0,1

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.

M. Khoshnevisan, manuscript, March 2003.

M. Khoshnevisan, "Generalized Smarandache Palindrome", Mathematics Magazine, Aurora, Canada, 10/2003.

M. Khoshnevisan, Proposed problem 1062, The PME Journal, USA, Vol. 11, No. 9, p. 501, 2003.

Charles Ashbacher, Lori Neirynck, The Density of Generalized Smarandache Palindromes.

G. Gregory, Generalized Smarandache Palindromes.

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.

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.

Adjacent sequences: A082458 A082459 A082460 this_sequence A082462 A082463 A082464

Sequence in context: A133584 A139059 A123156 this_sequence A071998 A043640 A076940

nonn

K. Ramsharan (ramsharan(AT)indiainfo.com), Apr 26 2003