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Primes p such that both p-2 and p+2 are not squarefree.
+10
9
2, 47, 173, 277, 727, 839, 929, 1181, 1423, 1447, 1523, 1627, 1811, 1847, 1861, 1973, 2207, 2297, 2423, 2693, 3323, 3701, 3719, 3877, 4327, 4363, 4457, 4673, 4691, 4903, 5227, 5573, 5821, 5927, 6173, 6221, 6323, 6473, 6577, 6653, 7027, 7103, 7477, 7823
MATHEMATICA
lst={}; Do[p=Prime[n]; If[ !SquareFreeQ[p-2]&&!SquareFreeQ[p+2], AppendTo[lst, p]], {n, 7!}]; lst
Primes of the form 2*p+1 where p is prime and p+1 is squarefree.
+10
7
5, 11, 59, 83, 227, 347, 563, 1019, 1283, 1307, 1523, 2459, 2579, 2819, 2963, 3803, 3947, 4259, 4547, 5387, 5483, 6779, 6827, 7187, 8147, 9587, 10667, 10883, 11003, 12107, 12227, 12539, 12659, 13043, 13163, 14243, 14387, 15683, 16139, 16187
EXAMPLE
For p = 2 (the only case with p+1 odd), 2*p+1 = 5 is prime and p+1 = 3 is squarefree, so 5 is in the sequence. For p = 3, 2*p+1 = 7 is prime and p+1 = 4 is not squarefree, so 7 is not in the sequence.
MATHEMATICA
lst = {}; Do[p = Prime[n]; If[PrimeQ[Floor[p/2]] && SquareFreeQ[Ceiling[p/2]], AppendTo[lst, p]], {n, 7!}]; lst
Select[2#+1&/@Select[Prime[Range[2000]], SquareFreeQ[#+1]&], PrimeQ] (* Harvey P. Dale, Aug 02 2024 *)
PROG
(Magma) [ q: p in PrimesUpTo(8100) | IsSquarefree(p+1) and IsPrime(q) where q is 2*p+1 ];
Primes of the form 2*p+1 where p is prime and p+1 is not squarefree.
+10
7
7, 23, 47, 107, 167, 179, 263, 359, 383, 467, 479, 503, 587, 719, 839, 863, 887, 983, 1187, 1319, 1367, 1439, 1487, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2879, 2903, 2999, 3023, 3119, 3167, 3203, 3467, 3623, 3779, 3863, 4007, 4079, 4127
EXAMPLE
For p = 2 (the only case with p+1 odd), 2*p+1 = 5 is prime but p+1 = 3 is squarefree, so 5 is not in the sequence. For p = 3, 2*p+1 = 7 is prime and p+1 = 4 is not squarefree, so 7 is in the sequence.
MATHEMATICA
lst = {}; Do[p = Prime[n]; If[PrimeQ[Floor[p/2]] && !SquareFreeQ[Ceiling[p/2]], AppendTo[lst, p]], {n, 7!}]; lst
2#+1&/@Select[Prime[Range[400]], !SquareFreeQ[#+1]&&PrimeQ[2#+1]&] (* Harvey P. Dale, Mar 17 2019 *)
PROG
(Magma) [ q: p in PrimesUpTo(2100) | not IsSquarefree(p+1) and IsPrime(q) where q is 2*p+1 ];
Primes of the form 2*p-1 where p is prime and p-1 is squarefree.
+10
6
3, 5, 13, 61, 157, 277, 421, 661, 733, 877, 997, 1093, 1213, 1237, 1381, 1933, 2797, 3253, 3517, 3733, 4021, 4261, 4621, 5413, 6037, 6133, 6637, 6781, 6997, 7213, 7477, 7933, 8053, 8221, 9013, 9133, 9277, 9661, 10357, 10453, 10861, 10957, 11317, 11677
EXAMPLE
For p = 2 (the only case with p-1 odd), 2*p-1 = 3 is prime and p-1 = 1 is squarefree, so 3 is in the sequence. For p = 19, 2*p-1 = 37 is prime and p-1 = 18 is not squarefree, so 37 is not in the sequence.
MATHEMATICA
lst={}; Do[p = Prime[n]; If[SquareFreeQ[Floor[p/2]] && PrimeQ[Ceiling[p/2]], AppendTo[lst, p]], {n, 7!}]; lst
PROG
(Magma) [ q: p in PrimesUpTo(5900) | IsSquarefree(p-1) and IsPrime(q) where q is 2*p-1 ];
Primes p such that none of p-2, p-1, p+1, and p+2 is squarefree.
+10
3
727, 1423, 1861, 3719, 6173, 9749, 11321, 13183, 19073, 20873, 23227, 23473, 23827, 26981, 27883, 34351, 35323, 41263, 42677, 44449, 45127, 45523, 47527, 48751, 49727, 52391, 53623, 53849, 68749, 71993, 72559, 78823, 83609, 89227, 92779
MATHEMATICA
<< NumberTheory`NumberTheoryFunctions` lst={}; Do[p=Prime[n]; If[ !SquareFreeQ[p-1]&&!SquareFreeQ[p+1]&&!SquareFreeQ[p-2]&&!SquareFreeQ[p+2], AppendTo[lst, p]], {n, 4*7!}]; lst
Primes p such that p+-2 and p+-3 are not squarefree.
+10
1
47, 1447, 1847, 3701, 6653, 11273, 14947, 15727, 17053, 18493, 21661, 24923, 26647, 29153, 32789, 33023, 38873, 39323, 42437, 42923, 44053, 47527, 47977, 49853, 52027, 52153, 56747, 56873, 59929, 71147, 74189, 79427, 80953, 99277, 99713
MATHEMATICA
lst={}; Do[p=Prime[n]; If[ !SquareFreeQ[p-2]&&!SquareFreeQ[p+2]&&!SquareFreeQ[p-3]&&!SquareFreeQ[p+3], AppendTo[lst, p]], {n, 3*7!}]; lst
Select[Prime[Range[10000]], NoneTrue[#+{-3, -2, 2, 3}, SquareFreeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 27 2019 *)
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