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A244552
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Palindromes j such that j +- the sum of digits of j are both palindromes.
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0
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1, 2, 3, 4, 181, 262, 343, 424, 767, 848, 929
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OFFSET
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1,2
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COMMENTS
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This sequence is finite. For any j, the largest digit sum possible is 9*A055642(j). Let j contain M digits. In order for j to be a palindrome and j + digsum(j) to be a palindrome, if digsum(j) affects the k-th digit of j, it must also affect the (M-k)-th digit of j for k = 1, 2, ..., M-1. For example, if j is 5 digits long and digsum(j) is 2 digits long, then j + digsum(j) and j - digsum(j) must affect the digit in the thousands place of j in order to produce a palindrome. This means that digsum(j) must be at least 4 digits long. Generally, we can say that A055642(digsum(j)) >= A055642(j) - A055642(digsum(j)) + 1 and thus, A055642(digsum(j)) >= (1/2)*(A055642(j)+1). This, however, fails when j > 3 digits. When j is 4 digits, the maximum that the digit sum could be is 36, a 2-digit number. Since it is a 2-digit number, it must affect the digit in the hundreds place of j when it is added to or subtracted from j. However, this is not possible since digsum(j) is only 2 digits long. For j > 4 digits the argument is similar.
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LINKS
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EXAMPLE
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181 is a palindrome, 181 + (1+8+1) = 191 is a palindrome, and 181 - (1+8+1) = 171 is a palindrome. Thus 181 is a member of this sequence.
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MATHEMATICA
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palQ[n_]:=Module[{t=Total[IntegerDigits[n]]}, AllTrue[{n, n+t, n-t}, PalindromeQ]]; Select[Range[1000], palQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 23 2018 *)
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PROG
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(PARI) rev(n)={r=""; for(i=1, #digits(n), r=concat(Str(digits(n)[i]), r)); return(eval(r))}
for(n=1, 10^7, if(rev(n)==n, dig=digits(n); s=sum(k=1, #dig, dig[k]); sm=n-s; la=n+s; if(rev(sm)==sm&&rev(la)==la, print1(n, ", "))))
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CROSSREFS
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KEYWORD
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nonn,base,fini,full
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AUTHOR
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STATUS
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approved
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