A057147 -id:A057147 - cp's OEIS frontend

A056992 Digital roots of square numbers A000290.

1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4, 9, 7, 7, 9

Offset: 1

Eric W. Weisstein

Cyclic with a period of nine. Note that (7, 9, 4, 1, 9, 1, 4, 9, 7) is palindromic.
a(n) is also the decimal expansion of 499264730/333333333. - Enrique Pérez Herrero, Jul 28 2009
a(n) is also the digital root of A002477(n). - Enrique Pérez Herrero, Dec 20 2009
First comment above by Enrique Pérez Herrero and his formula below together give the following identity: 1+Sum_{n>=2}(1+9*((n^2-1)/9-floor((n^2-1)/9)))/10^(n-1) = 499264730/333333333 = 1.49779419149779419149779419... - Alexander R. Povolotsky, Jun 14 2012

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Square Number.

Cf. A000290, A002477, A010888, A057147, A056991.

a056992 = a010888 . a000290  -- Reinhard Zumkeller, Mar 19 2014

DigitalRoot[n_Integer?NonNegative] := 1 + 9*FractionalPart[(n - 1)/9] A056992[n_]:=DigitalRoot[n^2] (* Enrique Pérez Herrero, Dec 20 2009 *)
Table[FixedPoint[Total[IntegerDigits[#]]&,n^2],{n,90}] (* Zak Seidov, Jun 13 2015 *)
PadRight[{},120,{1,4,9,7,7,9,4,1,9}] (* Harvey P. Dale, Apr 16 2022 *)

a(n) = 1+9*{(n^2-1)/9}, where the symbol {} means fractional part. - Enrique Pérez Herrero, Dec 20 2009
a(n) = 3(1 + cos(2n*Pi/3) + cos(4n*Pi/3)) + mod(3n^4+3n^6+4n^8,9). - Ant King, Oct 07 2009
G.f.: x*(1+4*x+9*x^2+7*x^3+7*x^4+9*x^5+4*x^6+x^7+9*x^8)/((1-x)*(1+x+x^2)*(1+x^3+x^6)). - Ant King, Oct 20 2009
a(n) = A010888(A057147(n)). - Reinhard Zumkeller, Mar 19 2014

A047892 a(1) = 2; for n > 0, a(n+1) = a(n) * sum of digits of a(n).

Original entry on oeis.org

2, 4, 16, 112, 448, 7168, 157696, 5361664, 166211584, 5651193856, 276908498944, 19383594926080, 1298700860047360, 79220752462888960, 6733763959345561600, 592571228422409420800, 45035413360103115980800

Offset: 1

Miklos SZABO (mike(AT)ludens.elte.hu)

a(n) mod 9 = A010712(n-1) for n > 1. - Reinhard Zumkeller, Sep 23 2007

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

Cf. A004207.
Cf. A007953.
Cf. A047912 (start=3), A047897 (start=5), A047898 (start=6), A047899 (start=7), A047900 (start=8), A047901 (start=9), A047902 (start=11).

a047892 n = a047892_list !! (n-1)
a047892_list = iterate a057147 2  -- Reinhard Zumkeller, Mar 19 2014

NestList[# Total[IntegerDigits[#]]&,2,20] (* Harvey P. Dale, Jul 18 2011 *)

a(n+1) = A057147(a(n)). - Reinhard Zumkeller, Mar 19 2014

Offset changed by Reinhard Zumkeller, Mar 19 2014

A047897 a(1) = 5; for n > 0, a(n+1) = a(n) * sum of digits of a(n).

Original entry on oeis.org

5, 25, 175, 2275, 36400, 473200, 7571200, 166566400, 5663257600, 226530304000, 5663257600000, 226530304000000, 5663257600000000, 226530304000000000, 5663257600000000000, 226530304000000000000, 5663257600000000000000

Offset: 1

Miklos SZABO (mike(AT)ludens.elte.hu)

After a(9), every second element has the same beginning. a(11+2k) = 40 * a(10+2k) = 40 * 25 * a(9+2k).

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

Cf. A047892 (start=2), A047912 (start=3), A047898 (start=6), A047899 (start=7), A047900 (start=8), A047901 (start=9), A047902 (start=11).

a047897 n = a047897_list !! (n-1)
a047897_list = iterate a057147 5  -- Reinhard Zumkeller, Mar 19 2014

NestList[#*Total[IntegerDigits[#]]&,5,20] (* Harvey P. Dale, Jan 25 2014 *)

a(n+1) = A057147(a(n)). - Reinhard Zumkeller, Mar 19 2014

A047900 a(1) = 8; for n > 0, a(n+1) = a(n) * sum of digits of a(n).

Original entry on oeis.org

8, 64, 640, 6400, 64000, 640000, 6400000, 64000000, 640000000, 6400000000, 64000000000, 640000000000, 6400000000000, 64000000000000, 640000000000000, 6400000000000000, 64000000000000000, 640000000000000000

Offset: 1

Miklos SZABO (mike(AT)ludens.elte.hu)

After the 2nd element, every element has the same beginning.

a(3+k) = 10 * a(2+k).

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

Cf. A047892 (start=2), A047912 (start=3), A047897 (start=5), A047898 (start=6), A047899 (start=7), A047901 (start=9), A047902 (start=11).

a047900 n = a047900_list !! (n-1)
a047900_list = iterate a057147 8  -- Reinhard Zumkeller, Mar 19 2014

NestList[# Total[IntegerDigits[#]]&,8,20] (* or *) Join[{8},NestList[ 10#&,64,20]] (* Harvey P. Dale, Jul 03 2020 *)

a(n+1) = A057147(a(n)). - Reinhard Zumkeller, Mar 19 2014

A047902 a(1) = 11; for n > 0, a(n+1) = a(n) * sum of digits of a(n).

Original entry on oeis.org

11, 22, 88, 1408, 18304, 292864, 9078784, 390387712, 15615508480, 671466864640, 38945078149120, 2375649767096320, 180549382299320320, 12638456760952422400, 960522713832384102400, 67236589968266887168000

Offset: 1

Miklos SZABO (mike(AT)ludens.elte.hu)

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

Cf. A047892 (start=2), A047912 (start=3), A047897 (start=5), A047898 (start=6), A047899 (start=7), A047900 (start=8), A047901 (start=9).

a047902 n = a047902_list !! (n-1)
a047902_list = iterate a057147 11  -- Reinhard Zumkeller, Mar 19 2014

a(n+1) = A057147(a(n)). - Reinhard Zumkeller, Mar 19 2014

A047898 a(1) = 6; for n > 0, a(n+1) = a(n) * (sum of digits of a(n)).

Original entry on oeis.org

6, 36, 324, 2916, 52488, 1417176, 38263752, 1377495072, 61987278240, 3347313024960, 150629086123200, 6778308875544000, 488038239039168000, 35138753210820096000, 2213741452281666048000, 159389384564279955456000

Offset: 1

Miklos SZABO (mike(AT)ludens.elte.hu)

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

Cf. A047892 (start=2), A047912 (start=3), A047897 (start=5), A047899 (start=7), A047900 (start=8), A047901 (start=9), A047902 (start=11).

a047898 n = a047898_list !! (n-1)
a047898_list = iterate a057147 6  -- Reinhard Zumkeller, Mar 19 2014

Nest[Append[#, # Total@ IntegerDigits@ # &@ Last[#]] &, {6}, 15] (* Michael De Vlieger, Jul 08 2019 *)

A047898_list, l = [6], 6
for _ in range(10**2):
    l *= sum(int(d) for d in str(l))
    A047898_list.append(l) # Chai Wah Wu, Jan 04 2015

a(n+1) = A057147(a(n)). - Reinhard Zumkeller, Mar 19 2014

A047899 a(1) = 7; for n > 0, a(n+1) = a(n) * sum of digits of a(n).

Original entry on oeis.org

7, 49, 637, 10192, 132496, 3312400, 43061200, 688979200, 33759980800, 1755519001600, 70220760064000, 2387505842176000, 138475338846208000, 9693273719234560000, 736688802661826560000, 64828614634240737280000

Offset: 1

Miklos SZABO (mike(AT)ludens.elte.hu)

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

Cf. A047892 (start=2), A047912 (start=3), A047897 (start=5), A047898 (start=6), A047900 (start=8), A047901 (start=9), A047902 (start=11).

a047899 n = a047899_list !! (n-1)
a047899_list = iterate a057147 7  -- Reinhard Zumkeller, Mar 19 2014

a(n+1) = A057147(a(n)). - Reinhard Zumkeller, Mar 19 2014

A047912 a(1) = 3; for n > 0, a(n+1) = a(n) * sum of digits of a(n).

Original entry on oeis.org

3, 9, 81, 729, 13122, 118098, 3188646, 114791256, 4132485216, 148769467776, 10711401679872, 578415690713088, 41645929731342336, 2998506940656648192, 296852187125008171008, 24045027157125661851648

Offset: 1

Miklos SZABO (mike(AT)ludens.elte.hu)

Apart from the first term, the same as A047901. - R. J. Mathar, Oct 18 2008

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

Cf. A047892 (start=2), A047897 (start=5), A047898 (start=6), A047899 (start=7), A047900 (start=8), A047901 (start=9), A047902 (start=11).

a047912 n = a047912_list !! (n-1)
a047912_list = iterate a057147 3  -- Reinhard Zumkeller, Mar 19 2014

NestList[# Total[IntegerDigits[#]]&,3,20]  (* Harvey P. Dale, Mar 21 2011 *)

a(n+1) = A057147(a(n)). - Reinhard Zumkeller, Mar 19 2014

A047901 a(1) = 9; a(n+1) = a(n) * sum of decimal digits of a(n).

Original entry on oeis.org

9, 81, 729, 13122, 118098, 3188646, 114791256, 4132485216, 148769467776, 10711401679872, 578415690713088, 41645929731342336, 2998506940656648192, 296852187125008171008, 24045027157125661851648, 2164052444141309566648320

Offset: 1

Miklos SZABO (mike(AT)ludens.elte.hu)

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

Cf. A047892 (start=2), A047912 (start=3), A047897 (start=5), A047898 (start=6), A047899 (start=7), A047900 (start=8), A047902 (start=11).

a047901 n = a047901_list !! (n-1)
a047901_list = iterate a057147 9  -- Reinhard Zumkeller, Mar 19 2014

NestList[# Total[IntegerDigits[#]]&,9,20] (* Harvey P. Dale, Feb 07 2022 *)

A047901_list, l = [9], 9
for _ in range(10**2):
    l *= sum(int(d) for d in str(l))
    A047901_list.append(l) # Chai Wah Wu, Jan 04 2015

a(n+1) = A057147(a(n)). - Reinhard Zumkeller, Mar 19 2014

Edited by Charles R Greathouse IV, Aug 02 2010

A245788 n times the number of 1's in the binary expansion of n.

Original entry on oeis.org

0, 1, 2, 6, 4, 10, 12, 21, 8, 18, 20, 33, 24, 39, 42, 60, 16, 34, 36, 57, 40, 63, 66, 92, 48, 75, 78, 108, 84, 116, 120, 155, 32, 66, 68, 105, 72, 111, 114, 156, 80, 123, 126, 172, 132, 180, 184, 235, 96, 147, 150, 204, 156, 212, 216, 275, 168, 228, 232, 295, 240

Offset: 0

Franklin T. Adams-Watters, Aug 01 2014

			G.f. = x + 2*x^2 + 6*x^3 + 4*x^4 + 10*x^5 + 12*x6 + 21*x^7 + 8*x^8 + 18*x^9 + ...

Jens Kruse Andersen, Table of n, a(n) for n = 0..1000
Project Euler, Problem 759, sequence f(n).

Cf. A000120 (number of 1's), A057147 (decimal version).

a:= n -> n * convert(convert(n,base,2),`+`):
seq(a(n),n=0..100); # Robert Israel, Aug 01 2014

Table[n*DigitCount[n,2,1],{n,0,100}] (* Harvey P. Dale, Dec 16 2014 *)

sumbit(n) = my(r);while(n>0,r+=n%2;n\=2);r
a(n) = n*sumbit(n)

{a(n) = if( n<0, 0, n * sumdigits(n, 2))}; /* Michael Somos, Aug 05 2014 */ /* since version 2.6.0 */

[n*bin(n)[2:].count('1') for n in range(1000)] # Chai Wah Wu, Aug 03 2014

a(2*n) = 2*a(n).
a(2*n+1) = 2*n + 1 + (2+1/n)*a(n). - Robert Israel, Aug 01 2014
G.f.: x * (d/dx) (1/(1 - x))*Sum_{k>=0} x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Mar 27 2018

cp's OEIS Frontend

A056992 Digital roots of square numbers A000290.

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A047892 a(1) = 2; for n > 0, a(n+1) = a(n) * sum of digits of a(n).

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A047897 a(1) = 5; for n > 0, a(n+1) = a(n) * sum of digits of a(n).

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A047900 a(1) = 8; for n > 0, a(n+1) = a(n) * sum of digits of a(n).

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A047902 a(1) = 11; for n > 0, a(n+1) = a(n) * sum of digits of a(n).

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Haskell

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A047898 a(1) = 6; for n > 0, a(n+1) = a(n) * (sum of digits of a(n)).

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A047899 a(1) = 7; for n > 0, a(n+1) = a(n) * sum of digits of a(n).

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Haskell

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A047912 a(1) = 3; for n > 0, a(n+1) = a(n) * sum of digits of a(n).

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