A016052 a(1) = 3; for n >= 1, a(n+1) = a(n) + sum of its digits.

3, 6, 12, 15, 21, 24, 30, 33, 39, 51, 57, 69, 84, 96, 111, 114, 120, 123, 129, 141, 147, 159, 174, 186, 201, 204, 210, 213, 219, 231, 237, 249, 264, 276, 291, 303, 309, 321, 327, 339, 354, 366, 381, 393, 408, 420, 426, 438, 453, 465, 480, 492

Offset: 1

Author

Robert G. Wilson v

Keywords

• nonn
• base
• easy

Comments

Mod 9 this sequence is 3, 6, 3, 6, 3, 6, ... This shows that this sequence is disjoint from A004207. - N. J. A. Sloane, Oct 15 2013

References

• D. R. Kaprekar, Puzzles of the Self-Numbers. 311 Devlali Camp, Devlali, India, 1959.
• D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately printed, 311 Devlali Camp, Devlali, India, 1963.
• G. E. Stevens and L. G. Hunsberger, A Result and a Conjecture on Digit Sum Sequences, J. Recreational Math. 27, no. 4 (1995), pp. 285-288.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 34-35.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• D. R. Kaprekar, The Mathematics of the New Self Numbers [annotated and scanned]
• Index entries for Colombian or self numbers and related sequences

Crossrefs

Cf. A004207, A007618, A006507, A016096, A230107, A230286, A230287, A084228.

Programs

Haskell

a016052 n = a016052_list !! (n-1)
a016052_list = iterate a062028 3  -- Reinhard Zumkeller, Oct 14 2013

Mathematica

NestList[# + Total[IntegerDigits[#]] &, 3, 51] (* Jayanta Basu, Aug 11 2013 *)
a[1] = 3; a[n_] := a[n] = a[n - 1] + Total@ IntegerDigits@ a[n - 1]; Array[a, 80] (* Robert G. Wilson v, Jun 27 2014 *)

PARI

a_list(nn) = { my(f(n, i) = n + vecsum(digits(n)), S=vector(nn+1)); S[1]=3; for(k=2, #S, S[k] = fold(f, S[1..k-1])); S[2..#S] } \\ Satish Bysany, Mar 04 2017

Python

from itertools import islice
def A016052_gen(): # generator of terms
    yield (a:=3)
    while True: yield (a:=a+sum(map(int,str(a))))
A016052_list = list(islice(A016052_gen(),20)) # Chai Wah Wu, Jun 16 2024

Formula

a(n) = A062028(a(n-1)) for n > 1. - Reinhard Zumkeller, Oct 14 2013

a(n) - a(n-1) = A084228(n+1). - Robert G. Wilson v, Jun 27 2014