Generalizedsmarandchepalindrome Sequence

Generalized Smarandache Palindromes which are not palindromes: 1010,1011,1021,1031,1041,1051,1061,1071,1081,1091,1101,1121,1131,1141,1151,1161,1171,1181,1191,1 201,1211,1212, 1231,1241,1251,… . There is no GSP of less than four digits which is not a palindrome. GSP of four digits are concatenated numbers of the forms: A(BC)A, where A is in {1,2,3,...,9} and both B,C in {0,1,2,...,9} with B different from C; or (AB)(AB), where A is in {1,2,3,...,9}and B in {0,1,2,...,9}, with A different from B. Charles Ashbacher, Lori Neirynck, The Density of Generalized Smarandache Palindromes, http://www.gallup.unm.edu/~smarandache/GeneralizedPalindromes.htm. G. Gregory, Generalized Smarandache Palindromes, http://www.gallup.unm.edu/~smarandache/GSP.htm. M. Khoshnevisan, Proposed Problem, manuscript sent K. Ramsharan, March 2003. Charles Ashbacher, Lori Neirynck, The Density of Generalized Smarandache Palindromes. G. Gregory, Generalized Smarandache Palindromes. A Generalized Smarandache Palindrome (GSP) is a concatenated number of the form: a1a2...anan...a2a1 or a1a2...an-1anan-1...a2a1, where all a1, a2, ..., an are positive integers of various number of digits. For example: a) 1235656312 is a Generalized Smarandache Palindrome (GSP) because we can group it as (12)(3)(56)(56)(3)(12), i.e. ABCCBA. b) Obviously, any palindromic number is a GSP number as well. c) Of course, any integer can be consider a GSP because we may consider the entire number as equal to a1, which is smarandachely palindromic; say N=176293 is GSP because we may take a1 = 176293 and thus N=a1. But one disregards this trivial case. A palindromic number of four digits has the concatenated form: abba, where a is in {1, 2, …, 9} and b is in {0, 1, 2, …, 9}. There are 9x10=90 palindromic numbers of four digits. For example, 1551, or 2002 are palindromic (and, of course, GSP too); yet 3753 is not palindromic but it is a GSP for 3753=3(75)3, i.e. of the form ABA; similarly 4646, for it can be organized as (46)(46), i.e. of the form CC. Therefore, a SGP, different from a palindromic number, should have the concatenated forms: 1) ABA, where A is in {1, 2, …, 9} and B is in {00, 01, 02, 03, …, 99}-{00, 11, 22, 33, …, 99}; 2) or CC, where C is in {10, 11, 12, …, 99}-{11, 22, 33, …, 99}. In the first case, one has 9x(100-10)=9x90=810. In the second case, one has 90-9=81. Total: 810+81=891 GSP numbers of four digits which are not palindromic. Submitted to njas ed K.Ramsharan, ep 26.04.03 Hi Mr. Rivera, I read your puzzle 56 about Honaker's constant the series 1/2 +...+1/palindrome = constant. Searching the Encyclopedia of Integer Sequences for palindromes I got the GSP. I wonder if a similar series (see the sequence below from the EIS) 1/1010 + 1/1011 + 1/1021 + ... is convergent or not?

I'd conjecture not, because the generalized smarandache palindromes are much more dense that than the normal palindromes. What do you think? K. Reddy, submitted to Crivera, primepuzzles, on 04.06.2003