A067386 -id:A067386 - 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",")))

A284037 Primes p such that p-1 and p+1 have two distinct prime factors.

Original entry on oeis.org

11, 13, 19, 23, 37, 47, 53, 73, 97, 107, 163, 193, 383, 487, 577, 863, 1153, 2593, 2917, 4373, 8747, 995327, 1492993, 1990657, 5308417, 28311553, 86093443, 6879707137, 1761205026817, 2348273369087, 5566277615617, 7421703487487, 21422803359743, 79164837199873

Offset: 1

Giuseppe Coppoletta, Mar 28 2017

nonn

Either p-1 or p+1 must be of the form 2^i * 3^j, since among three consecutive numbers exactly one is a multiple of 3. - Giovanni Resta, Mar 29 2017

Subsequence of A219528. See the previous comment. - Jason Yuen, Mar 08 2025

			7 is not a term because n + 1 = 8 has only one prime factor.
23 is a term because it is prime and n - 1 = 22 has two distinct prime factors (2, 11) and n + 1 = 24 has two distinct prime factors (2, 3).
43 is not a term because n - 1 = 42 has three distinct prime factors (2, 3, 7).

Robert Israel, Table of n, a(n) for n = 1..51 (all terms < 10^75)

Cf. A000668, A001221, A005105, A005109, A033845, A058383, A067386, A092506, A215504, A219528, A275598.

N:= 10^20: # To get all terms <= N
Res:= {}:
for i from 1 to ilog2(N) do
  for j from 1 to floor(log[3](N/2^i)) do
    q:= 2^i*3^j;
    if isprime(q-1) and nops(numtheory:-factorset((q-2)/2^padic:-ordp(q-2,2)))=1 then Res:= Res union {q-1} fi;
    if isprime(q+1) and nops(numtheory:-factorset((q+2)/2^padic:-ordp(q+2,2)))=1 then Res:= Res union {q+1} fi
od od:
sort(convert(Res,list)); # Robert Israel, Apr 16 2017

mx = 10^30; ok[t_] := PrimeQ[t] && PrimeNu[t-1]==2==PrimeNu[t+1]; Sort@ Reap[Do[ w = 2^i 3^j; Sow /@ Select[ w+ {1,-1}, ok], {i, Log2@ mx}, {j, 1, Log[3, mx/2^i]}]][[2, 1]] (* terms up to mx, Giovanni Resta, Mar 29 2017 *)

isok(n) = isprime(n) && (omega(n-1)==2) && (omega(n+1)==2); \\ Michel Marcus, Apr 17 2017

omega=sloane.A001221; [n for n in prime_range(10^6) if 2==omega(n-1)==omega(n+1)]

sorted([2^i*3^j+k for i in (1..40) for j in (1..20) for k in (-1,1) if is_prime(2^i*3^j+k) and sloane.A001221(2^i*3^j+2*k)==2])

A001221(a(n)) = 1 and A001221(a(n) - 1) = A001221(a(n) + 1) = 2.

a(33)-a(34) from Giovanni Resta, Mar 29 2017

A353123 a(n) is the first prime p for which the absolute value of the difference between the numbers of distinct prime factors of p+1 and p-1 is exactly n.

Original entry on oeis.org

3, 2, 31, 2309, 8191, 746129, 16546531, 300690389, 11823922111, 239378649509, 11003163441269, 304250263527209, 23293697005168589

Offset: 0

Yusuf Gurtas, May 08 2022

If a given prime p is less than a(n) then the numbers of distinct prime factors of p+1 and p-1 have a difference less than n.
From Daniel Suteu, May 11 2022: (Start)
a(13) <= 693386350578511591,
a(14) <= 42296567385289206991,
a(15) <= 3291505006196194517729,
a(16) <= 222099275340153625904489,
a(17) <= 12592092354842984193179971,
a(18) <= 873339227295479848905071071,
a(19) <= 54536351988824964540662450069,
a(20) <= 5513390541916364286137713664909. (End)
From Jon E. Schoenfield, May 11 2022: (Start)
For n > 1, if there exists any prime p < 2*prime(n+2)# such that the absolute difference of the numbers of distinct prime factors of p+1 and p-1 is exactly n, then (since a(n) <= p) it follows that a(n)+1 and a(n)-1, in some order, are either (1) a power of 2, and 2 times an odd number with n distinct prime factors, or (2) an odd prime times a power of 2, and the product of n+2 distinct primes. (In the first case, the numbers of distinct prime factors are 1 and n+1; in the second, they are 2 and n+2.)
E.g., for n = 6, given that p = 18888871 is a prime < 19399380 = 2*(2*3*5*7*11*13*17*19) and the prime factorizations of p+1 and p-1 are 2^3 * 2361109 and 2*3*5*7*11*13*17*37, then a(6)+1 must be too small to have more than 8 distinct prime factors, and also too small to have exactly 8 distinct prime factors with any factor having a multiplicity greater than 1. Thus, a(6) is the smallest prime p such that p+1 and p-1, in some order, are either (1) a power of 2, and 2 times an odd number with 6 distinct prime factors, or (2) a prime times a power of 2, and the product of 8 distinct primes. (As it turns out, a(6) = 16546531, so a(6) + 1 = 2^2 * 4136633 (2 distinct prime factors) and a(6) - 1 = 2*3*5*7*11*13*19*29 (8 distinct prime factors).)
For each n in 2..100, there exists such a prime p < 2*prime(n+2)#, so the numbers of distinct prime factors of a(n)+1 and a(n)-1 are, in some order, 1 and n+1 or 2 and n+2.
For n <= 100, the maximum value of a(n)/prime(n+2)# is a(16)/prime(18)# = 1.893617....
(End)

			a(2) = 31 because the number of distinct prime factors of 32 is 1 and the number of distinct prime factors of 30 is 3, giving a difference of 2. No prime less than 31 has this property.

Cf. A067386.

isok(p, n) = abs(omega(p-1)-omega(p+1)) == n;
a(n) = my(p=2); while (!isok(p,n), p=nextprime(p+1)); p; \\ Michel Marcus, May 09 2022

a(7)-a(8) from Amiram Eldar, May 08 2022
a(9)-a(10) from Yusuf Gurtas, May 08 2022
a(11) from Yusuf Gurtas, May 09 2022
a(9) corrected by Yusuf Gurtas, May 09 2022
a(12) from Yusuf Gurtas, May 09 2022

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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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A284037 Primes p such that p-1 and p+1 have two distinct prime factors.

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A353123 a(n) is the first prime p for which the absolute value of the difference between the numbers of distinct prime factors of p+1 and p-1 is exactly n.

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