A178316 -id:A178316 - cp's OEIS frontend

A275028 Squares whose digital rotation is also a square.

1, 121, 196, 529, 625, 961, 6889, 10201, 69169, 1002001, 1022121, 1212201, 5221225, 100020001, 100220121, 109181601, 121022001, 522808225, 602555209, 10000200001, 10002200121, 10020210201, 10201202001, 12100220001, 62188888129, 1000002000001

Offset: 1

Seiichi Manyama, Nov 12 2016

From Jon E. Schoenfield, Nov 13 2016: (Start)
It is assumed that the rotation changes each digit 6 to a 9 and vice versa, and that the digits 0, 1, 2, 5, and 8 are unchanged by the rotation, as is the case with a seven-segment display in which the digits are formed basically as follows:
.                                      
    | |  |   |  |  ||  |   |    |  ||  |_|
    ||  |  |   |    |   |  ||   |  ||   _|
(End)
Sequence is infinite since (10^m+1)^2 for m>0 are terms. Leading zeros after rotation are not allowed, as otherwise 10^m would be terms. All terms start and end with digits 1, 5, 6 or 9. - Chai Wah Wu, Apr 09 2024

			961 becomes 196 under such a rotation.

Chai Wah Wu, Table of n, a(n) for n = 1..164
Prime Curios, 31

Cf. A178316.

Select[Range[10^6]^2, If[Or[IntersectingQ[#, {3, 4, 7}], Last@# == 0], False, IntegerQ@ Sqrt@ FromDigits[Reverse@ # /. {6 -> 9, 9 -> 6}]] &@ IntegerDigits@ # &] (* Michael De Vlieger, Nov 14 2016 *)

from itertools import count, islice
from sympy.ntheory.primetest import is_square
def A275028_gen(): # generator of terms
    r, t = ''.maketrans('69','96'), set('0125689')
    for l in count(1):
        if l%10:
            m = l**2
            if set(s:=str(m)) <= t and is_square(int(s[::-1].translate(r))):
                yield m
A275028_list = list(islice(A275028_gen(),10)) # Chai Wah Wu, Apr 09 2024

A370447 Palindromic prime numbers that consist only of the digits {0,1,6,8,9} and which remain palindromic primes when their digits are rotated by 180 degrees.

Original entry on oeis.org

11, 101, 181, 16661, 18181, 19991, 1008001, 1160611, 1180811, 1190911, 1688861, 1880881, 1881881, 1988891, 100111001, 100888001, 101616101, 101919101, 106111601, 106191601, 108101801, 109111901, 109161901, 110111011, 111010111, 111181111, 116010611, 116696611

Offset: 1

Jean-Marc Rebert, Feb 18 2024

10886968801 is the least palindromic prime of this sequence for which the set of digits is {0,1,6,8,9}.

Terms must start and end with digit 1 and be of odd length for n > 1. - Michael S. Branicky, Feb 19 2024

			16661 becomes 19991 under such a rotation, and both are palindromic primes.

Michael S. Branicky, Table of n, a(n) for n = 1..16934 (all terms < 10^20)
Carlos Rivera, Puzzle 1163. Palindromic primes that are semiprimes when turned upside down, The Prime Puzzles & Problems Connection.

Subsequence of palindromes in A007597.

Cf. A002385, A178316.

rot(u)=my(v=[]);for(i=1,#u,my(x=u[i]);if(x==6,v=concat(9,v),x==9,v=concat(6,v),vecsearch([0,1,8],x)>0,v=concat(x,v)));v
is(x)=my(u=digits(x),su=Set(u));if(setintersect(su,Set([0,1,6,8,9]))!=su||!isprime(x)||Vecrev(u)!=u,return(0));my(y=fromdigits(rot(u)));return(isprime(y))

from sympy import isprime
from itertools import product, count, islice
def flip180(s): return s[::-1].translate({54:57, 57:54})
def agen(): # generator of terms
    yield 11
    for digits in count(3, 2):
        for rest in product("01689", repeat=digits//2-1):
            for mid in "01689":
                s = "".join(("1",)+rest+(mid,)+rest[::-1]+("1",))
                if isprime(t:=int(s)) and isprime(int(flip180(s))):
                    yield t
print(list(islice(agen(), 28))) # Michael S. Branicky, Feb 19 2024

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A275028 Squares whose digital rotation is also a square.

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