A227858 -id:A227858 - cp's OEIS frontend

A305719 Numbers whose squares have the same first and last digits.

1, 2, 3, 11, 22, 26, 39, 41, 68, 75, 97, 101, 109, 111, 119, 121, 129, 131, 139, 141, 202, 208, 212, 218, 222, 225, 235, 246, 254, 256, 264, 303, 307, 313, 319, 321, 329, 331, 339, 341, 349, 351, 359, 361, 369, 371, 379, 381, 389, 391, 399, 401, 409, 411, 419, 421, 429, 431, 439, 441, 638

Offset: 1

Neville Holmes, Jun 08 2018

			For k = 11, k^2 = 121;
for k = 26, k^2 = 676.

Cf. A227858, A116499, A116501.

Cf. A123912, A186438.

Select[Range[638], (d = IntegerDigits[#^2]; d[[1]] == d[[-1]]) &] (* Giovanni Resta, Jun 25 2018 *)

for(n=1, 10^3, my(d=digits(n^2)); if( d[1]==d[#d], print1(n,", "))); \\ Joerg Arndt, Jun 10 2018

def ok(n): s = str(n*n); return s[0] == s[-1]
print(list(filter(ok, range(1, 639)))) # Michael S. Branicky, Jul 16 2021

A240601 Recursive palindromes in base 10: palindromes n where each half of the digits of n is also a recursive palindrome.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494, 505, 515, 525, 535, 545, 555, 565, 575, 585, 595, 606, 616, 626, 636, 646, 656, 666, 676, 686, 696, 707, 717, 727, 737, 747, 757, 767, 777, 787, 797, 808, 818, 828, 838, 848, 858, 868, 878, 888, 898, 909, 919, 929, 939, 949, 959, 969, 979, 989, 999, 1111

Offset: 1

Lior Manor, Apr 09 2014

A number n with m digits in base 10 is a member if n is a palindrome, and the first floor(m/2) digits of n is already a previous term of a(n). All repdigit numbers are terms of a(n). Fast generation of new terms with 2m digits can be done by concatenating the previous terms with m digits twice. Fast generation of new terms with 2m+1 digits can be done by concatenating the previous terms with m digits twice with any single digit in the middle. The smallest palindrome which is not a member of a(n) is 1001.

			11011 is in the sequence since it is a palindrome of 5 digits, and the first floor(5/2) digits of it, 11, is also a term. 1001 and 10001 are not in the sequence since 10 is not in the sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Lior Manor)

Cf. A002113, A010785, A240602.

Cf. A227858 (first difference is a(110) = 1111, but A227858(109) = 1001). - Georg Fischer, Oct 23 2018

from itertools import product
def pals(d, base=10): # all d-digit palindromes as strings
  digits = "".join(str(i) for i in range(base))
  for p in product(digits, repeat=d//2):
    if d//2 > 0 and p[0] == "0": continue
    left = "".join(p); right = left[::-1]
    for mid in [[""], digits][d%2]: yield left + mid + right
def auptod(dd):
  for d in range(1, dd+1):
    for p in pals(d//2):
      if d//2 == 0: p = ""
      elif p[0] == "0": continue
      for mid in [[""], "0123456789"][d%2]: yield int(p+mid+p[::-1])
print([rp for rp in auptod(6)]) # Michael S. Branicky, May 22 2021

A077652 Primes whose initial and terminal decimal digits are identical.

Original entry on oeis.org

2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 1021, 1031, 1051, 1061, 1091, 1151, 1171, 1181, 1201, 1231, 1291, 1301, 1321, 1361, 1381, 1451, 1471, 1481, 1511, 1531, 1571, 1601, 1621, 1721, 1741, 1801, 1811

Offset: 1

Labos Elemer, Nov 19 2002

1021 is the smallest of these not to be palindromic. - Jonathan Vos Post, Nov 02 2013

All palindromic primes (A002385) except 11 have an odd number of digits, therefore all terms > 11 with an even number of digits are non-palindromic in this sequence. - M. F. Hasler, Nov 02 2013

Bruno Berselli, Table of n, a(n) for n = 1..1000

Cf. A000030, A000040, A002385, A227858.

[ p: p in PrimesUpTo(2000) | P[#P] eq P[1] where P is Intseq(p) ];  // Bruno Berselli, Jul 26 2011

Do[s1=First[IntegerDigits[Prime[n]]]; s2=Last[IntegerDigits[Prime[n]]]; If[Equal[s1, s2], Print[Prime[n]]], {n, 1, 1000}]
itdQ[n_]:=Module[{idn=IntegerDigits[n]},idn[[1]]==idn[[-1]]]; Select[Prime[ Range[ 500]], itdQ] (* Harvey P. Dale, Apr 12 2013 *)

is(n)=digits(n)[1]==n%10&&isprime(n) \\ M. F. Hasler, Nov 02 2013

A342826 Numbers k such that d(1)^0 + d(2)^1 + ... + d(p)^(p-1) = d(1)^(p-1) + d(2)^(p-2) + ... + d(p)^0, where d(i), i=1..p, are the digits of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464

Offset: 1

Carole Dubois, Mar 23 2021

This sequence starts off like other palindromic sequences such as A178354, A002113, A110751, and A227858 but it differs starting at the 110th term: 109th: 1001, 110th: 1011, 111th: 1101, ..., 119th: 1863, etc.
Differs from A297271 (which e.g. has 1021, 1031, 1041,.. 1091 which are absent here). - R. J. Mathar, Sep 27 2021
Contains the palindromes, and palindromes where pairs of digits have been substituted by d(i)=0, d(p-i)=1 or d(i)=1, d(p-1)=0, and "genuine" numbers like 1863, 2450, 2804, 2814, 3681, 4081, 4182, 103221, 113221, 122301, 122311, 142023,.. - R. J. Mathar, Sep 27 2021

			1863 is in this sequence because 1^0 + 8^1 + 6^2 + 3^3 = 1^3 + 8^2 + 6^1 + 3^0 = 72.

R. J. Mathar, Table of n, a(n) for n = 1..1137 (replacing older b-file which did not contain a(101))
Carole Dubois, Scatterplot of terms of A178354, A342826 vs Sum of powers in definition.

Cf. A002113 (subset), A179309, A110751, A227858.

isA342826 := proc(n)
    local dgs ;
    dgs := convert(n,base,10) ;
    if  add(op(i,dgs)^(i-1),i=1..nops(dgs)) = add(op(-i,dgs)^(i-1),i=1..nops(dgs)) then
        true;
    else
        false;
    end if;
end proc:
A342826 := proc(n)
    option remember ;
    if n =1 then
        1;
    else
        for a from procname(n-1)+ 1 do
            if isA342826(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Sep 27 2021

Select[Range[500],Mod[#,10]!=0&&Total[IntegerDigits[#]^Range[0,IntegerLength[ #]-1]]==Total[IntegerDigits[#]^Range[IntegerLength[#]-1,0,-1]]&] (* Harvey P. Dale, Jan 18 2023 *)

def digpow(s): return sum(int(d)**i for i, d in enumerate(s))
def aupto(limit):
  alst = []
  for k in range(1, limit+1):
    s = str(k)
    if digpow(s) == digpow(s[::-1]): alst.append(k)
  return alst
print(aupto(464)) # Michael S. Branicky, Mar 23 2021

A272623 Fibonacci numbers whose first digit is equal to its last digit.

Original entry on oeis.org

0, 1, 2, 3, 5, 8, 55, 17711, 63245986, 165580141, 498454011879264, 14472334024676221, 1100087778366101931, 298611126818977066918552, 781774079430987230203437, 14028366653498915298923761, 22698374052006863956975682, 36726740705505779255899443, 59425114757512643212875125

Offset: 1

Chai Wah Wu, May 03 2016

Chai Wah Wu, Table of n, a(n) for n = 1..513

Cf. A000045, A227858.

DeleteDuplicates@ Select[Fibonacci@ Range[0, 125], First@ # == Last@ # &@ IntegerDigits@ # &] (* Michael De Vlieger, May 04 2016 *)

isok(n) = my(d = digits(n)); d[1] == d[#d];
lista(nn) = print1(0, ", "); for (n=2, nn, if (isok(f=fibonacci(n)), print1(f, ", "))); \\ Michel Marcus, May 04 2016

A231278 Not necessarily palindromic primes of which initial and terminal digits are identical, as written in base 3.

Original entry on oeis.org

2, 111, 212, 1011, 1101, 1121, 2012, 2122, 10121, 10211, 11001, 11201, 12011, 12121, 12211, 20012, 20102, 20122, 21002, 21022, 22102, 22122, 22212, 101001, 101021, 101111, 102101, 102121, 110021, 110111, 110221, 111121, 111211, 112001, 112201, 120011, 120121

Offset: 1

Jonathan Vos Post, Nov 06 2013

Base-3 analog of what A077652 is for base 10.

			a(3) = 212, which starts and ends with "2", and in base 3 means 2*(3^2) + 1*(3^1) + 2*(3^0) = 18 + 3 + 2 = 23 (base 10), which is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..2500

Cf. A000030, A000040, A007089, A077652, A227858.

FromDigits/@Select[IntegerDigits[#,3]&/@Prime[Range[100]],#[[1]]==#[[-1]]&] (* Harvey P. Dale, Oct 23 2022 *)

More terms from Alois P. Heinz, Nov 07 2013

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A305719 Numbers whose squares have the same first and last digits.

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A240601 Recursive palindromes in base 10: palindromes n where each half of the digits of n is also a recursive palindrome.

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A077652 Primes whose initial and terminal decimal digits are identical.

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A342826 Numbers k such that d(1)^0 + d(2)^1 + ... + d(p)^(p-1) = d(1)^(p-1) + d(2)^(p-2) + ... + d(p)^0, where d(i), i=1..p, are the digits of k.

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A272623 Fibonacci numbers whose first digit is equal to its last digit.

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A231278 Not necessarily palindromic primes of which initial and terminal digits are identical, as written in base 3.

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