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

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

Author

Amarnath Murthy, Aug 11 2005

Keywords

• nonn
• base
• more

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