A064702 Nonnegative numbers such that additive and multiplicative digital roots coincide.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 123, 132, 137, 139, 168, 173, 179, 186, 188, 193, 197, 213, 231, 233, 267, 276, 299, 312, 317, 319, 321, 323, 332, 346, 364, 371, 389, 391, 398, 436, 463, 618, 627, 634, 643, 672, 681, 713, 719, 726, 731, 762, 791, 816, 818, 839

Offset

1,3

Comments

If k is in this sequence then all permutations of (the digits of) k are in this sequence.

A010888(a(n)) = A031347(a(n)). - Reinhard Zumkeller, Jul 10 2013

Links

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Digital Root

Eric Weisstein's World of Mathematics, Multiplicative Digital Root

Wikipedia, Digital root

Wikipedia, Multiplicative digital root

Index entries for 10-automatic sequences.

Maple

A007954 := proc(n) return mul(d, d=convert(n, base, 10)): end: A031347 := proc(n) local m: m:=n: while(length(m)>1)do m:=A007954(m): od: return m: end: A064702 := proc(n) option remember: local k: if(n=1)then return 1:fi: for k from procname(n-1)+1 do if(A031347(k)-1 = (k-1) mod 9)then return k: fi: od: end: seq(A064702(n), n=1..56); # Nathaniel Johnston, May 04 2011

Mathematica

okQ[n_]:=NestWhile[Times@@IntegerDigits[#]&, n, #>9&]== NestWhile[ Total[ IntegerDigits[ #]]&, n, #>9&]; Select[Range[1000], okQ]  (* Harvey P. Dale, Apr 20 2011 *)

Prog

(Haskell)
a064702 n = a064702_list !! (n-1)
a064702_list = filter (\x -> a010888 x == a031347 x) [1..]
-- Reinhard Zumkeller, Jul 10 2013
(PARI) is(n) = my(cn = n); while(cn > 9, d = digits(cn); cn = prod(i = 1, #d, d[i])); cn - 1 == (n-1)%9 \\ David A. Corneth, Aug 23 2018
(Python)
from math import prod
def A010888(n):
    while n > 9: n = sum(map(int, str(n)))
    return n
def A031347(n):
    while n > 9: n = prod(map(int, str(n)))
    return n
def ok(n): return A010888(n) == A031347(n)
print([k for k in range(840) if ok(k)]) # Michael S. Branicky, Sep 17 2022

Crossrefs

Cf. A010888, A031347, A239427.

Sequence in context: A249334 A338257 A064158 * A034710 A305257 A318273

Adjacent sequences: A064699 A064700 A064701 * A064703 A064704 A064705

Keyword

easy,nice,nonn,base

Author

Santi Spadaro, Oct 12 2001

Extensions

Definition rephrased by Reinhard Zumkeller, Jul 10 2013

Initial 0 added by Halfdan Skjerning, Aug 21 2018

Status

approved