A083829 -id:A083829 - cp's OEIS frontend

A083830 Palindromes of the form 3n + 1 where n is also a palindrome: palindromes arising in A083829.

4, 7, 22, 232, 424, 454, 484, 727, 757, 787, 2332, 23332, 42124, 42424, 42724, 45154, 45454, 45754, 48184, 48484, 48784, 72127, 72427, 72727, 75157, 75457, 75757, 78187, 78487, 78787, 233332, 2333332, 4212124, 4215124, 4218124, 4242424

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 09 2003

Cf. A083829.

Select[3#+1&/@Select[Range[15*10^5],PalindromeQ],PalindromeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 20 2020 *)

Corrected and extended by Ray Chandler, May 21 2003

A083455 Duplicate of A083829.

Original entry on oeis.org

1, 2, 7, 77, 141, 151, 161, 242, 252, 262, 777, 7777, 14041, 14141, 14241, 15051

Offset: 1

dead

A083832 Palindromes of the form 4n + 1 where n is also a palindrome. Palindromes arising in A083831.

Original entry on oeis.org

5, 9, 33, 353, 525, 565, 929, 969, 3553, 35553, 52125, 52525, 52925, 56165, 56565, 56965, 92129, 92529, 92929, 96169, 96569, 96969, 355553, 3555553, 5212125, 5216125, 5252525, 5256525, 5292925, 5296925, 5612165, 5616165, 5652565

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 09 2003

Cf. A083829, A083830, A083831.

4#+1&/@Select[Range[1500000],AllTrue[{#,4#+1},PalindromeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 06 2017 *)

Corrected and extended by Ray Chandler, May 21 2003

A083834 Palindromes of the form 5p + 1 where p is also a palindrome. Palindromes arising in A083833.

Original entry on oeis.org

6, 11, 111, 606, 656, 1111, 11111, 60106, 60606, 65156, 65656, 111111, 1111111, 6010106, 6015106, 6060606, 6065606, 6510156, 6515156, 6560656, 6565656, 11111111, 111111111, 601010106, 601060106, 601515106, 601565106, 606010606

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 09 2003

From Michael S. Branicky, Jun 13 2021: (Start)
All terms start and end with the digit 6, except those consisting of d+1 1's, which arise from palindromes of d 2's.
If a(n) > 6 starts with a 6, then its second and second-to-last digits are either both 0 or both 5; and if it has 5 or more digits, its third and third-to-last digits are either both 1 or both 6; thus, terms in the latter case only start with 111, 601, 606, 651, 656.
Using ^ to denote repeated concatenation, contains terms of the forms 1^(d+1), arising from palindromes of the form 2^d; (60)^d 6, arising from (12)^d 1; and (65)^d 6, arising from (13)^d 1; among other patterns.
(Conjectures)
All terms contain only the digits {0, 1, 5, 6}.
For d odd, the only term with d digits is 1^d; equivalently, all terms starting with 6 have odd length. (End)

Michael S. Branicky, Table of n, a(n) for n = 1..8217 (all terms where p has <= 26 digits)

Cf. A083829, A083830, A083832, A083833.

Select[5Select[Range@100000,PalindromeQ]+1,PalindromeQ] (* Giorgos Kalogeropoulos, Jun 11 2021 *)

is(n) = my(x=(n-1)/5, dn=digits(n), dx); if(x!=ceil(x), return(0)); dx=digits(x); dn==Vecrev(dn) && dx==Vecrev(dx) && n>1 \\ Felix Fröhlich, Jun 11 2021

from itertools import product
def ispal(n): s = str(n); return s == s[::-1]
def pals(d, base=10): # all positive d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d > 1 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]:
            t = int(left + mid + right)
            if t > 0: yield t
def ok(pal): return ispal(5*pal+1)
print([5*p+1 for d in range(1, 10) for p in pals(d) if ok(p)]) # Michael S. Branicky, Jun 11 2021

Corrected and extended by Ray Chandler, May 21 2003

A083835 Palindromes n such that 6n + 1 is also a palindrome.

Original entry on oeis.org

1, 9, 99, 121, 131, 999, 9999, 12021, 12121, 13031, 13131, 99999, 999999, 1202021, 1203021, 1212121, 1213121, 1302031, 1303031, 1312131, 1313131, 9999999, 99999999, 120202021, 120212021, 120303021, 120313021, 121202121, 121212121

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 09 2003

Cf. A083829, A083830, A083832, A083834, A083836.

Corrected and extended by Ray Chandler, May 21 2003

A083831 Palindromes n such that 4n + 1 is also a palindrome.

Original entry on oeis.org

1, 2, 8, 88, 131, 141, 232, 242, 888, 8888, 13031, 13131, 13231, 14041, 14141, 14241, 23032, 23132, 23232, 24042, 24142, 24242, 88888, 888888, 1303031, 1304031, 1313131, 1314131, 1323231, 1324231, 1403041, 1404041, 1413141, 1414141

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 09 2003

Among infinite subsequences are the repdigits 8...8 = 8*(10^k-1)/9. It appears that the only terms with an even number of digits are these for even k. - Robert Israel, Apr 04 2018

			13231 and 52925 are palindromes and 4*13231+1=52925, therefore 13231 is a term.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A083829, A083830, A083832.

N:= 100: # to get the first N terms
fe:= proc(x,d) local L;
   L:= convert(x,base,10);
   add(L[j]*(10^(d-j)+10^(d+j-1)),j=1..d)
end proc:
fo:= proc(x,d) local L;
  L:= convert(x,base,10);
  add(L[j]*(10^(d-j)+10^(d+j-2)),j=2..d) + L[1]*10^(d-1);
end proc:
ispali:= proc(n) local L;
   L:= convert(n,base,10);
   L = ListTools:-Reverse(L)
end proc:
count:= 0: Res:= NULL:
for d from 1 while count < N do
  for x from 10^(d-1) to 10^d-1 while count < N do
    y:= fo(x,d);
  if ispali(4*y+1) then
     count:= count+1; Res:= Res, y;
  fi
od:
for x from 10^(d-1) to 10^d-1 while count < N do
    y:= fe(x,d);
  if ispali(4*y+1) then
     count:= count+1; Res:= Res, y;
  fi
od:
od:
Res; # Robert Israel, Apr 04 2018

Select[Range[15*10^5],AllTrue[{#,4#+1},PalindromeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 08 2018 *)

isok(n) = my(dn = digits(n), dm = digits(4*n+1)); (Vecrev(dn) == dn) && (Vecrev(dm) == dm); \\ Michel Marcus, Apr 04 2018

Corrected and extended by Reinhard Zumkeller and Ray Chandler, May 18 2003

A083833 Palindromes p such that 5p + 1 is also a palindrome.

Original entry on oeis.org

1, 2, 22, 121, 131, 222, 2222, 12021, 12121, 13031, 13131, 22222, 222222, 1202021, 1203021, 1212121, 1213121, 1302031, 1303031, 1312131, 1313131, 2222222, 22222222, 120202021, 120212021, 120303021, 120313021, 121202121, 121212121

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 09 2003

From Michael S. Branicky, Jun 13 2021: (Start)
All terms start and end with the digit 1, except those consisting of d 2's, which lead to palindromes of d+1 1's.
If a(n) > 1 starts with 1, then its second and second-to-last digits are either both 2 or both 3; and if it has 5 or more digits, its third and third-to-last digits are either both 0 or both 1; thus, terms in the latter case only start with 120, 121, 130, 131, and 222.
Using ^ to denote repeated concatenation, contains terms of the forms 2^d, leading to palindromes of the form 1^(d+1); (12)^d 1, leading to (60)^d 6; and (13)^d 1, leading to (65)^d 6; among other patterns.
(Conjectures)
All terms contain only the digits {0, 1, 2, 3}.
For d even, the only term with d digits is 2^d; equivalently, all terms starting with 1 have odd length. (End)

Michael S. Branicky, Table of n, a(n) for n = 1..8217 (all terms where p has <= 26 digits)

Cf. A083829, A083830, A083832, A083834.

Select[Range[23*10^5],AllTrue[{#,5#+1},PalindromeQ]&] (* The program generates the first 22 terms of the sequence. *) (* Harvey P. Dale, Dec 17 2024 *)

from itertools import product
def ispal(n): s = str(n); return s == s[::-1]
def pals(d, base=10): # all positive d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d > 1 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]:
            t = int(left + mid + right)
            if t > 0: yield t
def ok(pal): return ispal(5*pal+1)
print([p for d in range(1, 10) for p in pals(d) if ok(p)]) # Michael S. Branicky, Jun 11 2021

Corrected and extended by Ray Chandler, May 21 2003

A083836 Palindromes of the form 6n + 1 where n is also a palindrome. palindromes arising in A083835.

Original entry on oeis.org

7, 55, 595, 727, 787, 5995, 59995, 72127, 72727, 78187, 78787, 599995, 5999995, 7212127, 7218127, 7272727, 7278727, 7812187, 7818187, 7872787, 7878787, 59999995, 599999995, 721212127, 721272127, 721818127, 721878127, 727212727

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 09 2003

Cf. A083829, A083830, A083832, A083834, A083835.

6#+1&/@Select[Range[12121*10^4],AllTrue[{#,6#+1},PalindromeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 15 2018 *)

More terms from Ray Chandler, May 21 2003

A083837 Smallest palindrome n such that kn + 1 is a palindrome, or 0 if no such number exists.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 55, 0, 5, 7, 8, 2, 0, 3, 4, 5, 1, 5, 0, 5, 4, 5, 2, 5, 3, 4, 0, 1, 7, 5, 4, 5, 0, 2, 8, 3, 0, 0, 1, 8, 4, 7, 9, 0, 2, 2, 0, 6, 0, 1, 2, 7, 6, 161, 7, 2, 0, 6, 8, 0, 1, 9, 3, 8, 7, 2, 0, 7, 8, 0, 2, 1, 3, 4, 0, 2, 0, 7, 4, 6, 2, 0, 1, 4, 6, 2, 0, 8, 4, 6, 2, 0, 3, 1, 6, 1, 0, 9, 0, 3

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 09 2003

0 in sequence indicates no palindrome < 10^15 - Ray Chandler, Jul 13 2003

			a(3) = 1: 3*1 + 1 = 4; a(11) = 55, 55*11+1=606.

Cf. A083829, A083830, A083832, A083834, A083835, A083838.

Corrected and extended by Ray Chandler, May 22 2003

A083838 Smallest palindrome of the form nk+1 where k is also a palindrome:, or 0 if no such number exists; palindromes arising in A083837.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 55, 11, 606, 0, 66, 99, 121, 33, 0, 55, 77, 101, 22, 111, 0, 121, 101, 131, 55, 141, 88, 121, 0, 33, 232, 171, 141, 181, 0, 77, 313, 121, 0, 0, 44, 353, 181, 323, 424, 0, 99, 101, 0, 313, 0, 55, 111, 393, 343, 9339, 414, 121, 0, 373, 505, 0, 66, 595

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 09 2003

0 in sequence indicates no palindrome < 10^15 - Ray Chandler, Jul 13 2003

			a(9)=6*9+1=55, a(11)=55*11+1=606.

Cf. A083829, A083830, A083832, A083834, A083835, A083837.

Corrected and extended by Ray Chandler, May 22 2003

cp's OEIS Frontend

A083830 Palindromes of the form 3n + 1 where n is also a palindrome: palindromes arising in A083829.

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A083455 Duplicate of A083829.

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A083832 Palindromes of the form 4n + 1 where n is also a palindrome. Palindromes arising in A083831.

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A083834 Palindromes of the form 5p + 1 where p is also a palindrome. Palindromes arising in A083833.

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A083835 Palindromes n such that 6n + 1 is also a palindrome.

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A083831 Palindromes n such that 4n + 1 is also a palindrome.

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A083833 Palindromes p such that 5p + 1 is also a palindrome.

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A083836 Palindromes of the form 6n + 1 where n is also a palindrome. palindromes arising in A083835.

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A083837 Smallest palindrome n such that kn + 1 is a palindrome, or 0 if no such number exists.

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A083838 Smallest palindrome of the form nk+1 where k is also a palindrome:, or 0 if no such number exists; palindromes arising in A083837.