A331031
The prime numbers that are prime-indexed primes and whose digit sum, adjacent digit sum concatenation, and adjacent digit difference concatenation are also primes.
Original entry on oeis.org
41, 83, 401, 2063, 6863, 10909, 20063, 26489, 44621, 105229, 187067, 205507, 233267, 238547, 240047, 243301, 256307, 346763, 367021, 376003, 395581, 555707, 562181, 563467, 600203, 613243, 644843, 675263, 689789, 785801, 787601, 837667, 845381, 954263, 959389, 1070203
Offset: 1
a(4) = 2063 as 2063 is the 311th prime, 2+0+6+3 = 11, '2+0'+'0+6'+'6+3' = 269, '|2-0|'+'|0-6|'+'|6-3|' = 263, and 2063, 311, 11, 269, 263 are all prime.
A329527
The prime numbers that are prime-indexed primes and whose reversal, digit sum, sum of digits to their own power, adjacent digit sum concatenation, and adjacent digit difference concatenation are also primes.
Original entry on oeis.org
10180481, 11245547, 18486581, 35015063, 72042701, 72466367, 112823743, 113135621, 171199663, 304000381, 308486107, 318827167, 370257067, 382355443, 722948621, 731621629, 765348167, 771649421, 775786489, 776751581, 916132267, 963985829, 965521463, 980165701, 1002471581
Offset: 1
a(1) = 10180481, as 10180481 is the 675797th prime, 10180481 in reversal is 18408101, 1+0+1+8+0+4+8+1=23, 1^1+0^0+1^1+8^8+0^0+4^4+8^8+1^1=33554693, '1+0'+'0+1'+'1+8'+'8+0'+'0+4'+'4+8'+'8+1'=11984129, '|1-0|'+'|0-1|'+'|1-8|'+'|8-0|'+'|0-4|'+'|4-8|'+'|8-1|'=1178447, and 10180481, 675797, 18408101, 23, 33554693, 11984129, 1178447 are all prime.
Cf.
A000040,
A006450,
A330653,
A331031,
A007953,
A040115,
A053392,
A046704,
A007500,
A045503,
A004086.
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