cp's OEIS Frontend

A338039 Numbers m such that A338038(m) = A338038(A004086(m)) where A004086(i) is i read backwards and A338038(i) is the sum of the primes and exponents in the prime factorization of i ignoring 1-exponents; palindromes and multiples of 10 are excluded.

%I A338039 #48 May 10 2024 02:42:20
%S A338039 18,81,198,576,675,819,891,918,1131,1304,1311,1818,1998,2262,2622,
%T A338039 3393,3933,4031,4154,4514,4636,6364,8181,8749,8991,9478,12441,14269,
%U A338039 14344,14421,15167,15602,16237,18018,18449,18977,19998,20651,23843,24882,26677,26892,27225
%N A338039 Numbers m such that A338038(m) = A338038(A004086(m)) where A004086(i) is i read backwards and A338038(i) is the sum of the primes and exponents in the prime factorization of i ignoring 1-exponents; palindromes and multiples of 10 are excluded.
%C A338039 Palindromes (A002113) are excluded from the sequence because they obviously satisfy the condition.
%C A338039 Sequence is infinite since it includes 18, 1818, 181818, .... See link.
%C A338039 There are many cases of terms that are the repeated concatenation of integers like: 1818, 8181, 181818, ... , but also 131313131313131313131313131313 and more. See A338166.
%C A338039 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
%H A338039 Michel Marcus, <a href="/A338039/b338039.txt">Table of n, a(n) for n = 1..2998</a>
%H A338039 Chris Bispels, Muhammet Boran, Steven J. Miller, Eliel Sosis, and Daniel Tsai, <a href="https://arxiv.org/abs/2405.05267">v-Palindromes: An Analogy to the Palindromes</a>, arXiv:2405.05267 [math.HO], 2024.
%H A338039 Muhammet Boran, Garam Choi, Steven J. Miller, Jesse Purice, and Daniel Tsai, <a href="https://arxiv.org/abs/2307.00770">A characterization of prime v-palindromes</a>, arXiv:2307.00770 [math.NT], 2023.
%H A338039 Daniel Tsai, <a href="https://arxiv.org/abs/2010.03151">A recurring pattern in natural numbers of a certain property</a>, arXiv:2010.03151 [math.NT], 2020.
%H A338039 Daniel Tsai, <a href="http://math.colgate.edu/~integers/v32/v32.mail.html">A recurring pattern in natural numbers of a certain property</a>, Integers (2021) Vol. 21, Article #A32.
%H A338039 Daniel Tsai, <a href="https://arxiv.org/abs/2111.10211">v-palindromes: an analogy to the palindromes</a>, arXiv:2111.10211 [math.HO], 2021.
%e A338039 For m = 18 = 2*3^2, A338038(18) = 2 + (3+2) = 7 and for m = 81 = 3^4, A338038(81) = 7, so 18 and 81 are terms.
%p A338039 rev:= proc(n) local L,i;
%p A338039   L:= convert(n,base,10);
%p A338039   add(L[-i]*10^(i-1),i=1..nops(L))
%p A338039 end proc:
%p A338039 g:= proc(n) local t;
%p A338039   add(t[1]+t[2],t=subs(1=0,ifactors(n)[2]))
%p A338039 end proc:
%p A338039 filter:= proc(n) local r;
%p A338039   if n mod 10 = 0 then return false fi;
%p A338039   r:= rev(n);
%p A338039   r <> n and g(r)=g(n)
%p A338039 end proc:
%p A338039 select(filter, [$1..30000]); # _Robert Israel_, Oct 13 2020
%t A338039 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 *)
%o A338039 (PARI) f(n) = my(f=factor(n)); vecsum(f[,1]) + sum(k=1, #f~, if (f[k,2]!=1, f[k,2])); \\ A338038
%o A338039 isok(m) = my(r=fromdigits(Vecrev(digits(m)))); (m % 10) && (m != r) && (f(r) == f(m));
%Y A338039 Cf. A004086 (read n backwards), A002113, A029742 (non-palindromes), A338038, A338166.
%K A338039 nonn,base
%O A338039 1,1
%A A338039 _Michel Marcus_, Oct 08 2020