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 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. 4
 18, 81, 198, 576, 675, 819, 891, 918, 1131, 1304, 1311, 1818, 1998, 2262, 2622, 3393, 3933, 4031, 4154, 4514, 4636, 6364, 8181, 8749, 8991, 9478, 12441, 14269, 14344, 14421, 15167, 15602, 16237, 18018, 18449, 18977, 19998, 20651, 23843, 24882, 26677, 26892, 27225 (list; graph; refs; listen; history; text; internal format)
 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 Daniel Tsai, A recurring pattern in natural numbers of a certain property, arXiv:2010.03151 [math.NT], 2020. EXAMPLE 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. 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 Cf. A004086 (read n backwards), A002113, A029742 (non-palindromes), A338038, A338166. Sequence in context: A039408 A043231 A044011 * A085504 A214531 A271502 Adjacent sequences:  A338036 A338037 A338038 * A338040 A338041 A338042 KEYWORD nonn,base AUTHOR Michel Marcus, Oct 08 2020 STATUS approved

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Last modified July 24 05:05 EDT 2021. Contains 346273 sequences. (Running on oeis4.)