A239480 Palindromes such that additive and multiplicative persistences coincide.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 99, 101, 111, 121, 131, 141, 151, 161, 171, 202, 212, 222, 262, 282, 303, 313, 393, 404, 424, 454, 474, 525, 545, 565, 585, 595, 636, 656, 676, 757, 838, 858, 959, 1001, 1111, 1221, 1331, 1441, 1991, 2002, 2112, 2552

Offset

1,3

Comments

Palindromes k for which A031286(k) = A031346(k).

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Additive Persistence

Eric Weisstein's World of Mathematics, Multiplicative Persistence

Index entries for sequences related to palindromes

Formula

A002113 INTERSECT A239427.

Example

99 -> 18 -> 9 has additive persistence 2. 99 -> 81 -> 8 has multiplicative persistence 2. The palindromic number 99 is therefore in the sequence.

Prog

(PARI) for(n=0, 2552, s=Vec(Str(n)); if(s==vecextract(s, "-1..1"), 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, ", "))));
(Python)
from math import prod
from itertools import count, islice, product
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 is_pal(n): return (s:=str(n)) == s[::-1]
def pals(base=10): # all d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for d in count(1):
        for p in product(digits, repeat=d//2):
            if d > 1 and p[0] == "0": continue
            left = "".join(p); right = left[::-1]
            for mid in [[""], digits][d%2]: yield int(left + mid + right)
def ok(n): return is_pal(n) and A031286(n) == A031346(n)
def agen(): yield from filter(ok, pals())
print(list(islice(agen(), 20))) # Michael S. Branicky, Jun 22 2023

Crossrefs

Cf. A002113, A031286, A031346, A239427.

Sequence in context: A082208 A201061 A099165 * A117055 A117056 A082207

Adjacent sequences: A239477 A239478 A239479 * A239481 A239482 A239483

Keyword

nonn,base

Author

Arkadiusz Wesolowski, Mar 20 2014

Status

approved