A353073 -id:A353073 - 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

A353074 Numbers that differ from their prime neighbors by distinct squares.

Original entry on oeis.org

140, 148, 182, 190, 242, 250, 284, 292, 338, 346, 410, 418, 422, 430, 548, 556, 578, 586, 632, 640, 692, 700, 710, 718, 788, 796, 812, 820, 830, 838, 891, 903, 920, 928, 1022, 1030, 1040, 1048, 1052, 1060, 1154, 1162, 1172, 1180, 1250, 1258, 1336, 1352, 1400

Offset: 1

Tanya Khovanova, Apr 21 2022

nonn

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

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

Cf. A353072, A353073.

q:= n-> (s-> nops(s)=2 and andmap(issqr, s))({n-prevprime(n), nextprime(n)-n}):
select(q, [$3..2000])[];  # Alois P. Heinz, Apr 22 2022

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

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

A353089 Least number which differs from both of its prime neighbors by n^2, and -1 if no such number exists.

Original entry on oeis.org

4, 93, 532, 5607, 31932, 31433, 604122, 3851523, 39175298, 378044079, 367876650, 383204683, 22076314482

Offset: 1

Jean-Marc Rebert, Apr 22 2022

a(12) < 1294268635 is the first term where the first formula is a strict inequality. - Michael S. Branicky, Apr 22 2022

			a(1) = 4, because 3 and 5 are the prime neighbors of 4, and 5 - 4 = 4 - 3 = 1 = 1^2 and no number less than 4 differs from both of its prime neighbors by 1^2.
a(2) = 93, because 97 and 89 are the prime neighbors of 93, and 97 - 93 = 93 - 89 = 4 = 2^2 and no number less than 93 differs from both of its prime neighbors by 2^2.

Cf. A000040, A000230, A000290, A038664, A353073, A055380, A282690.

a[n_] := a[n] = Module[{diff, diff2, p, q, r},
     {diff, diff2, p} = {n*n, 2*n*n, NextPrime[1 + n^2]};
     q = NextPrime[p];
     r = NextPrime[q];
     While[!(q - p == diff2 || (q - p == diff && r - q == diff)),
          {p, q, r} = {q, r, NextPrime[r]}];
     Return[If[q - p == diff2, Floor[(q + p)/2], q]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 10}] (* Jean-François Alcover, Jun 07 2022, after Michael S. Branicky's code *)

a(n) = my(k=2); while (((nextprime(k+1)-k) != n^2) || ((k-precprime(k-1)) != n^2), k++); k; \\ Michel Marcus, Jul 10 2022

from sympy import nextprime
def a(n):
    diff, diff2, p = n*n, 2*n*n, nextprime(1+n**2)
    q = nextprime(p)
    r = nextprime(q)
    while not (q-p == diff2 or (q-p == diff and r-q == diff)):
        p, q, r = q, r, nextprime(r)
    return (q+p)//2 if q-p == diff2 else q
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Apr 22 2022

a(n) <= A000040(A038664(n^2)) + n^2. - Alois P. Heinz, Apr 22 2022
a(n) <= A000230(n^2) + n^2. - David A. Corneth, May 02 2022
a(n) = A282690(n^2). - Michel Marcus, Jul 10 2022

a(13) from Michael S. Branicky, Apr 24 2022

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A353088 Primes having square prime gaps to both neighbor primes.

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A353074 Numbers that differ from their prime neighbors by distinct squares.

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Maple

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A353089 Least number which differs from both of its prime neighbors by n^2, and -1 if no such number exists.

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