Bui Quang Tuan - cp's OEIS frontend

A256495 Palindromes i such that 2*i^2 is a palindrome.

0, 1, 2, 11, 101, 111, 1001, 1111, 10001, 10101, 11011, 100001, 101101, 110011, 1000001, 1001001, 1010101, 1100011, 10000001, 10011001, 10100101, 11000011, 100000001, 100010001, 100101001, 101000101, 110000011, 1000000001, 1000110001, 1001001001, 1010000101

Offset: 1

Bui Quang Tuan, Mar 31 2015

Subsequence of palindromes of A256437.
The sequence contains all positive integers of the form: m*10^(i + NumberOfDigit(m)) + m where i is any nonnegative integer and m is any term of A000533.
Also contains 1 + 10^i and 1 + 10^i + 10^(2*i) for all i >= 1.  Are there any members with more than four 1's, or any members other than 2 with digits other than 0's and 1's? - Robert Israel, Apr 13 2015

			Palindrome 11 is in the sequence because 2*11^2 = 242, a palindrome.

Lars Blomberg, Table of n, a(n) for n = 1..91

Cf. A256437.

dmax:= 11: # to get all terms with at most dmax digits
revdigs:= proc(n)
  local L,i;
  L:= convert(n,base,10);
  add(10^(i-1)*L[-i],i=1..nops(L));
end proc:
filter:= proc(n) local L;
  L:= convert(2*n^2,base,10);
  L = ListTools:-Reverse(L)
end proc:
A:= {}:
for d from 1 to dmax do
  if d::even then
     A:= A union select(filter, {seq(10^(d/2)*x + revdigs(x), x=10^(d/2-1)..10^(d/2)-1)})
  else
     m:= (d-1)/2;
     A:= A union select(filter, {seq(seq(10^(m+1)*x + y*10^m + revdigs(x), y=0..9),x=10^(m-1)..10^m-1)})
  fi
od:
A;  # if using Maple 11 or earlier, uncomment the next line
# sort(convert(A,list)); # Robert Israel, Apr 13 2015

palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d]; Select[
Range@ 10000000, palQ@ # && palQ[#^2 + FromDigits[Reverse@ IntegerDigits@ #]^2] &] (* Michael De Vlieger, Mar 31 2015 *)
Select[Range[0,10101*10^5],AllTrue[{#,2#^2},PalindromeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 26 2020 *)

ispal(n) = my(d = digits(n)); Vecrev(d) == d;
lista(nn) = {for (n=0, nn, if (ispal(n) && ispal(2*n^2), print1(n, ", ")););} \\ Michel Marcus, Mar 31 2015

a(19)-a(22) from Michel Marcus, Mar 31 2015

a(23)-a(31) from Lars Blomberg, Apr 13 2015

A256437 Nonnegative integers i such that i^2 + reverse(i)^2 is a palindrome.

Original entry on oeis.org

0, 1, 2, 10, 11, 12, 20, 21, 30, 100, 101, 102, 110, 111, 120, 200, 201, 210, 220, 300, 310, 1000, 1001, 1002, 1010, 1011, 1012, 1020, 1021, 1022, 1100, 1101, 1102, 1110, 1111, 1120, 1200, 1201, 1210, 1300, 2000, 2001, 2010, 2011, 2100, 2101, 2110, 2200, 2201

Offset: 1

Bui Quang Tuan, Mar 29 2015

This sequence generates A256398.

			12 is in the sequence because 12^2 + 21^2 = 585, a palindrome.

Bui Quang Tuan, Table of n, a(n) for n = 1..554

Cf. A002113, A004086, A256398.

palQ[n_] := Reverse@ IntegerDigits@ n == IntegerDigits@ n; Select[
Range@2210, palQ[#^2 + FromDigits[Reverse[IntegerDigits@ #]]^2] &] (* Michael De Vlieger, Mar 29 2015 *)

rev(n)=r=""; d=digits(n); for(i=1, #d, r=concat(Str(d[i]), r)); eval(r);
ispal(n) = d = digits(n); Vecrev(d) == d;
isok(n) = ispal(n^2+rev(n)^2) \\ Michel Marcus, Apr 01 2015

A256437_list = [i for i in range(10**6) if str(i**2 + int(str(i)[::-1])**2) == str(i**2 + int(str(i)[::-1])**2)[::-1]] # Chai Wah Wu, Apr 09 2015

A256515 Nonpalindromic positive integers k such that the absolute value of k^2 - reverse(k)^2 is a square.

Original entry on oeis.org

56, 65, 5265, 5625, 5656, 6565, 12705, 44370, 50721, 51557, 55517, 56056, 59248, 65065, 71555, 75515, 84295, 139755, 273728, 360145, 481610, 523908, 541063, 557931, 560056, 560439, 565656, 606056, 621770, 650065, 650606, 656565, 697996, 699796, 809325, 827372

Offset: 1

Bui Quang Tuan, Apr 01 2015

			The nonpalindromic number 5265 is a term because abs(5265^2 - 5625^2) = 1980^2.

Michael S. Branicky, Table of n, a(n) for n = 1..425 (all terms < 10^11; terms 1..68 from Bui Quang Tuan)

Cf. A004086 (digit reversal), A202386, A068536.

[n: n in [0..10^6] | Intseq(n) ne Reverse(Intseq(n)) and IsSquare(s) where s is Abs(n^2-Seqint(Reverse(Intseq(n)))^2)]; // Bruno Berselli, Apr 01 2015

Select[Range[200000], ! PalindromeQ@ # && IntegerQ@ Sqrt@ Abs[#^2 - IntegerReverse[#]^2] &] (* Michael De Vlieger, Mar 02 2022 *)

from sympy.ntheory.primetest import is_square
def R(n): return int(str(n)[::-1])
def ok(n): Rn = R(n); return n != Rn and is_square(abs(n**2 - Rn**2))
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Mar 02 2022

A256398 Palindromes of the form i^2 + reverse(i)^2.

Original entry on oeis.org

0, 2, 8, 101, 242, 404, 585, 909, 10001, 12221, 14841, 20402, 24642, 40004, 44244, 48884, 50805, 90009, 96269, 1000001, 1030301, 1080801, 1210121, 1244421, 1298921, 1440441, 1478741, 1690961, 2004002, 2234322, 2468642, 2484842, 4000004, 4050504, 4410144

Offset: 1

Bui Quang Tuan, Mar 28 2015

Is 864666666468 the only term in this sequence that has an even number of digits? - Jon E. Schoenfield, Mar 30 2015

The next terms with an even number of digits are 5807785995877085, 56359464311346495365, and 943614966934439669416349, which are obtained for i = 37939066, 3553782166, 529145826418 (and their reverses). - Giovanni Resta, Aug 22 2025

			Palindrome 585 is in the sequence because 585 = 12^2 + 21^2.
The smallest term that can be obtained in more than one way is 125484521 = 11020^2 + 2011^2 = 11200^2 + 211^2. Are there any terms that can be obtained in more than two ways? - _Jon E. Schoenfield_, Mar 30 2015

Bui Quang Tuan, Table of n, a(n) for n = 1..458

Cf. A002113 (palindromes), A056964 (n+rev(n)).

Cf. A256437.

Sort@ DeleteDuplicates@ Select[Table[n^2 + FromDigits[Reverse[IntegerDigits@ n]]^2, {n, 10000}], Reverse@ IntegerDigits@ # == IntegerDigits@ # &] (* Michael De Vlieger, Mar 28 2015 *)

rev(n)=r="";d=digits(n);for(i=1,#d,r=concat(Str(d[i]),r));eval(r)
v=[];for(n=0,10^4,if(rev(P=(n^2+rev(n)^2))==P,v=concat(v,P)));vecsort(v,,8) \\ Derek Orr, Mar 29 2015

Data corrected by Derek Orr, Mar 29 2015

A256370 Positive integers n such that n^4 + (n+1)^4 + (n+2)^4 + (n+3)^4 + (n+4)^4 is prime.

Original entry on oeis.org

7, 25, 97, 115, 145, 169, 223, 247, 343, 379, 385, 421, 541, 577, 601, 607, 673, 691, 751, 847, 895, 961, 997, 1111, 1129, 1237, 1267, 1303, 1327, 1459, 1489, 1555, 1615, 1639, 1657, 1663, 1741, 1765, 1771, 1807, 1819, 1831, 1873, 1903, 1927, 1945, 1951, 1963

Offset: 1

Bui Quang Tuan, Mar 26 2015

nonn

NK(n,k) conjecture:
  If k + 1 is prime then there are infinitely many primes of form:
    NK(n,k) = n^k + (n+1)^k + (n+2)^k + ... + (n+k-1)^k + (n+k)^k
  If k + 1 is not prime then gcd(NK(n,k), k + 1) > 1 with any positive integer n.
Some examples in the OEIS:
  k = 1, primes of form NK(n,1) are all odd primes A065091.
  k = 2, primes of form NK(n,2) is A027864.
  k = 4, this sequence generates all primes of form NK(n,4).
All terms == 1 (mod 6). Bunyakovsky's conjecture implies that the sequence is infinite. - Robert Israel, Mar 29 2015

			7 is in the sequence because 7^4 + 8^4 + 9^4 + 10^4 + 11^4 = 37699 which is prime.

Bui Quang Tuan, Table of n, a(n) for n = 1..1000
Wikipedia, Bunyakovsky conjecture

Cf. A027864.

[n: n in [0..2*10^3] | IsPrime( n^4 + (n+1)^4 + (n+2)^4 + (n+3)^4 + (n+4)^4)]; // Vincenzo Librandi, Mar 27 2015

F:= unapply(expand(add((n+i)^4,i=0..4)), n):
select(isprime, [seq(6*i+1,i=1..1000)]); # Robert Israel, Mar 29 2015

Select[Range@ 2000, PrimeQ[#^4 + (# + 1)^4 + (# + 2)^4 + (# + 3)^4 + (# + 4)^4] &] (* Michael De Vlieger, Mar 26 2015 *)
Position[Partition[Range[2000]^4,5,1],?(PrimeQ[Total[#]]&)]//Flatten (* _Harvey P. Dale, Apr 28 2022 *)

from gmpy2 import is_prime
A256370_list = [n for n in range(1,10**6) if is_prime(5*n*(n*(n*(n + 8) + 36) + 80) + 354)] # Chai Wah Wu, Mar 29 2015

A256176 Primes formed by concatenating n with n+1 and by concatenating n+2 with n+3.

Original entry on oeis.org

67, 89, 7879, 8081, 9091, 9293, 186187, 188189, 276277, 278279, 426427, 428429, 438439, 440441, 450451, 452453, 600601, 602603, 606607, 608609, 798799, 800801, 816817, 818819, 858859, 860861, 936937, 938939, 960961, 962963, 11401141, 11421143

Offset: 1

Bui Quang Tuan, Mar 18 2015

Subsequence of A030458.

First bisection: A156121.

			67, 89 are in the sequence because they are primes and 6, 7, 8, 9 are four consecutive integers.
7879, 8081 are in the sequence because they are primes and 78, 79, 80, 81 are four consecutive integers.
186187, 188189 are in the sequence because they are primes and 186, 187, 188, 189 are four consecutive integers.

Bui Quang Tuan, Table of n, a(n) for n = 1..100

f[n_] := FromDigits@ Flatten[IntegerDigits /@ Range[n, n + 1]]; {f@ #, f[# + 2]} & /@ Select[Range@ 1200, AllTrue[{f@ #, f[# + 2]}, PrimeQ] &] // Flatten (* Michael De Vlieger, Mar 18 2015 *)
fd[{a_,b_}]:=FromDigits[Join[IntegerDigits[a],IntegerDigits[b]]]; Select[ {fd[ Take[#,2]],fd[Take[#,-2]]}&/@Partition[Range[1500],4,1],AllTrue[ #,PrimeQ]&]//Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 17 2018 *)

lista(nn) = {for (n=1, nn, if (isprime(p=eval(concat(Str(n), Str(n+1)))) && isprime(q=eval(concat(Str(n+2), Str(n+3)))), print1(p, ", ", q, ", ")););} \\ Michel Marcus, Mar 18 2015

A253188 Minimal positive integer k such that n^n >= (n-k)^(n+k).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 10, 11, 12, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59

Offset: 1

Bui Quang Tuan, Mar 24 2015

nonn

			a(4) = 1 because 4^4 = 256 > 243 = (4-1)^(4+1).
a(5) = 2 because 5^5 = 3125 > 2187 = (5-2)^(5+2) (but < 4096 = (5-1)^(5+1)).

Bui Quang Tuan, Table of n, a(n) for n = 1..1000

f[n_] := Block[{k = 1}, While[n^n < (n - k)^(n + k), k++]; k]; Array[f, 120] (* Michael De Vlieger, Mar 24 2015 *)

a(n) = {k = 1; npn = n^n; while(npn < (n-k)^(n+k), k++); k;} \\ Michel Marcus, Mar 24 2015

A256162 Positive integers a(n) such that number of digits in decimal expansion of a(n)^a(n) is divisible by a(n).

Original entry on oeis.org

1, 8, 9, 98, 99, 998, 999, 9998, 9999, 99998, 99999, 999998, 999999, 9999998, 9999999, 99999998, 99999999, 999999998, 999999999, 9999999998, 9999999999, 99999999998, 99999999999, 999999999998, 999999999999, 9999999999998, 9999999999999

Offset: 1

Bui Quang Tuan, Mar 17 2015

A055642(a(n)^a(n)) = A055642(a(n))*a(n).

1 + floor(log_10(a(n)^a(n))) = a(n)*(1 + floor(log_10(a(n)))).

			1^1 = 1 has 1 digit, and 1 is divisible by 1.
8^8 = 16777216 has 8 digits, and 8 is divisible by 8.
98^98 has 196 digits, and 196 is divisible by 98.

Bui Quang Tuan, Table of n, a(n) for n = 1..101

Cf. A055642 (Number of digits in decimal expansion of n).

[1] cat [10^Floor((n+1)/2)-2*Floor((n+1)/2)+n-1: n in [1..30]]; // Vincenzo Librandi, Mar 18 2015

Select[Range@10000, Mod[IntegerLength[#^#], #] == 0 &] (* Michael De Vlieger, Mar 17 2015 *)
Join[{1}, Table[(10^Floor[n/2] - 2 Floor[n/2] + n - 2), {n, 2, 30}]] (* Vincenzo Librandi, Mar 18 2015 *)

isok(n) = !(#digits(n^n) % n); \\ Michel Marcus, Mar 17 2015

a(n) = 10^floor(n/2) - 2*floor(n/2) + n - 2 = 10^floor(n/2)-(1+(-1)^n)/2 - 1 for n>1, a(1) = 1.

cp's OEIS Frontend

User: Bui Quang Tuan

A256495 Palindromes i such that 2*i^2 is a palindrome.

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A256437 Nonnegative integers i such that i^2 + reverse(i)^2 is a palindrome.

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A256515 Nonpalindromic positive integers k such that the absolute value of k^2 - reverse(k)^2 is a square.

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A256398 Palindromes of the form i^2 + reverse(i)^2.

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A256370 Positive integers n such that n^4 + (n+1)^4 + (n+2)^4 + (n+3)^4 + (n+4)^4 is prime.

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A256176 Primes formed by concatenating n with n+1 and by concatenating n+2 with n+3.

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A253188 Minimal positive integer k such that n^n >= (n-k)^(n+k).

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