A082935 Smallest palindrome beginning with n and a digit sum of n at some stage.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10801, 11711, 12621, 13531, 14441, 15351, 16261, 17171, 1881, 1949999999999999999999491, 208802, 2139312, 227722, 2329232, 246642, 2519152, 265562, 27972, 28882, 29792, 3088803, 3179713, 3278723, 3369633, 3468643

Offset: 1

Author

Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 16 2003

Keywords

• base
• nonn

Comments

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.

Checked the conjecture above to n=100. - Robert G. Wilson v

Examples

			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.
		

Crossrefs

Cf. A082217.

Programs

Mathematica

(*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;
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}]

Extensions

Edited, corrected and extended by Robert G. Wilson v, Jun 27 2003