A343675 Undulating alternating palindromic primes.

2, 3, 5, 7, 101, 181, 383, 727, 787, 929, 10301, 10501, 14341, 16361, 16561, 18181, 30103, 30703, 32323, 36563, 38183, 38783, 70507, 72727, 74747, 78787, 90709, 94949, 96769, 1074701, 1092901, 1212121, 1218121, 1412141, 1616161, 1658561, 1856581, 1878781, 3072703

Offset: 1

Author

Chai Wah Wu, Apr 25 2021

Keywords

• nonn
• base

Comments

All terms have an odd number of decimal digits.

For n > 3, a(n) is odd and not divisible by 5.

Intersection of A002385, A030144 and A059168.

Subsequence of A343590.

Examples

			16361 is a term as it is a palindromic prime, the digits 1, 6, 3, 6 and 1 have odd and even parity alternately, and also alternately rise and fall.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002385, A030144, A059168, A343590.

Programs

Mathematica

Union@Flatten[{{2,3,5,7},Array[Select[FromDigits/@Riffle@@@Tuples[{Tuples[{1,3,5,7,9},#],Tuples[{0,2,4,6,8},#-1]}],(s=Union@Partition[Sign@Differences@IntegerDigits@#,2];(s=={{1,-1}}||s=={{-1,1}})&&PrimeQ@#&&PalindromeQ@#)&]&,4]}] (* Giorgos Kalogeropoulos, Apr 26 2021 *)

Python

from sympy import isprime
def f(w):
    for s in w:
        for t in range(int(s[-1])+1,10,2):
            yield s+str(t)
def g(w):
    for s in w:
        for t in range(1-int(s[-1])%2,int(s[-1]),2):
            yield s+str(t)
A343675_list = [2,3,5,7]
for l in range(1,9):
    for d in '1379':
        x = d
        for i in range(1,l+1):
            x = g(x) if i % 2 else f(x)
        A343675_list.extend([int(p+p[-2::-1]) for p in x if isprime(int(p+p[-2::-1]))])
        y = d
        for i in range(1,l+1):
            y = f(y) if i % 2 else g(y)
        A343675_list.extend([int(p+p[-2::-1]) for p in y if isprime(int(p+p[-2::-1]))])