A118880 -id:A118880 - cp's OEIS frontend

A231688 a(n) = Sum_{i=0..n} digsum(i)^3, where digsum(i) = A007953(i).

0, 1, 9, 36, 100, 225, 441, 784, 1296, 2025, 2026, 2034, 2061, 2125, 2250, 2466, 2809, 3321, 4050, 5050, 5058, 5085, 5149, 5274, 5490, 5833, 6345, 7074, 8074, 9405, 9432, 9496, 9621, 9837, 10180, 10692, 11421, 12421, 13752, 15480, 15544, 15669, 15885, 16228, 16740, 17469, 18469, 19800, 21528, 23725, 23850, 24066, 24409, 24921, 25650, 26650, 27981, 29709, 31906, 34650

Offset: 0

N. J. A. Sloane, Nov 13 2013

Grabner, P. J.; Kirschenhofer, P.; Prodinger, H.; Tichy, R. F.; On the moments of the sum-of-digits function. Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), 263-271, Kluwer Acad. Publ., Dordrecht, 1993.

Indranil Ghosh, Table of n, a(n) for n = 0..10000
J. Coquet, Power sums of digital sums, J. Number Theory 22 (1986), no. 2, 161-176.
R. E. Kennedy and C. N. Cooper, An extension of a theorem by Cheo and Yien concerning digital sums, Fibonacci Q. 29, No. 2, 145-149 (1991).
J.-L. Mauclaire and Leo Murata, On q-additive functions, I. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.
J.-L. Mauclaire and Leo Murata, On q-additive functions, II. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 9, 441-444.
H. Riede, Asymptotic estimation of a sum of digits, Fibonacci Q. 36, No. 1, 72-75 (1998).
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag. 41 1968 21-25.

Cf. A007953, A037123, A074784, A231689.

Partial sums of A118880.

Maple
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See A037123.
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Accumulate[Table[Total[IntegerDigits[n]]^3,{n,0,60}]] (* Harvey P. Dale, Aug 06 2021 *)

a(n) = sum(i=0, n, sumdigits(i)^3); \\ Michel Marcus, Jan 07 2017

A182128 Number of iterations of the map n -> (sum of the decimal digits of n)^3 before reaching the last number of the cycle.

Original entry on oeis.org

0, 0, 2, 3, 3, 2, 3, 3, 1, 2, 1, 2, 3, 3, 2, 3, 3, 1, 2, 2, 2, 3, 3, 2, 3, 3, 1, 2, 2, 2, 3, 3, 2, 3, 3, 1, 2, 2, 2, 2, 3, 2, 3, 3, 1, 2, 2, 2, 2, 3, 2, 3, 3, 1, 2, 2, 2, 2, 3, 2, 3, 3, 1, 2, 2, 2, 2, 3, 2, 2, 3, 1, 2, 2, 2, 2, 3, 2, 2, 3, 1, 2, 2, 2, 2, 3, 2

Offset: 0

Michel Lagneau, Apr 13 2012

a(n) is the number of times that the cube of the sum of the digits must be calculated before reaching the last number of the cycle.

			0 is in the sequence twice because 0 -> 0 and 1 -> 1;
a(3) = 3:
    3 ->       3^3 = 27;
   27 ->   (2+7)^3 = 729;
  729 -> (7+2+9)^3 = 18^3 = 5832 is the end of the map because 5832 -> (5+8+3+2)^3 = 18^3 is already in the trajectory. Hence we obtain the map: 3 -> 27 -> 729 -> 5832 with 3 iterations.

Cf. A177148, A178481, A118880.

A182128 := proc(n)
        local traj ,c;
        traj := n ;
        c := [n] ;
        while true do
                traj := A118880(traj) ;
                if member(traj,c) then
                        return nops(c)-1 ;
                end if;
                c := [op(c),traj] ;
        end do:
end proc:
seq(A182128(n),n=0..80) ; # R. J. Mathar, Jul 08 2012

A162840 Numbers k such that the cube of the sum of digits of k equals the product of digits of k.

Original entry on oeis.org

0, 1, 666666, 1377789, 1377798, 1377879, 1377897, 1377978, 1377987, 1378779, 1378797, 1378977, 1379778, 1379787, 1379877, 1387779, 1387797, 1387977, 1389777, 1397778, 1397787, 1397877, 1398777, 1555888, 1558588, 1558858

Offset: 0

Boris Hostnik (megpplus(AT)siol.net), Jul 14 2009

			666666 is in the sequence because (1) cubed sum of its digits is (6+6+6+6+6+6)^3 = 46656, (2) the product of its digits is 6*6*6*6*6*6=46656; 46656=46656.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A007954, A118880, A117720, A034710. - R. J. Mathar, Jul 19 2009

A007953 := proc(n) add(d,d=convert(n,base,10)) ; end: A007954 := proc(n) mul(d,d=convert(n,base,10)) ; end: A118880 := proc(n) (A007953(n))^3; end: for n from 1 to 2000000 do if A118880(n) = A007954(n) then printf("%d,\n",n) ; fi; od: # R. J. Mathar, Jul 19 2009

Select[Range[0,156*10^4],Total[IntegerDigits[#]]^3==Times@@IntegerDigits[#]&] (* Harvey P. Dale, Jul 07 2022 *)

{n: A118880(n)=A007954(n)}. - R. J. Mathar, Jul 19 2009

cp's OEIS Frontend

A231688 a(n) = Sum_{i=0..n} digsum(i)^3, where digsum(i) = A007953(i).

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A182128 Number of iterations of the map n -> (sum of the decimal digits of n)^3 before reaching the last number of the cycle.

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Maple

A162840 Numbers k such that the cube of the sum of digits of k equals the product of digits of k.

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Author

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Programs

Maple

Mathematica

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