A353136 -id:A353136 - cp's OEIS frontend

A353088 Primes having square prime gaps to both neighbor primes.

9551, 12889, 22193, 22307, 27143, 29917, 32261, 40423, 42863, 46807, 46993, 47981, 57637, 60041, 60493, 71597, 72613, 73819, 77137, 84263, 88427, 89153, 90583, 93463, 97463, 97613, 97883, 112543, 115057, 118931, 126307, 127877, 131321, 134093, 137873, 144883

Offset: 1

Alois P. Heinz, Apr 22 2022

nonn

			Prime 9551 is a term, the gap to the previous prime 9547 is 4 and the gap to the next prime 9587 is 36 and both gaps are squares.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Intersection of A000040 and A353073.

Cf. A000290, A001223, A016742, A138198, A353135, A353136, A353137, A353550.

q:= n-> isprime(n) and andmap(issqr, [n-prevprime(n), nextprime(n)-n]):
select(q, [$3..200000])[];

q[n_] := PrimeQ[n] && IntegerQ@Sqrt[n-NextPrime[n, -1]] && IntegerQ@ Sqrt[NextPrime[n]-n];
Select[Range[3, 200000], q] (* Jean-François Alcover, May 14 2022, after Alois P. Heinz *)
Select[Prime[Range[2,15000]],AllTrue[{Sqrt[#-NextPrime[#,-1]],Sqrt[NextPrime[#]-#]},IntegerQ]&] (* Harvey P. Dale, Jan 22 2024 *)

from itertools import islice
from sympy import nextprime, integer_nthroot
def A353088_gen(): # generator of terms
    p, q, g, h = 3, 5, True, False
    while True:
        if g and h:
            yield p
        p, q = q, nextprime(q)
        g, h = h, integer_nthroot(q-p,2)[1]
A353088_list = list(islice(A353088_gen(),30)) # Chai Wah Wu, Apr 22 2022

A353135 Primes having Fibonacci prime gaps to both neighbor primes.

Original entry on oeis.org

3, 5, 10007, 11777, 12163, 17291, 20443, 20477, 37781, 41333, 47743, 47777, 49991, 59887, 59921, 61091, 61331, 64271, 77417, 88177, 88609, 88643, 89363, 91639, 93337, 97073, 105863, 106453, 107507, 108463, 108497, 112363, 113383, 113717, 125149, 133631, 134293

Offset: 1

Alois P. Heinz, Apr 25 2022

nonn

			Prime 10007 is a term, the gap to the previous prime 9973 is 34 and the gap to the next prime 10009 is 2 and both gaps are Fibonacci numbers.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000040, A000045, A014445, A353088, A353136, A353137.

f:= proc(n) option remember; (t-> issqr(t+4) or issqr(t-4))(5*n^2) end:
q:= n-> isprime(n) and andmap(f, [n-prevprime(n), nextprime(n)-n]):
select(q, [$3..150000])[];

f[n_] := f[n] = With[{t = 5n^2}, IntegerQ@Sqrt[t+4] || IntegerQ@Sqrt[t-4]];
q[n_] := PrimeQ[n] && f[n-NextPrime[n, -1]] && f[NextPrime[n]-n];
Select[Range[3, 150000], q] (* Jean-François Alcover, May 14 2022, after Alois P. Heinz *)

A353137 Primes whose gaps to both neighbor primes are powers of two.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 41, 43, 71, 97, 101, 103, 107, 109, 193, 197, 227, 229, 281, 311, 313, 349, 397, 401, 457, 461, 463, 487, 491, 499, 617, 643, 743, 761, 769, 823, 827, 857, 859, 881, 883, 911, 937, 1091, 1093, 1279, 1301, 1303, 1427, 1429, 1447, 1451

Offset: 1

Alois P. Heinz, Apr 25 2022

nonn

			Prime 1447 is a term, the gap to the previous prime 1439 is 8 and the gap to the next prime 1451 is 4 and both gaps are powers of two.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000040, A000079, A353088, A353135, A353136.

p2:= proc(n) option remember; is(n=2^ilog2(n)) end:
q:= n-> isprime(n) and andmap(p2, [n-prevprime(n), nextprime(n)-n]):
select(q, [$3..3000])[];

p2[n_] := n == 2^Floor[Log2[n]];
q[n_] := PrimeQ[n] && p2[n-NextPrime[n, -1]] && p2[NextPrime[n]-n];
Select[Range[3, 3000], q] (* Jean-François Alcover, May 14 2022, after Alois P. Heinz *)

A353550 Primes having cube prime gaps to both neighbor primes.

Original entry on oeis.org

89689, 107441, 367957, 368021, 725209, 803749, 832583, 919511, 1070753, 1315151, 1333027, 1353487, 1414913, 1843357, 2001911, 2038039, 2201273, 2207783, 2269537, 2356699, 2356763, 2670817, 2696843, 2715071, 2717929, 2731493, 2906887, 2971841, 3032467, 3184177, 3252217

Offset: 1

Karl-Heinz Hofmann, Apr 25 2022

nonn

Up to prime 669763117 all gaps are 8 and 64 or 64 and 8. Prime 669763117 is the first one with gaps 8 and 216. Possible gaps must be in A016743.

			a(2) = 107441; previous prime is 107377 and the gap is 64 (a cube); next prime is 107449 and the gap is 8 (a cube too).

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A000040, A000578, A016743, A353088 (square gaps), A163112 (gaps > 20).
Cf. A353137 (gaps are a power of 2), A353135 (Fibonacci gaps).
Cf. A353136 (triangular numbers gaps).

iscube:= proc(n) option remember; is(n=iroot(n, 3)^3) end:
q:= n-> isprime(n) and andmap(iscube, [n-prevprime(n), nextprime(n)-n]):
select(q, [$3..3500000])[];  # Alois P. Heinz, Apr 25 2022

p = Prime[Range[3*10^5]]; pos = Position[Differences[p], ?(IntegerQ@Surd[#, 3] &)] // Flatten; p[[pos[[Position[Differences[pos], 1] // Flatten]] + 1]] (* _Amiram Eldar, Apr 26 2022 *)

from sympy import sieve as p
def A016743(totest): return (totest % 2 == 0 and round(totest**(1/3))**3 == totest)
print([p[n] for n in range(2,235000) if A016743(p[n]-p[n-1]) and A016743(p[n+1]-p[n])])

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A353088 Primes having square prime gaps to both neighbor primes.

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A353135 Primes having Fibonacci prime gaps to both neighbor primes.

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A353137 Primes whose gaps to both neighbor primes are powers of two.

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A353550 Primes having cube prime gaps to both neighbor primes.

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