OFFSET
1,1
COMMENTS
Palindromes (A002113) are excluded from the sequence because they obviously satisfy the condition.
Sequence is infinite since it includes 18, 1818, 181818, .... See link.
There are many cases of terms that are the repeated concatenation of integers like: 1818, 8181, 181818, ... , but also 131313131313131313131313131313 and more. See A338166.
If n is in the sequence and has d digits, and gcd(n, x) = gcd(A004086(n), x) where x = (10^((k+1)*d)-1)/(10^d-1), then the concatenation of k copies of n is also in the sequence. - Robert Israel, Oct 13 2020
LINKS
Michel Marcus, Table of n, a(n) for n = 1..2998
Chris Bispels, Muhammet Boran, Steven J. Miller, Eliel Sosis, and Daniel Tsai, v-Palindromes: An Analogy to the Palindromes, arXiv:2405.05267 [math.HO], 2024.
Muhammet Boran, Garam Choi, Steven J. Miller, Jesse Purice, and Daniel Tsai, A characterization of prime v-palindromes, arXiv:2307.00770 [math.NT], 2023.
Daniel Tsai, A recurring pattern in natural numbers of a certain property, arXiv:2010.03151 [math.NT], 2020.
Daniel Tsai, A recurring pattern in natural numbers of a certain property, Integers (2021) Vol. 21, Article #A32.
Daniel Tsai, v-palindromes: an analogy to the palindromes, arXiv:2111.10211 [math.HO], 2021.
EXAMPLE
MAPLE
rev:= proc(n) local L, i;
L:= convert(n, base, 10);
add(L[-i]*10^(i-1), i=1..nops(L))
end proc:
g:= proc(n) local t;
add(t[1]+t[2], t=subs(1=0, ifactors(n)[2]))
end proc:
filter:= proc(n) local r;
if n mod 10 = 0 then return false fi;
r:= rev(n);
r <> n and g(r)=g(n)
end proc:
select(filter, [$1..30000]); # Robert Israel, Oct 13 2020
MATHEMATICA
s[1] = 0; s[n_] := Plus @@ First /@ (f = FactorInteger[n]) + Plus @@ Select[Last /@ f, # > 1 &]; Select[Range[30000], !Divisible[#, 10] && (r = IntegerReverse[#]) != # && s[#] == s[r] &] (* Amiram Eldar, Oct 08 2020 *)
PROG
(PARI) f(n) = my(f=factor(n)); vecsum(f[, 1]) + sum(k=1, #f~, if (f[k, 2]!=1, f[k, 2])); \\ A338038
isok(m) = my(r=fromdigits(Vecrev(digits(m)))); (m % 10) && (m != r) && (f(r) == f(m));
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Michel Marcus, Oct 08 2020
STATUS
approved