 8, 24, 16, 64, 8, 8, 32, 384, 48, 64, 128, 128
Offset: 1

Views

Author

Amarnath Murthy, Aug 11 2005

Keywords

Examples

			a(3) = tau(321) = 4.

Links

Crossrefs

Cf. A000005, A000422, A176024.
Cf. A110756, A110758, A110759, A110760.

Programs

  • Mathematica
    s = ""; Do[s = ToString[n] <> s; Print[DivisorSigma[0, ToExpression[s]]], {n, 1, 45}] (* Ryan Propper, Sep 23 2005 *)
Table[DivisorSigma[0,FromDigits[Flatten[IntegerDigits/@Range[n,1,-1]]]],{n,50}] (* The program takes a long time to run. *) (* Harvey P. Dale, Jun 06 2018 *)

Formula

a(n) = A000005(A000422(n)). - Jinyuan Wang, May 23 2020

Extensions

More terms from Ryan Propper, Sep 23 2005
a(46)-a(66) from Jinyuan Wang, May 23 2020

A110758 a(n) is the number of divisors of N, where N = concatenation of n taken n times.

Original entry on oeis.org

1, 4, 6, 12, 8, 96, 8, 64, 20, 256, 96, 2304, 64, 512, 12288, 5120, 64, 5120, 8, 6144, 24576, 3072, 64, 24576, 1536, 1024, 7168, 12288, 256, 3145728, 32, 98304, 36864, 2048, 8192, 491520, 128, 128, 49152, 131072, 128, 6291456, 256, 73728, 5242880
Offset: 1

Views

Author

Amarnath Murthy, Aug 11 2005

Keywords

Examples

			a(3) = tau(333) = 6.

Links

Crossrefs

Cf. A000005, A000461.
Cf. A110756, A110757, A110759, A110760.

Programs

  • Mathematica
    f[n_] := Sum[n*10^(i * Length[IntegerDigits[n]]), {i, 0, n - 1}]; Do[Print[DivisorSigma[0, f[n]]], {n, 100}] (* Ryan Propper, Jul 21 2006 *)
Table[DivisorSigma[0,FromDigits[Flatten[IntegerDigits/@PadRight[{},n,n]]]],{n,50}] (* Harvey P. Dale, Apr 10 2023 *)
  • PARI
    a(n) = numdiv(eval(concat(apply(x->Str(x), vector(n, k, n))))); \\ Michel Marcus, Feb 12 2023

Formula

a(n) = A000005(A000461(n)). - Michel Marcus, Nov 18 2018

Extensions

Corrected and extended by Ryan Propper, Jul 21 2006

A110759 a(n) = tau(N), where N = concatenation 1,2,3,...,n,...,3,2,1. E.g., for n = 4, N = 1234321.

Original entry on oeis.org

1, 3, 9, 9, 9, 243, 9, 81, 45, 2, 4, 18, 8, 64, 96, 16, 24, 48, 64, 4, 48, 8, 16, 384, 4, 64, 640, 4, 16, 768, 16, 512, 144, 64, 64, 448, 8, 48, 192, 16, 64, 96, 8, 64, 896, 128, 64, 192, 128, 128, 384, 32, 64, 1280, 16, 64, 192, 16, 24, 192, 32, 16
Offset: 1

Views

Author

Amarnath Murthy, Aug 11 2005

Keywords

Comments

First 9 terms are odd as corresponding N are perfect squares.
Factorization of the larger N values:
f(25) = 989931671244066864878631629*p53
f(26) = 7*3209*17627*1322221*554840431325362973971*p48
f(27) = 3^4*7*223*28807*108727*5439394515032275997*361855463775135800641*p34
f(28) = 149*p89
f(29) = 7*317310923*296879723071339*p72
f(30) = 3^2*7*167*761*133337*431911*273884231501*4950715302671*p58
f(31) = 827*1141296551*10940622359204560200188943089306257*p58
f(32) = 7*31*5537737*42583813*62231909*19871693507*1441602757913*15884064847039967*p44
f(33) = 3^2*7^2*281*743580875118413*177233764237488717892587862569137279765057*p50
f(34) = 197*509*17780359481*34117699655579*22315348168833851*p70
f(35) = 7*10243*73778819*217751506979*815234955828637451*p78

Examples

			a(3) = tau(12321) = 9.

Crossrefs

Cf. A000005, A110756, A110757, A110758, A110760, A173426.

Programs

  • Maple
    A055642 := proc(n) 1+floor(log10(n)) ; end; A000005 := proc(n) numtheory[tau](n) ; end ; rep := proc(n) local a ; a := 1 ; for i from 2 to n do a := a*10^A055642(i)+i ; end; for i from n-1 to 1 by -1 do a := a*10^A055642(i)+i ; end; RETURN(a) ; end; A110759 := proc(n) A000005(rep(n)) ; end; for n from 1 to 50 do printf("%d %d ",n,A110759(n)) ; od ; # R. J. Mathar, Feb 10 2007
  • Mathematica
    Table[DivisorSigma[0,FromDigits[Join[Flatten[IntegerDigits/@Range[n]], Flatten[ IntegerDigits/@ Range[n-1,1,-1]]]]],{n,40}] (* Harvey P. Dale, Nov 17 2017 *)

Formula

a(n) = A000005(A173426(n)). - Georg Fischer, Feb 28 2023

Extensions

More terms from R. J. Mathar, Feb 10 2007
a(21)-a(35) from Robert Gerbicz, Nov 27 2010
a(36)-a(44) from Jinyuan Wang, May 17 2020
a(45)-a(58) from Tyler Busby, Feb 13 2023
a(59)-a(62) from Tyler Busby, Mar 04 2025

A366954 The sum of the divisors of the concatenation of 1,2,3,...,n.

Original entry on oeis.org

1, 28, 168, 1854, 19776, 327152, 1244416, 27319968, 178422816, 22222222056, 2075415810048, 308768621226000, 12455031810211128, 2469135782022242640, 197530862561779410288, 21604938101329359719880, 1821571286217135606177024, 270250398197557076360997936
Offset: 1

Views

Author

Sean A. Irvine, Oct 29 2023

Keywords

Examples

			a(3)=168 because the divisors of 123 are {1, 3, 41, 123}.

Crossrefs

Cf. A007908, A000203, A110756, A075019, A075022.

Formula

a(n) = sigma(A007908(n)) = A000203(A007908(n)).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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