A323537 -id:A323537 - cp's OEIS frontend

A115103 Primes p such that p-1 and p+1 have the same number of prime factors with multiplicity.

5, 19, 29, 43, 67, 89, 151, 173, 197, 233, 271, 283, 307, 317, 349, 461, 491, 569, 571, 593, 653, 701, 739, 751, 787, 857, 859, 907, 919, 1013, 1061, 1097, 1277, 1291, 1303, 1483, 1667, 1747, 1831, 1867, 1889, 1913, 1973, 2003, 2083, 2131, 2311, 2357, 2393

Offset: 1

Cino Hilliard, Mar 02 2006

			19-1 = 2*3*3 has 3 factors. 19+1 = 2*2*5 has 3 factors. So 19 is in the table.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

Cf. A067386 (without multiplicity), A323498, A323536, A323537.

isA115103 := proc(n)
    if not type(n,prime) then
        return false;
    end if;
    if numtheory[bigomega](n-1) <> numtheory[bigomega](n+1) then
        false;
    else
        true ;
    end if ;
end proc:
for n from 2 to 3000 do
    if isA115103(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Feb 13 2019
# second Maple program:
q:= p-> isprime(p) and (f-> f(p+1)=f(p-1))(numtheory[bigomega]):
select(q, [$1..3000])[];  # Alois P. Heinz, May 08 2022

Select[Prime[Range[400]],PrimeOmega[#-1]==PrimeOmega[#+1]&] (* Harvey P. Dale, Apr 26 2014 *)

g(n) = forprime(x=1,n,p1=bigomega(x-1);p2=bigomega(x+1);if(p1==p2,print1(x",")))

A323498 Primes p such that p - k and p + k have the same number of prime factors (with multiplicity), for k = 1..6.

Original entry on oeis.org

2131991, 2917927, 3776273, 4742407, 6853409, 16850609, 21789233, 24095791, 24810251, 26316233, 27470537, 27667529, 28962127, 29896439, 30949327, 31289527, 36123853, 36443893, 38824913, 40941233, 41660009, 42533551, 44233193, 45868967, 48313567, 49265009, 51135991

Offset: 1

Zak Seidov, Jan 16 2019

nonn

At least one of p - k and p + k must be composite for each k in for k = 1..5.

Proof: If k = 3 then p - k and p + k are even. If k isn't three then exactly one of p - k, p and p + k is divisible by 3. QED. - David A. Corneth, Jan 18 2019

			For p = 2131991 is in the sequence because for k=1, p - 1 = 2*5*7*7*9*229 and p + 1 = 2*2*2*3*3*29611 are both 6-almost primes, for k=2, p - 2 = 3*710663 and p + 2 = 29*73517 are both semiprimes, etc.

Cf. A115103 (k=1), A323536 (k=7), A323537 (k=8).

upto(n) = {my(res = List(), q = 5); forprime(p = 7, n, t = 1; for(m = 1, 2, for(i = 0, 2, if(bigomega(p + 2*i + m) != bigomega(p - 2*i - m), t = 0; next(2) ) ) ); if(t == 1, listput(res, p)); q = p; ); res } \\ David A. Corneth, Jan 17 2019

is(n) = if(!isprime(n) || n < 7, return(0)); for(k = 1, 6, if(bigomega(n + k) != bigomega(n - k), return(0))); 1 \\ David A. Corneth, Jan 17 2019

use ntheory ':all'; for (my($p,$k)=(2,6); $p <= 10**7; $p = next_prime($p)) { print "$p\n" if vecall {factor($p-$) == factor($p+$)} 1..$k } # Daniel Suteu, Jan 17 2019

a(23)-a(27) from David A. Corneth, Jan 17 2019

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A115103 Primes p such that p-1 and p+1 have the same number of prime factors with multiplicity.

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A323498 Primes p such that p - k and p + k have the same number of prime factors (with multiplicity), for k = 1..6.

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