A358657 -id:A358657 - cp's OEIS frontend

A378265 Terms k of A358657 such that lpf(k-1) and lpf(k+1) are twin primes pair, where lpf(k) = A020639(k) is the least prime dividing k.

216, 1591596, 2046456, 2051496, 3108204, 8933184, 10496844, 10630836, 13579236, 20866644, 26666856, 27288036, 30398544, 30538404, 33949656, 34851384, 35722044, 36657180, 38588544, 48634956, 67747896, 81982116, 82130796, 87172884, 87865056, 98639100, 100473444

Offset: 1

Amiram Eldar, Nov 21 2024

nonn

Iannucci (2004-2005) called the three numbers before each term and the three numbers after each term (i.e., {k-3, k-2, k-1} and {k+1, k+2, k+3}) "almost prime twin prime triplet twins" (APTPTTs for short), and found that there are 126 terms below 10^9.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Douglas E. Iannucci, Almost prime twin prime triplet twins, Journal of Recreational Mathematics, Vol. 33, No. 2 (2004-2005), pp. 125-129.

Cf. A001097, A020639, A358657.

lista(lim) = {my(p = 2, f1, f2); forprime(q = 3, lim/2, if(q == p+2 && factor(2*p-1)[,2] == [1,1]~ && factor(2*q+1)[,2] == [1,1]~, f1 = factor(2*p+1); f2 = factor(2*q-1); if(f1[,2] == [1,1]~ && f2[,2] == [1,1]~ && abs(f1[1,1] - f2[1,1]) ==  2, print1(2*p+2, ", "))); p = q);}

a(n) == 0 (mod 36). - Hugo Pfoertner, Nov 21 2024

A358666 Numbers such that the two numbers before and the two numbers after are squarefree semiprimes.

Original entry on oeis.org

144, 204, 216, 300, 696, 1140, 1764, 2604, 3240, 3900, 4536, 4764, 5316, 5460, 6000, 6504, 7116, 7836, 7860, 8004, 8484, 9300, 9864, 9936, 10020, 11760, 12180, 13140, 13656, 14256, 15096, 16020, 16440, 16860, 18000, 19536, 20016, 20136, 20280, 21780, 22116, 22236, 23940

Offset: 1

Tanya Khovanova and Massimo Kofler, Nov 25 2022

nonn

All numbers in this sequence are divisible by 12. Proof: Suppose n is odd and in this sequence, then either n-1 or n+1 is divisible by 4, creating a contradiction. Suppose n is even, but not divisible by 4, then n-2 is divisible by 4, creating a contradiction. Suppose n is not divisible by 3. Then there exist x such that 3x and 3(x+1) are among squarefree semiprimes surrounding n; one of them is divisible by 6, creating a contradiction.

			The following numbers are squarefree semiprimes: 214 = 2*107, 215 = 5*43, 217 = 7*31, and 218 = 2*109. Thus, 216 is in this sequence.

Cf. A001358, A358657, A358665.

Select[Range[100000], Transpose[FactorInteger[# - 2]][[2]] == {1, 1} && Transpose[FactorInteger[# - 1]][[2]] == {1, 1} && Transpose[FactorInteger[# + 2]][[2]] == {1, 1} && Transpose[FactorInteger[# + 1]][[2]] == {1, 1} &]

from itertools import count, islice
from sympy import isprime, factorint
def issfsemiprime(n): return list(factorint(n).values()) == [1, 1] if n&1 else isprime(n//2)
def ok(n): return all(issfsemiprime(n+i) for i in (-2, 2, -1, 1))
def agen(): yield from (k for k in count(12, 12) if ok(k))
print(list(islice(agen(), 43))) # Michael S. Branicky, Nov 26 2022

cp's OEIS Frontend

A378265 Terms k of A358657 such that lpf(k-1) and lpf(k+1) are twin primes pair, where lpf(k) = A020639(k) is the least prime dividing k.

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