Displaying 1-10 of 24 results found.
Digital roots of square numbers A000290.
+10
18
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
COMMENTS
Cyclic with a period of nine. Note that (7, 9, 4, 1, 9, 1, 4, 9, 7) is palindromic.
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 14 201
FORMULA
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+4x+9x^2+7x^3+7x^4+9x^5+4x^6+x^7+9x^8)/((1-x)(1+x+x^2)(1+x^3+x^6)). - Ant King, Oct 20 2009
MATHEMATICA
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(1) = 2; for n > 0, a(n+1) = a(n) * sum of digits of a(n).
+10
15
2, 4, 16, 112, 448, 7168, 157696, 5361664, 166211584, 5651193856, 276908498944, 19383594926080, 1298700860047360, 79220752462888960, 6733763959345561600, 592571228422409420800, 45035413360103115980800
MATHEMATICA
NestList[# Total[IntegerDigits[#]]&, 2, 20] (* Harvey P. Dale, Jul 18 2011 *)
PROG
(Haskell)
a047892 n = a047892_list !! (n-1)
AUTHOR
Miklos SZABO (mike(AT)ludens.elte.hu)
a(1) = 5; for n > 0, a(n+1) = a(n) * sum of digits of a(n).
+10
11
5, 25, 175, 2275, 36400, 473200, 7571200, 166566400, 5663257600, 226530304000, 5663257600000, 226530304000000, 5663257600000000, 226530304000000000, 5663257600000000000, 226530304000000000000, 5663257600000000000000
COMMENTS
After a(9), every second element has the same beginning. a(11+2k) = 40 * a(10+2k) = 40 * 25 * a(9+2k).
MATHEMATICA
NestList[#*Total[IntegerDigits[#]]&, 5, 20] (* Harvey P. Dale, Jan 25 2014 *)
PROG
(Haskell)
a047897 n = a047897_list !! (n-1)
AUTHOR
Miklos SZABO (mike(AT)ludens.elte.hu)
a(1) = 8; for n > 0, a(n+1) = a(n) * sum of digits of a(n).
+10
11
8, 64, 640, 6400, 64000, 640000, 6400000, 64000000, 640000000, 6400000000, 64000000000, 640000000000, 6400000000000, 64000000000000, 640000000000000, 6400000000000000, 64000000000000000, 640000000000000000
COMMENTS
After the 2nd element, every element has the same beginning.
a(3+k) = 10 * a(2+k).
MATHEMATICA
NestList[# Total[IntegerDigits[#]]&, 8, 20] (* or *) Join[{8}, NestList[ 10#&, 64, 20]] (* Harvey P. Dale, Jul 03 2020 *)
PROG
(Haskell)
a047900 n = a047900_list !! (n-1)
AUTHOR
Miklos SZABO (mike(AT)ludens.elte.hu)
a(1) = 11; for n > 0, a(n+1) = a(n) * sum of digits of a(n).
+10
11
11, 22, 88, 1408, 18304, 292864, 9078784, 390387712, 15615508480, 671466864640, 38945078149120, 2375649767096320, 180549382299320320, 12638456760952422400, 960522713832384102400, 67236589968266887168000
PROG
(Haskell)
a047902 n = a047902_list !! (n-1)
AUTHOR
Miklos SZABO (mike(AT)ludens.elte.hu)
a(1) = 6; for n > 0, a(n+1) = a(n) * (sum of digits of a(n)).
+10
10
6, 36, 324, 2916, 52488, 1417176, 38263752, 1377495072, 61987278240, 3347313024960, 150629086123200, 6778308875544000, 488038239039168000, 35138753210820096000, 2213741452281666048000, 159389384564279955456000
MATHEMATICA
Nest[Append[#, # Total@ IntegerDigits@ # &@ Last[#]] &, {6}, 15] (* Michael De Vlieger, Jul 08 2019 *)
PROG
(Haskell)
a047898 n = a047898_list !! (n-1)
(Python)
for _ in range(10**2):
....l *= sum(int(d) for d in str(l))
AUTHOR
Miklos SZABO (mike(AT)ludens.elte.hu)
a(1) = 7; for n > 0, a(n+1) = a(n) * sum of digits of a(n).
+10
10
7, 49, 637, 10192, 132496, 3312400, 43061200, 688979200, 33759980800, 1755519001600, 70220760064000, 2387505842176000, 138475338846208000, 9693273719234560000, 736688802661826560000, 64828614634240737280000
PROG
(Haskell)
a047899 n = a047899_list !! (n-1)
AUTHOR
Miklos SZABO (mike(AT)ludens.elte.hu)
a(1) = 3; for n > 0, a(n+1) = a(n) * sum of digits of a(n).
+10
10
3, 9, 81, 729, 13122, 118098, 3188646, 114791256, 4132485216, 148769467776, 10711401679872, 578415690713088, 41645929731342336, 2998506940656648192, 296852187125008171008, 24045027157125661851648
MATHEMATICA
NestList[# Total[IntegerDigits[#]]&, 3, 20] (* Harvey P. Dale, Mar 21 2011 *)
PROG
(Haskell)
a047912 n = a047912_list !! (n-1)
AUTHOR
Miklos SZABO (mike(AT)ludens.elte.hu)
a(1) = 9; a(n+1) = a(n) * sum of decimal digits of a(n).
+10
9
9, 81, 729, 13122, 118098, 3188646, 114791256, 4132485216, 148769467776, 10711401679872, 578415690713088, 41645929731342336, 2998506940656648192, 296852187125008171008, 24045027157125661851648, 2164052444141309566648320
MATHEMATICA
NestList[# Total[IntegerDigits[#]]&, 9, 20] (* Harvey P. Dale, Feb 07 2022 *)
PROG
(Haskell)
a047901 n = a047901_list !! (n-1)
(Python)
for _ in range(10**2):
....l *= sum(int(d) for d in str(l))
AUTHOR
Miklos SZABO (mike(AT)ludens.elte.hu)
n times the number of 1's in the binary expansion of n.
+10
6
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
FORMULA
a(2*n) = 2*a(n).
G.f.: x * (d/dx) (1/(1 - x))*Sum_{k>=0} x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Mar 27 2018
EXAMPLE
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 + ...
MAPLE
a:= n -> n * convert(convert(n, base, 2), `+`):
MATHEMATICA
Table[n*DigitCount[n, 2, 1], {n, 0, 100}] (* Harvey P. Dale, Dec 16 2014 *)
PROG
(PARI) sumbit(n) = my(r); while(n>0, r+=n%2; n\=2); r
a(n) = n*sumbit(n)
(PARI) {a(n) = if( n<0, 0, n * sumdigits(n, 2))}; /* Michael Somos, Aug 05 2014 */ /* since version 2.6.0 */ (Python) [n*bin(n)[2:].count('1') for n in range(1000)] # Chai Wah Wu, Aug 03 2014
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