A110760 -id:A110760 - cp's OEIS frontend

A110756 a(n) = tau(N), where N = concatenation 1,2,3,...,n.

1, 6, 4, 4, 8, 28, 4, 24, 12, 8, 64, 32, 8, 8, 8, 12, 24, 48, 64, 192, 256, 64, 16, 48, 24, 160, 96, 64, 16, 128, 8, 48, 64, 4, 192, 120, 16, 16, 64, 384, 32, 128, 16, 48, 768, 128, 32, 192, 64, 768, 8, 32, 64, 1792, 32, 24, 64, 16, 16, 128, 8, 192, 24, 768, 64

Offset: 1

Amarnath Murthy, Aug 11 2005

			a(3) = tau(123) = 4.

Tyler Busby, Table of n, a(n) for n = 1..117

Cf. A000005, A007908, A110757, A110758, A110759, A110760.

A055642 := proc(n) 1+floor(log10(n)) ; end; A000005 := proc(n) numtheory[tau](n) ; end ; A007908 := proc(n) local a ; a := 1 ; for i from 2 to n do a := a*10^A055642(i)+i ; end; RETURN(a) ; end; A110756 := proc(n) A000005(A007908(n)) ; end; for n from 1 to 50 do printf("%d %d ",n,A110756(n)) ; od ; # R. J. Mathar, Feb 10 2007

Table[DivisorSigma[0,FromDigits[Flatten[IntegerDigits/@Range[n]]]],{n,60}] (* Harvey P. Dale, Apr 01 2024 *)

a(n) = A000005(A007908(n)). - R. J. Mathar, Feb 10 2007

More terms from R. J. Mathar, Feb 10 2007
More terms from David Wasserman, Dec 22 2008
a(57)-a(65) from Jinyuan Wang, May 23 2020

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

Amarnath Murthy, Aug 11 2005

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

Tyler Busby, Table of n, a(n) for n = 1..109

Cf. A000005, A000422, A176024.

Cf. A110756, A110758, A110759, A110760.

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 *)

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

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

Amarnath Murthy, Aug 11 2005

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

Max Alekseyev, Table of n, a(n) for n = 1..158

Cf. A000005, A000461.

Cf. A110756, A110757, A110759, A110760.

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 *)

a(n) = numdiv(eval(concat(apply(x->Str(x), vector(n, k, n))))); \\ Michel Marcus, Feb 12 2023

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

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

Amarnath Murthy, Aug 11 2005

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

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

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

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

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

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

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

A360736 Number of prime divisors of A007942(n) = decimal concatenation of sequence (n, n-1, ..., 2, 1, 2, ..., n-1, n) counted with multiplicity.

Original entry on oeis.org

0, 3, 3, 2, 5, 8, 3, 4, 3, 3, 3, 5, 1, 4, 6, 3, 2, 3, 4, 11, 4, 8, 2, 4, 5, 6, 5, 9, 5, 6, 6, 4, 5, 7, 4, 8, 8, 5, 7, 7, 3, 3, 7, 9, 7, 7, 10, 8, 6, 7, 7, 10, 5, 5

Offset: 1

Bernard Schott, Mar 18 2023

For n <= 1530, only a(13) = 1 (answer to Smaradanche problem 19).

First semiprimes appear in A007942 at indices 4, 17, 23 since a(4) = a(17) = a(23) = 2.

			a(4) = 2 since 4321234 = 2 * 2160617;
a(6) = 8 since 65432123456 = 2^6 * 7 * 146053847;
a(12) = 5 since 12111098765432123456789101112 = 2^3*60800821*24899126702236725259;
a(13) = 1 since 131211109876543212345678910111213 is prime.

M. Fleuren, Factoring of the Smarandache Mirror Sequence.
F. Smarandache, Only Problems, Not Solutions!, Mirror sequence, problem 19, page 20.

Cf. A001222, A007942, A110760, A361624.

from sympy import factorint
def A360736(n): return sum(factorint(int(''.join(map(str,range(n,1,-1)))+''.join(map(str,range(1,n+1))))).values()) # Chai Wah Wu, Mar 21 2023

a(n) = A001222(A007942(n)).

a(36)-a(54) from Amiram Eldar, Mar 19 2023

A361624 Number of distinct prime factors in decimal concatenation of integer (n, n-1, ..., 2, 1, 2, ..., n-1, n) = A007942(n).

Original entry on oeis.org

0, 2, 3, 2, 5, 3, 3, 4, 3, 3, 3, 3, 1, 4, 6, 2, 2, 3, 4, 7, 4, 8, 2, 3, 4, 6, 5, 7, 5, 6, 6, 3, 5, 7, 4, 5, 8, 5, 6, 6, 3, 3, 7, 7, 7, 7, 10, 7, 6, 6, 7, 4, 5, 5, 7

Offset: 1

Bernard Schott, Mar 18 2023

a(n) < A360736(n) when n > 10 is a multiple of 4 or of 25, since, for these indices, A007942(n) is divisible by 2^2 or 5^2; but this inequality holds also, for other indices: for n = 6 (see example) and n = 39 where A007942(39) = 29 * 617^2 * 10185403128074353 * ...

			a(4) = 2 since 4321234 = 2 * 2160617;
a(6) = 3 since 65432123456 = 2^6 * 7 * 146053847.

M. Fleuren, Factoring of the Smarandache Mirror Sequence.
F. Smarandache, Only Problems, Not Solutions!, Mirror sequence, problem 19, page 20.

Cf. A001221, A007942, A110760, A360736.

from sympy import primefactors
def A361624(n): return len(primefactors(int(''.join(map(str,range(n,1,-1)))+''.join(map(str,range(1,n+1)))))) # Chai Wah Wu, Mar 21 2023

a(n) = A001221(A007942(n)).

a(36)-a(54) from Amiram Eldar, Mar 19 2023
a(42) corrected by Sean A. Irvine, Sep 26 2023
a(55) from Sean A. Irvine, Oct 16 2023

cp's OEIS Frontend

A110756 a(n) = tau(N), where N = concatenation 1,2,3,...,n.

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

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

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

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A360736 Number of prime divisors of A007942(n) = decimal concatenation of sequence (n, n-1, ..., 2, 1, 2, ..., n-1, n) counted with multiplicity.

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A361624 Number of distinct prime factors in decimal concatenation of integer (n, n-1, ..., 2, 1, 2, ..., n-1, n) = A007942(n).

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