A229621 -id:A229621 - cp's OEIS frontend

A244551 Numbers k such that k +- (sum of digits of k) are both palindromes.

1, 2, 3, 4, 10, 100, 105, 181, 262, 267, 343, 348, 424, 429, 681, 762, 767, 843, 848, 924, 929, 1000, 10000, 100000, 1000000, 10000000, 63999991, 72999982, 81999973, 90999964, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 10000000000000000

Offset: 1

Derek Orr, Jun 29 2014

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.
Intersection of A229545 and A229621. - Michel Marcus, Jun 30 2014
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

			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.

Derek Orr, More terms in the sequence

Cf. A229545, A229621, A007953.

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,", ")))

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

a(27)-a(32) from Michel Marcus, Jun 30 2014

a(33)-a(39) from Chai Wah Wu, Aug 24 2015

A244573 Numbers n such that 10n + d - digsum(10n + d) is a palindrome for any d in {0,1,2,3,4,5,6,7,8,9}.

Original entry on oeis.org

1, 10, 18, 26, 34, 42, 68, 76, 84, 92, 100, 279, 368, 457, 546, 635, 724, 813, 902, 1000, 1071, 1152, 1233, 1314, 1486, 1567, 1648, 1729, 1981, 2051, 2132, 2213, 2385, 2466, 2547, 2628, 2709, 2880, 2961, 3031, 3112, 3284, 3365, 3446, 3527, 3608, 3699, 3860, 3941, 4011, 4183, 4264

Offset: 1

Derek Orr, Jun 30 2014

			180 - (1+8+0) = 171, a palindrome. By adding {1,2,3,4,5,6,7,8,9} to 180 and subtracting that number's digsum, it will still be 171, a palindrome. Since 180 = 18*10, 18 is a member of this sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A229621, A066568.

palQ[n_]:=AnyTrue[Table[10n+d-Total[IntegerDigits[10n+d]],{d,0,9}],PalindromeQ]; Select[Range[4300],palQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 13 2021 *)

rev(n)={r="";for(i=1,#digits(n),r=concat(Str(digits(n)[i]),r));return(eval(r))}
for(n=1,10^4,s=sum(i=1,#digits(10*n),digits(10*n)[i]);if(rev(10*n-s)==10*n-s,print1(n,", ")))

A229623 Palindromes m such that m - sum_of_digits(m) is also a palindrome.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 101, 181, 262, 343, 424, 686, 767, 848, 929, 1001, 10001, 100001, 1000001, 10000001, 100000001, 1000000001, 10000000001, 100000000001, 1000000000001

Offset: 1

Derek Orr, Sep 26 2013

It is conjectured that a(n) = 10^(n-18) + 1 for all n > 20. - Derek Orr, Apr 05 2015

Palindromes in the sequence A229621.

			767 - (7+6+7) = 747 (another palindrome). So, 767 is in this sequence.

Cf. A229621, A002113.

palQ[n_]:=Module[{idn=IntegerDigits[n], idn2}, idn2=IntegerDigits[n - Total[idn]]; idn==Reverse[idn]&&idn2==Reverse[idn2]]; Select[Range[0, 2 10^6], palQ] (* Vincenzo Librandi, Apr 06 2015 *)

b(n)={my(d, i, r); r=vector(#digits(n-10^(#digits(n\11)))+#digits(n\11)); n=n-10^(#digits(n\11)); d=digits(n); for(i=1, #d, r[i]=d[i]; r[#r+1-i]=d[i]); sum(i=1, #r, 10^(#r-i)*r[i])} \\ Code from David A. Corneth in A002113, Jun 06 2014
pal(n)=my(d=digits(n));Vecrev(d)==d
for(n=1,10^7,my(m=b(n), s=sumdigits(m));if(pal(m-s),print1(m,", "))) \\ Derek Orr, Apr 05 2015

def pal(n):
  r = ''
  for i in str(n):
    r = i + r
  return r == str(n)
def DS(n):
  s = 0
  for i in str(n):
    s += int(i)
  return s
{print(n, end=', ') for n in range(10**6) if pal(n) and pal(n-DS(n))}
## Simplified by Derek Orr, Apr 05 2015

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A244551 Numbers k such that k +- (sum of digits of k) are both palindromes.

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A244573 Numbers n such that 10n + d - digsum(10n + d) is a palindrome for any d in {0,1,2,3,4,5,6,7,8,9}.

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A229623 Palindromes m such that m - sum_of_digits(m) is also a palindrome.

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cp's OEIS Frontend

A244551 Numbers k such that k +- (sum of digits of k) are both palindromes.

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Extensions

A244573 Numbers n such that 10*n + d - digsum(10*n + d) is a palindrome for any d in {0,1,2,3,4,5,6,7,8,9}.

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A229623 Palindromes m such that m - sum_of_digits(m) is also a palindrome.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python

A244573 Numbers n such that 10n + d - digsum(10n + d) is a palindrome for any d in {0,1,2,3,4,5,6,7,8,9}.