A143958 -id:A143958 - cp's OEIS frontend

A143957 An integer >= 2 is included if {the difference between the largest and smallest primes dividing n} divides n+1.

6, 12, 14, 15, 18, 20, 24, 35, 36, 39, 44, 45, 48, 50, 54, 63, 72, 75, 80, 84, 96, 108, 119, 135, 143, 144, 147, 152, 153, 155, 162, 175, 192, 200, 208, 216, 224, 225, 230, 231, 242, 245, 275, 279, 288, 294, 299, 315, 320, 323, 324, 374, 375, 384, 399, 405, 429

Offset: 1

Leroy Quet, Sep 05 2008

nonn

			The largest prime dividing 14 is 7. The smallest prime dividing 14 is 2. 7-2=5 divides 14+1=15. So 14 is in the sequence.

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

Cf. A143958. Includes A037074.

filter:= proc(n) local P;
  P:= numtheory:-factorset(n);
  if nops(P) = 1 then return false fi;
  n+1 mod (max(P)-min(P)) = 0
end proc:
select(filter, [$2..1000]); # Robert Israel, Nov 04 2020

Select[Range[2, 429], If[Or[PrimeQ[#], PrimePowerQ[#]], False, Mod[#1, Last[#2] - First[#2]] == 0 & @@ {# + 1, FactorInteger[#][[All, 1]]}] &] (* Michael De Vlieger, Nov 04 2020 *)

Extended by Ray Chandler, Nov 07 2008

A295798 a(n) is the number of divisors d of prime(n)^2 - 1 such that prime(n) + d is prime.

Original entry on oeis.org

2, 3, 5, 5, 8, 6, 7, 11, 7, 13, 13, 8, 16, 11, 7, 11, 11, 8, 9, 23, 5, 13, 9, 22, 12, 14, 15, 8, 20, 12, 20, 19, 8, 20, 14, 19, 5, 12, 10, 7, 12, 33, 24, 7, 18, 28, 20, 13, 9, 15, 21, 27, 20, 29, 12, 11, 14, 30, 5, 25, 4, 10, 33, 19, 7, 3, 12, 18, 9, 26, 13, 19, 13, 12, 40, 9, 15, 12, 24, 17, 37, 17

Offset: 1

Robert Israel, Nov 27 2017

nonn

a(n) is the number of semiprimes in A143958 whose least prime factor is prime(n).

The least n for which a(n)=0 is 7511.

			For n=3, prime(3)=5; 5^2-1 has 5 divisors d such that 5+d is prime, namely 2, 6, 8, 12, 24.  Thus a(3)=5.

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

Cf. A000040, A143958.

f:= proc(p) nops(select(t -> isprime(p+t), numtheory:-divisors(p^2-1))) end proc:
map(f, [seq(ithprime(i),i=1..100)]);

Table[DivisorSum[p^2 - 1, 1 &, PrimeQ[p + #] &], {p, Prime@ Range@ 82}] (* Michael De Vlieger, Nov 27 2017 *)

a(n) = sumdiv(prime(n)^2-1, d, isprime(prime(n)+d)); \\ Michel Marcus, Nov 30 2017

A295818 Primes p such that p+d is composite for all divisors d of p^2-1.

Original entry on oeis.org

76333, 80077, 106243, 115603, 155333, 175963, 224683, 279353, 281207, 299653, 309157, 312253, 314267, 318523, 361093, 413053, 428083, 526997, 533893, 573829, 580093, 605533, 625237, 637243, 655987, 661993, 706633, 708403, 716747, 723803, 737293, 740227, 741347, 741373, 766967, 775757, 776267, 776317

Offset: 1

Robert Israel, Nov 27 2017

nonn

Primes n such that A295798(A000720(n)) = 0.

Primes n such that no semiprime with least prime factor n is in A143958.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 2000 from Robert Israel)

Cf. A000720, A143958, A295798.

filter:= p -> isprime(p) and andmap(t -> not isprime(p+t), numtheory:-divisors(p^2-1)):
select(filter, [seq(i,i=3..10^6,2)]);

is(n) = if(isprime(n), fordiv(n^2-1, d, if(isprime(n+d), return(0))); 1, 0) \\ Iain Fox, Nov 27 2017

is(n)=my(o=valuation(n+1,2), f1=factor((n-1)<>o, f), f=Mat(vecsort(Vec(concat(f1~,f2~)),1))~); fordiv(f, d, if(isprime(n+d), return(0))); isprime(n) \\ Charles R Greathouse IV, Nov 28 2017

cp's OEIS Frontend

A143957 An integer >= 2 is included if {the difference between the largest and smallest primes dividing n} divides n+1.

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A295798 a(n) is the number of divisors d of prime(n)^2 - 1 such that prime(n) + d is prime.

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A295818 Primes p such that p+d is composite for all divisors d of p^2-1.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

PARI