OFFSET
1,2
COMMENTS
For k = 0, 1, 2, ..., 10^k is a member of this sequence, so A011557 is a subsequence.
Also, floor(a(n)/10) must be in A244573.
The corresponding digit sums are: 1, 2, 3, 4, 1, 1, 6, 10, 10, 15, 10, 15, 10, 15, 15, 15, 20, 15, 20, 15, 20, 1, 1, 1, 1, 1, 55, 55, 55, 55, 1, 1, .... - Derek Orr, Jun 30 2014
When the digits sums are 1, the terms are a power of 10 terms, with corresponding palindromes being 2 apart (consecutive palindromes). The next instances of consecutive palindromes appear when digit sums are 55, so that the palindromes are 110 apart (see sequence A104459 that lists the possible differences between adjacent palindromes). - Michel Marcus, Jul 01 2014
There are infinitely many terms in this sequence that are not of the form 10^k for some k. Take the number 180 {59 9's} 631. This is a 65-digit number (180999...999631). This number is a member of this sequence. From this number, we can generate 34 other numbers. Keeping the 59 9's there, to preserve its properties, subtract 9 from 180 and add 90 to 631. Now we have 171 {59 9's} 721. This is also a member. The 59 9's are only to make the digit sum = 550. Thus, the two palindromes are 1100 apart, they are consecutive. If we keep doing this arithmetic (subtract 9 from the first number and add 90 to the second), we get 162 {59 9's} 811, 153 {59 9's} 901, 144 {58 9's} 991. The last number only has 58 9's in order to make sure the digit sum stays at 550. Other numbers and patterns to them have been listed in an a-file. Similarly, patterns like this will appear when considering digit sums of 5500 and larger. - Derek Orr, Jul 01 2014
LINKS
EXAMPLE
267 - (2+6+7) = 252 is a palindrome and 267 + (2+6+7) = 282 is also a palindrome. Thus 252 is a member of this sequence.
PROG
(PARI) rev(n)={r=""; for(i=1, #digits(n), r=concat(Str(digits(n)[i]), r)); return(eval(r))}
for(n=1, 10^7, 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, ", ")))
(Python)
def palgen(l, b=10): # generator of palindromes in base b of length <= 2*l
if l > 0:
yield 0
for x in range(1, l+1):
n = b**(x-1)
n2 = n*b
for y in range(n, n2):
k, m = y//b, 0
while k >= b:
k, r = divmod(k, b)
m = b*m + r
yield y*n + b*m + k
for y in range(n, n2):
k, m = y, 0
while k >= b:
k, r = divmod(k, b)
m = b*m + r
yield y*n2 + b*m + k
A244551_list = []
for p in palgen(9):
l = len(str(p))
for i in range(1, l*9+1):
n = p-i
if n > 0:
if sum((int(d) for d in str(n))) == i:
s = str(n-i)
if s == s[::-1]:
A244551_list.append(n) # Chai Wah Wu, Aug 24 2015
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Derek Orr, Jun 29 2014
EXTENSIONS
a(27)-a(32) from Michel Marcus, Jun 30 2014
a(33)-a(39) from Chai Wah Wu, Aug 24 2015
STATUS
approved