A078931 -id:A078931 - cp's OEIS frontend

A162567 Primes p such that pi(p) divides p-1 and/or p+1, where pi(p) is the number of primes <= p.

2, 3, 5, 7, 11, 13, 29, 37, 43, 349, 359, 1087, 1091, 3079, 8423, 64579, 64591, 64601, 64609, 64661, 64709, 481043, 481067, 1304707, 3523969, 3524249, 3524317, 3524387, 9558541, 9559799, 9560009, 9560039, 25874767, 70115921, 189962009

Offset: 1

Leroy Quet, Jul 06 2009

nonn

			The 10th prime is 29. Since 10 divides 29+1 = 30, 29 is in the sequence.
The 12th prime is 37. Since 12 divides 37-1 = 36, 37 is in the sequence.

Robert G. Wilson v, Table of n, a(n) for n = 1..199 (data obtained from A048891 and A052013; sorted by Amiram Eldar Sep 05 2024)

Union of A048891 and A052013. - Michel Marcus, Mar 04 2019

Cf. A000040, A078931.

isA162567 := proc(p) RETURN ( (p-1) mod numtheory[pi](p) = 0 or (p+1) mod numtheory[pi](p) = 0 ) ; end: for n from 1 to 50000 do p := ithprime(n) ; if isA162567(p) then printf("%d,",p) ; fi; od: # R. J. Mathar, Jul 30 2009
with(numtheory): a := proc (n) if `mod`(ithprime(n)-1, pi(ithprime(n))) = 0 or `mod`(ithprime(n)+1, pi(ithprime(n))) = 0 then ithprime(n) else end if end proc: seq(a(n), n = 1 .. 250000); # Emeric Deutsch, Jul 31 2009

Select[Prime[Range[11000000]],Or@@Divisible[{#-1,#+1},PrimePi[#]]&] (* Harvey P. Dale, Sep 08 2012 *)

a(n) = A000040(A078931(n)). - Alois P. Heinz, Feb 20 2023

a(10)-a(35) from Donovan Johnson, Jul 29 2009

A225318 Numbers n such that either prime(n-1) == -1 (mod n) or prime(n+1) == -1 (mod n) but not both.

Original entry on oeis.org

2, 4, 7, 8, 14, 16, 26, 27, 32, 33, 35, 76, 78, 169, 170, 172, 175, 177, 183, 184, 185, 434, 446, 1054, 1056, 2638, 2702, 6468, 15930, 40069, 40070, 40080, 40112, 40115, 40157, 251721, 251758, 251767, 251770, 251788, 637286, 4124464, 4124704

Offset: 1

Irina Gerasimova, May 05 2013

nonn

			2nd prime is 3 and 2 is a member because 1st prime, 2, is congruent to 0 mod 2 and 3rd prime, 5, is congruent to -1 mod 2;
6th prime is 11 and 6 is not a member because 5th prime, 11, is congruent to -1 mod 6 and 7th prime, 17, is congruent to -1 mod 6;
7th prime is 17 and 7 is a member because 6th prime, 13, is congruent to -1 mod 7 and 8th prime, 19, is congruent to 1 mod 6;
14th prime is 43 and 14 is a member because 13th prime, 41, is congruent to -1 mod 14 and 15th prime, 47, is congruent to 5 mod 14.

Cf. A038700, A045924, A078931, A219109.

for n from 2 to 100000 do
    if modp(ithprime(n-1),n) = modp(-1,n) then
        pn := true ;
    else
        pn := false ;
    end if;
    if modp(ithprime(n+1),n) = modp(-1,n) then
        pm := true ;
    else
        pm := false ;
    end if;
    if pn <> pm then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, May 09 2013

is(n)=my(p=prime(n-1),q=nextprime(nextprime(p+1)+1),v=[p+1,q+1]%n); !vecmin(v) && vecmax(v) \\ Charles R Greathouse IV, Mar 18 2014

Corrected by R. J. Mathar, May 09 2013

a(36)-a(43) from Alois P. Heinz, May 18 2013

cp's OEIS Frontend

A162567 Primes p such that pi(p) divides p-1 and/or p+1, where pi(p) is the number of primes <= p.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions