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%I #10 Mar 21 2015 01:36:02
%S 1,2,3,4,5,6,7,8,9,10801,11711,12621,13531,14441,15351,16261,17171,
%T 1881,1949999999999999999999491,208802,2139312,227722,2329232,246642,
%U 2519152,265562,27972,28882,29792,3088803,3179713,3278723,3369633,3468643
%N Smallest palindrome beginning with n and a digit sum of n at some stage.
%C In most cases (perhaps in all other) except for n = 19 the digit sum in the first round itself is n. In case of 19 the first round of digit sum is 199 and the second round digit sum is 19.
%C Checked the conjecture above to n=100. - _Robert G. Wilson v_
%e a(19)=1949999999999999999999491. The smallest such number is 194 followed by 19 nines followed by 491. The first digit sum would be 199 and the next sum is 19.
%t (*This code works for all numbers up to 100 except 19*) NextPalindrome[n_] := Block[{l = Floor[Log[10, n] + 1], idn = IntegerDigits[n]}, If[ Union[idn] == {9}, Return[n + 2], If[l < 2, Return[n + 1], If[ FromDigits[ Reverse[ Take[idn, Ceiling[l/2]]]] FromDigits[ Take[idn, -Ceiling[l/2]]], FromDigits[ Join[ Take[idn, Ceiling[l/2]], Reverse[ Take[idn, Floor[l/2]]]]], idfhn = FromDigits[ Take[idn, Ceiling[l/2]]] + 1;
%t idp = FromDigits[ Join[ IntegerDigits[idfhn], Drop[ Reverse[ IntegerDigits[ idfhn]], Mod[l, 2]]]]]]]]; f[n_] := Block[{k = 1, dn = IntegerDigits[n]}, sdn = 2*Plus @@ dn; If[sdn == 2n, n, If[sdn == n, FromDigits[ Join[dn, Reverse[dn]]], If[sdn > n, 0, k = 10^Floor[(n - sdn)/9] - 1;; While[Plus @@ IntegerDigits[k] + sdn != n, k = NextPalindrome[k]]; FromDigits[ Join[dn, IntegerDigits[k], Reverse[dn]]]]]]]; Table[ f[n], {n, 1, 35}]
%Y Cf. A082217.
%K base,nonn
%O 1,2
%A Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 16 2003
%E Edited, corrected and extended by _Robert G. Wilson v_, Jun 27 2003
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