A304373 -id:A304373 - cp's OEIS frontend

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

Eric W. Weisstein

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

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

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

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

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

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

Corrected by Reinhard Zumkeller, Feb 05 2009

A304366 Numbers with additive persistence = 1.

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, 32, 33, 34, 35, 36, 40, 41, 42, 43, 44, 45, 50, 51, 52, 53, 54, 60, 61, 62, 63, 70, 71, 72, 80, 81, 90, 100, 101, 102, 103, 104, 105, 106, 107, 108, 110, 111, 112, 113, 114, 115, 116

Offset: 1

Jaroslav Krizek, May 11 2018

For d >= 2, there are A000581(d+8) terms with d digits. - Robert Israel, Dec 28 2023

			Adding the digits of 10 gives 1, a single-digit number, so 10 is a member. Adding the digits of 39 gives 12, which is a 2-digit number, so 39 is not a member. - _Michael B. Porter_, May 16 2018

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Additive Persistence.

Cf. A000581, A031286.

Cf. Numbers with additive persistence k: A304367 (k=2), A304368 (k=3), A304373 (k=4).

select(t -> convert(convert(t,base,10),`+`) < 10, [$10 .. 200]); # Robert Israel, Dec 28 2023

Select[Range@ 120, Length@ FixedPointList[Total@ IntegerDigits@ # &, #] == 3 &] (* Michael De Vlieger, May 14 2018 *)

nb(n) = {my(nba = 0); while (n > 9, n = sumdigits(n); nba++); nba;}
isok(n) = nb(n) == 1; \\ Michel Marcus, May 13 2018

A031286(a(n)) = 1.

A304367 Numbers n with additive persistence = 2.

Original entry on oeis.org

19, 28, 29, 37, 38, 39, 46, 47, 48, 49, 55, 56, 57, 58, 59, 64, 65, 66, 67, 68, 69, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 109, 118, 119, 127, 128, 129, 136, 137, 138, 139, 145, 146, 147, 148, 149, 154

Offset: 1

Jaroslav Krizek, May 11 2018

First deviation from A298638 is at a(161); a(161) = 299, A298638(161) = 307.

			Repeatedly taking the sum of digits starting with 19 gives 10 and then 1. There are two steps, so the additive persistence is 2, and 19 is a member. - _Michael B. Porter_, May 16 2018

Eric Weisstein's World of Mathematics, Additive Persistence.

Cf. A031286.

Cf. Numbers with additive persistence k: A304366 (k=1), A304368 (k=3), A304373 (k=4).

Select[Range@ 160, Length@ FixedPointList[Total@ IntegerDigits@ # &, #] == 4 &] (* Michael De Vlieger, May 14 2018 *)

nb(n) = {my(nba = 0); while (n > 9, n = sumdigits(n); nba++); nba;}
isok(n) = nb(n) == 2; \\ Michel Marcus, May 13 2018

A031286(a(n)) = 2.

A304368 Numbers n with additive persistence = 3.

Original entry on oeis.org

199, 289, 298, 379, 388, 397, 469, 478, 487, 496, 559, 568, 577, 586, 595, 649, 658, 667, 676, 685, 694, 739, 748, 757, 766, 775, 784, 793, 829, 838, 847, 856, 865, 874, 883, 892, 919, 928, 937, 946, 955, 964, 973, 982, 991, 1099, 1189, 1198, 1279, 1288, 1297

Offset: 1

Jaroslav Krizek, May 11 2018

First deviation from A166459 is at a(101); a(101) = 1999, A166459(101) = 2089.

			Repeatedly taking the sum of digits starting with 199 gives 19, 10, and then 1. There are three steps, so the additive persistence is 3, and 199 is a member. - _Michael B. Porter_, May 16 2018

Eric Weisstein's World of Mathematics, Additive Persistence.

Cf. A031286.

Cf. Numbers with additive persistence k: A304366 (k=1), A304367 (k=2), A304373 (k=4).

Select[Range@ 1300, Length@ FixedPointList[Total@ IntegerDigits@ # &, #] == 5 &] (* Michael De Vlieger, May 14 2018 *)

nb(n) = {my(nba = 0); while (n > 9, n = sumdigits(n); nba++); nba;}
isok(n) = nb(n) == 3; \\ Michel Marcus, May 13 2018

A031286(a(n)) = 3.

cp's OEIS Frontend

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

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A304366 Numbers with additive persistence = 1.

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A304367 Numbers n with additive persistence = 2.

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A304368 Numbers n with additive persistence = 3.

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