A072060 -id:A072060 - cp's OEIS frontend

A072055 a(n) = 2*prime(n)+1.

5, 7, 11, 15, 23, 27, 35, 39, 47, 59, 63, 75, 83, 87, 95, 107, 119, 123, 135, 143, 147, 159, 167, 179, 195, 203, 207, 215, 219, 227, 255, 263, 275, 279, 299, 303, 315, 327, 335, 347, 359, 363, 383, 387, 395, 399, 423, 447, 455, 459, 467, 479

Offset: 1

Reinhard Zumkeller, Jun 11 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. P. Boas & N. J. A. Sloane, Correspondence, 1974

Cf. A000040, A005385, A023589, A023590, A023591, A023592, A023593, A072059, A023595, A072060, A072056, A072057, A072058, A072192, A165286, A278230.
One less than A089241. After the initial term equal to A166496.
Row 4 of A286625, column 4 of A286623.

a072055 = (+ 1) . (* 2) . a000040  -- Reinhard Zumkeller, Oct 10 2013

2*Prime[Range[60]]+1  (* Harvey P. Dale, Mar 31 2011 *)

a(n) = A089241(n)-1.

A072059 Smallest prime p such that 2*p+1 has n distinct prime factors.

Original entry on oeis.org

2, 7, 97, 577, 7507, 217717, 5232727, 75172597, 1617423307, 59844662377, 2750790860317, 109455887488447, 4621264673452927, 218071376383127767, 10914293640945722527, 662082573402158125717, 41249727342503299116997

Offset: 1

Reinhard Zumkeller, Jun 11 2002

nonn

Note that for each n=1,...,8, the product of the smallest n-1 distinct prime factors of 2*a(n)+1 is p(n)#/2, where p(n)# is the primorial (A002110) of the n-th prime - and the n-th distinct prime factor >= p(n+1). - Rick L. Shepherd, Jul 06 2002

			a(4)=577=A000040(106): 2*577+1 = 1155 = 11*7*5*3, 4 distinct factors.

Cf. A001221, A023589, A072055, A072060.

for (n=1,8, p=1; until(isprime(p) && omega(2*p+1)==n, p++); print1(p,","))

More terms from Rick L. Shepherd, Jul 06 2002

More terms from Don Reble, Apr 15 2003

A367687 a(n) is the first prime p such that n*p+1 is the product of n primes counted with multiplicity.

Original entry on oeis.org

2, 7, 17, 47, 79, 9479, 41, 5923, 199, 33461, 2141, 69173177, 11579, 7655281, 20753, 64869017, 233231, 2622816297743, 341477, 14508897313, 8138947, 24565981007, 27445337, 90698401133219401, 313566167, 2552728502809, 229909997, 23451738297083, 948780491, 20677177107714198558766009, 3390080033

Offset: 1

Zak Seidov and Robert Israel, Nov 26 2023

			a(3) = 17 because 17 is prime and 3 * 17 + 1 = 52 = 2^2 * 13 is the product of 3 primes, and no smaller prime works.

Robert Israel, Table of n, a(n) for n = 1..500

Cf. A001222, A072060.

f:= proc(n)
  uses priqueue;
local Q,t,q,i;
  initialize(Q);
  q:= 2;
  while n mod q = 0 do q:= nextprime(q) od:
  insert([-q^n,q,n],Q);
  do
    t:= extract(Q);
    if -t[1]-1 mod n = 0 and isprime((-t[1]-1)/n) then return (-t[1]-1)/n fi;
    q:= nextprime(t[2]);
    while n mod q = 0 do q:= nextprime(q) od;
    for i from 1 to t[3] do
      insert([t[1]*(q/t[2])^i,q,i],Q);
    od
  od;
end proc:
map(f, [$1..40]);

A001222(n*a(n)+1) = n.

cp's OEIS Frontend

A072055 a(n) = 2*prime(n)+1.

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A072059 Smallest prime p such that 2*p+1 has n distinct prime factors.

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A367687 a(n) is the first prime p such that n*p+1 is the product of n primes counted with multiplicity.

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