A031286 Additive persistence: number of summations of digits needed to obtain a single digit (the additive digital root).

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

Offset

0,20

Links

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

Antonios Meimaris, On the additive persistence of a number in base p, Preprint, 2015.

N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.

Eric Weisstein's World of Mathematics, Additive Persistence

Maple

read("transforms") ;
A031286 := proc(n)
    local a, nper;
    nper := n ;
    a := 0 ;
    while nper > 9 do
        nper := digsum(nper) ;
        a := a+1 ;
    end do:
    a ;
end proc:
seq(A031286(n), n=0..80) ; # R. J. Mathar, Jan 02 2018

Mathematica

lst = {}; Do[s = 0; While[n > 9, s++; n = Plus @@ IntegerDigits[n]]; AppendTo[lst, s], {n, 0, 98}]; lst (* Arkadiusz Wesolowski, Oct 17 2012 *)

Prog

(PARI) dsum(n)=my(s); while(n, s+=n%10; n\=10); s
a(n)=my(s); while(n>9, s++; n=dsum(n)); s \\ Charles R Greathouse IV, Sep 13 2012
(Python)
def A031286(n):
    ap = 0
    while n > 9:
        n = sum(int(d) for d in str(n))
        ap += 1
    return ap
# Chai Wah Wu, Aug 23 2014

Crossrefs

Cf. A010888 (additive digital root of n).

Cf. A031347 (multiplicative digital root of n).

Cf. A031346 (multiplicative persistence of n).

Cf. also A006050, A045646.

Cf. Numbers with additive persistence k: A304366 (k=1), A304367 (k=2), A304368 (k=3), A304373 (k=4). - Jaroslav Krizek, May 28 2018

Sequence in context: A308479 A031280 A134870 * A031276 A305080 A261794

Adjacent sequences: A031283 A031284 A031285 * A031287 A031288 A031289

Keyword

nonn,base

Author

Eric W. Weisstein

Extensions

Corrected by Reinhard Zumkeller, Feb 05 2009

Status

approved