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Numbers that differ from their prime neighbors by distinct squares.
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%I #18 Jan 10 2023 20:32:53

%S 140,148,182,190,242,250,284,292,338,346,410,418,422,430,548,556,578,

%T 586,632,640,692,700,710,718,788,796,812,820,830,838,891,903,920,928,

%U 1022,1030,1040,1048,1052,1060,1154,1162,1172,1180,1250,1258,1336,1352,1400

%N Numbers that differ from their prime neighbors by distinct squares.

%H Alois P. Heinz, <a href="/A353074/b353074.txt">Table of n, a(n) for n = 1..10000</a>

%e Prime neighbors of 891 are 887 and 907, with different square differences 4 and 16. Thus, 891 is in this sequence.

%p q:= n-> (s-> nops(s)=2 and andmap(issqr, s))({n-prevprime(n), nextprime(n)-n}):

%p select(q, [$3..2000])[]; # _Alois P. Heinz_, Apr 22 2022

%t Select[Range[3, 2000], IntegerQ[Sqrt[NextPrime[#] - #]] && IntegerQ[Sqrt[# - Prime[PrimePi[NextPrime[# - 1]] - 1]]] && NextPrime[#] - # != # - Prime[PrimePi[NextPrime[# - 1]] - 1] &]

%o (PARI) isok(k) = my(d,dd); (k>1) && issquare(nextprime(k+1)-k, &d) && issquare(k-precprime(k-1), &dd) && (d!=dd); \\ _Michel Marcus_, Apr 22 2022

%Y Cf. A353072, A353073.

%K nonn

%O 1,1

%A _Tanya Khovanova_, Apr 21 2022