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a(n) = tau(N), where N = concatenation 1,2,3,...,n,...,3,2,1. E.g., for n = 4, N = 1234321.
5

%I #46 Mar 29 2023 10:58:07

%S 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,

%T 640,4,16,768,16,512,144,64,64,448,8,48,192,16,64,96,8,64,896,128,64,

%U 192,128,128,384,32,64,1280,16,64,192,16

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

%C First 9 terms are odd as corresponding N are perfect squares.

%C Factorization of the larger N values:

%C f(25) = 989931671244066864878631629*p53

%C f(26) = 7*3209*17627*1322221*554840431325362973971*p48

%C f(27) = 3^4*7*223*28807*108727*5439394515032275997*361855463775135800641*p34

%C f(28) = 149*p89

%C f(29) = 7*317310923*296879723071339*p72

%C f(30) = 3^2*7*167*761*133337*431911*273884231501*4950715302671*p58

%C f(31) = 827*1141296551*10940622359204560200188943089306257*p58

%C f(32) = 7*31*5537737*42583813*62231909*19871693507*1441602757913*15884064847039967*p44

%C f(33) = 3^2*7^2*281*743580875118413*177233764237488717892587862569137279765057*p50

%C f(34) = 197*509*17780359481*34117699655579*22315348168833851*p70

%C f(35) = 7*10243*73778819*217751506979*815234955828637451*p78

%F a(n) = A000005(A173426(n)). - _Georg Fischer_, Feb 28 2023

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

%p 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

%t Table[DivisorSigma[0,FromDigits[Join[Flatten[IntegerDigits/@Range[n]], Flatten[ IntegerDigits/@ Range[n-1,1,-1]]]]],{n,40}] (* _Harvey P. Dale_, Nov 17 2017 *)

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

%K nonn,base,more

%O 1,2

%A _Amarnath Murthy_, Aug 11 2005

%E More terms from _R. J. Mathar_, Feb 10 2007

%E a(21)-a(35) from _Robert Gerbicz_, Nov 27 2010

%E a(36)-a(44) from _Jinyuan Wang_, May 17 2020

%E a(45)-a(58) from _Tyler Busby_, Feb 13 2023