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

Author

Santi Spadaro, Oct 12 2001

Keywords

• easy
• nice
• nonn
• base

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.

Crossrefs

Cf. A010888, A031347, A239427.

Programs

Haskell

a064702 n = a064702_list !! (n-1)
a064702_list = filter (\x -> a010888 x == a031347 x) [1..]
-- Reinhard Zumkeller, Jul 10 2013

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 *)

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

Extensions

Definition rephrased by Reinhard Zumkeller, Jul 10 2013

Initial 0 added by Halfdan Skjerning, Aug 21 2018