A064702 -id:A064702 - cp's OEIS frontend

A239427 Numbers such that additive and multiplicative persistences coincide.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 28, 29, 30, 31, 32, 33, 37, 38, 40, 41, 42, 46, 48, 50, 51, 56, 58, 60, 61, 64, 65, 67, 70, 71, 73, 76, 80, 81, 82, 83, 84, 85, 90, 92, 99, 100, 101, 102, 103, 104, 105, 106

Offset: 1

Arkadiusz Wesolowski, Mar 19 2014

Numbers n for which A031286(n) = A031346(n).

			28 -> 10 -> 1 has additive persistence 2. 28 -> 16 -> 6 has multiplicative persistence 2. 28 is therefore in the sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..66 from Arkadiusz Wesolowski)
Eric Weisstein's World of Mathematics, Additive Persistence
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Supersequence of A239480. Cf. A031286, A031346, A064702.

for(n=0, 106, v=n; a=0; while(n>9, a++; n=sumdigits(n)); n=v; m=0; while(n>9, m++; d=digits(n); n=prod(k=1, #d, d[k])); n=v; if(a==m, print1(n, ", ")));

from math import prod
def A031286(n):
    ap = 0
    while n > 9: n, ap = sum(map(int, str(n))), ap+1
    return ap
def A031346(n):
    mp = 0
    while n > 9: n, mp = prod(map(int, str(n))), mp+1
    return mp
def ok(n): return A031286(n) == A031346(n)
print([k for k in range(107) if ok(k)]) # Michael S. Branicky, Sep 17 2022

A272029 Least number k > n such that the multiplicative and additive digital roots of k are both equal to n.

Original entry on oeis.org

1111111111, 137, 1111111113, 22, 1157, 123, 1111111117, 179, 11133

Offset: 1

Altug Alkan, Jul 13 2016

Sequence is trivially finite by definition.

How is the distribution of corresponding digital roots in A064702? This sequence is the result of related observation in limited range.

			a(5) = 1157 because 1+1+5+7 = 14, 1+4 = 5 and 1*1*5*7 = 35, 3*5 = 15, 1*5 = 5. 1157 is the least number that is greater than 5 with this property.

Cf. A010888, A031347, A064702.

A318273 Numbers with digits in nondecreasing order such that additive and multiplicative digital roots coincide.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 123, 137, 139, 168, 179, 188, 233, 267, 299, 346, 389, 899, 1124, 1157, 1347, 1355, 1469, 1779, 1788, 2236, 2346, 2348, 2778, 3335, 3779, 11126, 11133, 11148, 11177, 11222, 11238, 11279, 11339, 11369, 11579, 11666, 11677, 11679, 11699

Offset: 1

David A. Corneth, Aug 23 2018

A299690 can be used to find terms for this sequence below some bound by prepending ones to terms while staying below that bound so the additive and multiplicative root that term matches.

For example, 27 is in A299690 and has multiplicative root 4. 27 has the additive root 9. Prepending 4 ones gives the number 111127 which has multiplicative root 4, the same as 27 has, but it also has an additive root of 4. Furthermore, the digits are in nondecreasing order hence is in this sequence.

Cf. A009994, A010888, A031347, A064702, A277061, A299690.

is(n) = my(cn=n); d=digits(n); if(d!=vecsort(d), return(0)); while(cn>9, d=digits(cn); cn=prod(i=1, #d, d[i])); cn-1 == (n-1)%9 || n == 0

A357131 Numbers m such that A010888(m) = A031347(m) = A031286(m) = A031346(m); only the least of the anagrams are considered.

Original entry on oeis.org

0, 137, 11126, 111134, 111278, 1111223, 11111447, 111112247, 1111122227, 111111111137, 11111111111126, 111111111111134, 1111111111111223, 111111111111111111111111111111111111111111111111111111111111111111111111111111111111278

Offset: 1

Mohammed Yaseen, Sep 14 2022

The infinite sequence {(R_1)37, (R_91)37, (R_991)37, (R_9991)37, (R_99991)37, ...} is a subsequence, where R_k is the repunit of length k. Hence this sequence is infinite.
a(14) <= (R_84)278.
Some additional terms < 10^100: (R_84)278, (R_86)447, (R_86)2247, (R_86)22227, (R_91)37, (R_93)26, (R_94)34, (R_93)278, (R_94)223, (R_95)447, (R_95)2247, (R_95)22227.
If a term k > 0 then k cannot contain a digit 0 as if it does A031347(k) = 0 while A031346(k) = 1, contradicting equality. - David A. Corneth, Sep 15 2022
a(14) > 10^50. - Michael S. Branicky, Sep 16 2022
From Michael S. Branicky, Sep 17 2022: (Start)
(R_{10^k})37 and (R_{2*10^k - 10})37 also form infinite subsequences for k >= 0.
Indeed, terms of the form (R_k)e form infinite subsequences for each e in 26, 34, 37, 223, 278, 447, 2247, 22227 for k such that A007953(k + A007953(e)) = 2.
a(2)-a(14) and all terms of the forms above have 2 = A010888(m) = A031347(m) = A031286(m) = A031346(m).
(R_{t})277 where t+2+7+7 = 4 followed by 55555 9's is a term with 4 = A010888(m) = A031347(m) = A031286(m) = A031346(m).
Likewise, there exists a term of the form (R_{t})5579 with 5 = A010888(m) = A031347(m) = A031286(m) = A031346(m), where t+26 is part of the additive persistence chain ending ..., 5999999, 59, 14, 5. Likewise for 888899 and 6, and so on.
However, there are no terms with A010888(m) = A031347(m) = A031286(m) = A031346(m) = 1 or 3. (End)

			137 is in the sequence as A010888(137) = 137 mod 9 = 2, A031347(137) = 2 via 1*3*7 = 21 -> 2*1 = 2 < 10, A031286(137) = 2 via 1+3+7 = 11 -> 1+1 = 2 so two steps, A031346(137) = 2 via 1*3*7 = 21 -> 2*1 = 2 so two steps. As all these outcomes are 2, 137 is a term. - _David A. Corneth_, Sep 15 2022

Eric Weisstein's World of Mathematics, Additive Persistence
Eric Weisstein's World of Mathematics, Digital Root
Eric Weisstein's World of Mathematics, Multiplicative Digital Root
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Subsequence of A179239.
Cf. A010888, A031347, A031286, A031346.
Cf. A064702, A239427.

a(8)-a(13) from Pontus von Brömssen, Sep 14 2022

a(14) confirmed by Michael S. Branicky, Sep 17 2022

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A239427 Numbers such that additive and multiplicative persistences coincide.

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A272029 Least number k > n such that the multiplicative and additive digital roots of k are both equal to n.

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A318273 Numbers with digits in nondecreasing order such that additive and multiplicative digital roots coincide.

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A357131 Numbers m such that A010888(m) = A031347(m) = A031286(m) = A031346(m); only the least of the anagrams are considered.

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