A371622 -id:A371622

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A371651

a(n) is the first prime p such that p - 2 and p + 2 both have exactly n prime factors, counted with multiplicity.

+10
2

5, 23, 173, 2693, 32587, 495637, 4447627, 35303123, 717591877, 928090627, 69692326373, 745041171877, 5012236328123, 64215009765623, 945336806640623, 8885812685546873

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(n) is the first prime p such that A001222(p - 2) = A001222(p + 2) = n.

3*10^9 < a(13) <= 5012236328123.

3*10^9 < a(14) <= 64215009765623.

LINKS

Table of n, a(n) for n=1..16.

FORMULA

a(n) > 2*A154704(n) for n > 1.

EXAMPLE

a(3) = 173 because 173 is prime, 173 - 2 = 171 = 3^2 * 19 and 173 + 2 = 175 = 5^2 * 7 are both products of 3 primes with multiplicity, and no smaller number works.

MAPLE

V:= Vector(8):

p:= 3: count:= 0:

while count < 8 do

p:= nextprime(p);

i:= numtheory:-bigomega(p-2);

if i <= 8 and V[i] = 0 and numtheory:-bigomega(p+2) = i

then V[i]:= p; count:= count+1

od:

convert(V, list);

PROG

(Python)

from sympy import primeomega, nextprime

def A371651(n):

p = 3

while True:

if n == primeomega(p-2) == primeomega(p+2):

return p

p = nextprime(p) # Chai Wah Wu, Apr 02 2024

(PARI)

generate(A, B, n) = A=max(A, 2^n); (f(m, p, j) = my(list=List()); if(j==1, forprime(q=max(p, ceil(A/m)), B\m, my(t=m*q); if(isprime(t-2) && bigomega(t-4) == n, listput(list, t-2))), forprime(q = p, sqrtnint(B\m, j), list=concat(list, f(m*q, q, j-1)))); list); vecsort(Vec(f(1, 3, n)));

a(n) = my(x=2^n, y=2*x); while(1, my(v=generate(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Apr 13 2024

CROSSREFS

Cf. A001222. Contained in A371622.

Cf. A115186, A154704.

KEYWORD

nonn,more

AUTHOR

Robert Israel, Apr 01 2024

EXTENSIONS

a(11) from Michael S. Branicky, Apr 01 2024

a(12) from Michael S. Branicky, Apr 02 2024

a(13) from Chai Wah Wu, Apr 04 2024

a(14)-a(16) from Daniel Suteu, Apr 13 2024

STATUS

approved

A371656

Numbers k such that k - 2 and k + 2 have the same number of prime factors, counted with multiplicity.

+10
2

5, 8, 9, 10, 12, 15, 21, 23, 24, 36, 37, 38, 39, 45, 53, 58, 60, 67, 68, 69, 81, 84, 86, 89, 93, 99, 100, 102, 105, 110, 111, 112, 113, 117, 120, 121, 129, 131, 134, 138, 143, 144, 154, 157, 165, 172, 173, 178, 184, 185, 188, 195, 203, 204, 207, 211, 215, 216, 217, 219, 225, 230, 231, 240, 244

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Numbers k such that A001222(k - 2) = A001222(k + 2).

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

EXAMPLE

a(4) = 10 is a term because 10 - 2 = 8 = 2^3 and 10 + 2 = 12 = 2^2 * 3 are both products of 3 primes, counted with multiplicity.

MAPLE

M:= map(numtheory:-bigomega, [$1..10^3]):

select(k -> M[k-2] = M[k+2], [$3 .. 10^3 - 2]);

MATHEMATICA

Select[Range[3, 245], PrimeOmega[#-2]==PrimeOmega[#+2]&] (* Stefano Spezia, Apr 01 2024 *)

CROSSREFS

Cf. A001222. Contains A371622.