A079148 -id:A079148 - cp's OEIS frontend

A005385 Safe primes p: (p-1)/2 is also prime.

5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2579, 2819, 2879, 2903, 2963

Offset: 1

N. J. A. Sloane

Then (p-1)/2 is called a Sophie Germain prime: see A005384.
Or, primes of the form 2p+1 where p is prime.
Primes p such that denominator(Bernoulli(p-1) + 1/p) = 6. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 10 2004
Primes p such that p-1 is a semiprime. - Zak Seidov, Jul 01 2005
A156659(a(n)) = 1; A156875 gives numbers of safe primes <= n. - Reinhard Zumkeller, Feb 18 2009
From Daniel Forgues, Jul 31 2009: (Start)
A safe prime p is 7 or of the form 6k-1, k >= 1, i.e., p == 5 (mod 6).
A prime p of the form 6k+1, k >= 2, i.e., p = 1 (mod 6), cannot be a safe prime since (p-1)/2 is composite and divisible by 3. (End)
If k is the product of the n-th safe prime p and its corresponding Sophie Germain prime (p-1)/2, then a(n) = 2(k-phi(k))/3 + 1, where phi is Euler's totient function. - Wesley Ivan Hurt, Oct 03 2013
From Bob Selcoe, Apr 14 2014: (Start)
When the n-th prime is divided by all primes up to the (n-1)-th prime, safe primes (p) have remainders of 1 when divided by 2 and (p-1)/2 and no other primes. That is, p(mod j)=1 iff j={2,(p-1)/2}; p>j, {p,j}=>prime.  Explanation: Generally, x(mod y)=1 iff x=y'+1, where y' is the set of divisors of y, y'>1. Since safe primes (p) are of the form p(mod j)=1 iff p and j are prime, then j={j'}. That is, since j is prime, there are no divisors of j (greater than 1) other than j.  Therefore, no primes other than j exist which satisfy the equation p(mod j)=1.
Except primes of the form 2^n+1 (n>=0), all non-safe primes (p') will have at least one prime (p") greater than 2 and less than (p-1)/2 such that p'(mod p")=1. Explanation: Non-safe primes (p') are of the form p'(mod k)=1 where k is composite. This means prime divisors of k exist, and p" is the set of prime divisors of k (example p'=89: k=44; p"={2,11}).  The exception applies because p"={2} iff p'=2^n+1.
Refer to the rows in triangle A207409 for illustration and further explanation. (End)
Conjecture: there is a strengthening of the Bertrand postulate for n >= 24: the interval (n, 2*n) contains a safe prime. It has been tested by Peter J. C. Moses up to n = 10^7. - Vladimir Shevelev, Jul 06 2015
The six known safe primes p such that (p-1)/2 is a Fibonacci prime are in A263880. - Jonathan Sondow, Nov 04 2015
The only term in common with A005383 is 5. - Zak Seidov, Dec 31 2015
From the fourth entry onward, do these correspond to Smarandache's problem 34 (see A007931 link), specifically values which cannot be used (do not meet conditions) to confirm the conjecture? - Bill McEachen, Sep 29 2016
Primes p with the property that there is a prime q such that p+q^2 is a square. - Zak Seidov, Feb 16 2017
It is conjectured that there are infinitely many safe primes, and their estimated asymptotic density ~ 2C/(log n)^2 (where C = 0.66... is the twin prime constant A005597) converges to the actual value as far as we know. - M. F. Hasler, Jun 14 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
A. R. Ashrafi and F. Koorepazan-Moftakhar, Towards the Classification of Finite Simple Groups with exactly Three or Four Supercharacter Theories, arXiv preprint arXiv:1605.08971 [math.GR], 2016.
R. P. Boas & N. J. A. Sloane, Correspondence, 1974.
Siji Chen and Sheng Chen, Connectedness of digraphs from quadratic polynomials, Involve (2020) Vol. 13, No. 2, 357-360.
Benoit Cloitre, On the fractal behavior of primes, 2011.
L. H. Gallardo and O. Rahavandrainy, There are finitely many even perfect polynomials over F_p with p+1 irreducible divisors, Acta Mathematica Universitatis Comenianae, Vol. 83, No. 2, 2016, 261-275.
David Naccache, Double-Speed Safe Prime Generation, IACR, Report 2003/175, 2003.
Planetmath, Safe prime.
Michael J. Wiener, Safe Prime Generation with a Combined Sieve, IACR, Report 2003/186, 2003.
Wikipedia, Safe prime.

Cf. A007700, A023272, A023302, A023330, A057331, A005602, A207409, A263880.
Except for the initial term, this is identical to A079148.
Cf. A005384, A005383.
Subsequence of A088707.
Primes in A072055.

a005385 n = a005385_list !! (n-1)
a005385_list = filter ((== 1) . a010051 . (`div` 2)) a000040_list
-- Reinhard Zumkeller, Sep 18 2011

[p: p in PrimesUpTo(3000) | IsPrime((p-1) div 2)]; // Vincenzo Librandi, Jul 06 2015

with(numtheory); [ seq(safeprime(i),i=1..3000) ]: convert(%,set); convert(%,list); sort(%);
A005385_list := n->select(i->isprime(iquo(i,2)),select(i->isprime(i),[$1..n])): # Peter Luschny, Nov 08 2010

Select[Prime[Range[1000]],PrimeQ[(#-1)/2]&] (* Zak Seidov, Jan 26 2011 *)

g(n) = forprime(x=2,n,y=x+x+1;if(isprime(y),print1(y","))) \\ Cino Hilliard, Sep 12 2004

[x|x<-primes(10^3), bigomega(x-1)==2] \\ Altug Alkan, Nov 04 2015

from sympy import isprime, primerange
def aupto(limit):
  alst = []
  for p in primerange(1, limit+1):
    if isprime((p-1)//2): alst.append(p)
  return alst
print(aupto(2963)) # Michael S. Branicky, May 07 2021

a(n) = 2 * A005384(n) + 1.

More terms from Larry Reeves (larryr(AT)acm.org), Feb 15 2001

A109287 4-almost primes equal to p*q + 1, where p and q are (not necessarily distinct) primes.

Original entry on oeis.org

16, 36, 40, 56, 88, 135, 156, 184, 204, 210, 220, 248, 250, 260, 296, 306, 315, 328, 330, 340, 342, 372, 414, 459, 472, 490, 516, 536, 546, 580, 584, 636, 650, 686, 690, 708, 714, 732, 735, 738, 804, 808, 819, 836, 850, 852, 870, 872, 940, 950, 966, 975, 996

Offset: 1

Giovanni Teofilatto and Jonathan Vos Post, Aug 20 2005

4-almost primes of the form semiprime + 1.

			a(1) = 16 because (3*5+1)=(2^4) = 16.
a(2) = 36 because (5*7+1)=((2^2)*(3^2)) = 36.
a(3) = 40 because (3*13+1)=((2^3)*5) = 40.
a(4) = 56 because (5*11+1)=((2^3)*7) = 56.
a(5) = 88 because (3*29+1)=((2^3)*11) = 88.
a(6) = 135 because (2*67+1)=((3^3)*5) = 135.
a(7) = 156 because (5*31+1)=((2^2)*3*13) = 156.
a(8) = 184 because (3*61+1)=((2^3)*23) = 184.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Primes are in A000040. Semiprimes are in A001358. 4-almost primes are in A014613.
Primes of the form semiprime + 1 are in A005385 (safe primes).
Semiprimes of the form semiprime + 1 are in A109373.
3-almost primes of the form semiprime + 1 are in A109067.
4-almost primes of the form semiprime + 1 are in this sequence.
5-almost primes of the form semiprime + 1 are in A109383.
Least n-almost prime of the form semiprime + 1 are in A128665.
Similar to A076153; after A076153(0)=3 next difference is A076153(100)=1792 whereas A109287(100)=1794.
Cf. A077065, A079148, A092307, A109288, A109289, A109290.

bo[n_] := Plus @@ Last /@ FactorInteger[n]; Select[Range[1000], bo[ # ] == 4 && bo[ # - 1] == 2 &] (* Ray Chandler, Aug 27 2005 *)

is(n)=bigomega(n)==4 && bigomega(n-1)==2 \\ Charles R Greathouse IV, Sep 16 2015

a(n) is in this sequence iff a(n) is in A014613 and (a(n)-1) is in A001358.

Extended by Ray Chandler, Aug 27 2005

Edited by Ray Chandler, Mar 20 2007

A109373 Semiprimes of the form semiprime + 1.

Original entry on oeis.org

10, 15, 22, 26, 34, 35, 39, 58, 86, 87, 94, 95, 119, 122, 123, 134, 142, 143, 146, 159, 178, 202, 203, 206, 214, 215, 218, 219, 254, 299, 302, 303, 327, 335, 362, 382, 394, 395, 446, 447, 454, 482, 502, 515, 527, 538, 543, 554, 566, 623, 634, 635, 695, 698

Offset: 1

Jonathan Vos Post, Aug 24 2005

			a(1) = 10 because (3*3+1)=(2*5) = 10.
a(2) = 15 because (2*7+1)=(3*5) = 15.
a(3) = 22 because (3*7+1)=(2*11) = 22.
a(4) = 26 because (5*5+1)=(2*13) = 26.
a(5) = 34 because (3*11+1)=(2*17) = 34.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Primes are in A000040. Semiprimes are in A001358.
Primes of the form semiprime + 1 are in A005385 (safe primes).
Semiprimes of the form semiprime + 1 are in this sequence.
3-almost primes of the form semiprime + 1 are in A109067.
4-almost primes of the form semiprime + 1 are in A109287.
5-almost primes of the form semiprime + 1 are in A109383.
Least n-almost prime of the form semiprime + 1 are in A128665.
Cf. A056809, A077065, A079148, A086005, A086006, A092307.
Subsequence of A088707; A064911.

a109373 n = a109373_list !! (n-1)
a109373_list = filter ((== 1) . a064911) a088707_list
-- Reinhard Zumkeller, Feb 20 2012

fQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ Range[ 700], fQ[ # - 1] && fQ[ # ] &] (* Robert G. Wilson v *)
With[{sps=Select[Range[700],PrimeOmega[#]==2&]},Transpose[Select[ Partition[ sps,2,1],#[[2]]-#[[1]]==1&]][[2]]] (* Harvey P. Dale, Sep 05 2012 *)

is(n)=bigomega(n)==2 && bigomega(n-1)==2 \\ Charles R Greathouse IV, Jan 31 2017

a(n) is in this sequence iff a(n) is in A001358 and (a(n)-1) is in A001358.

a(n) = A070552(n) + 1.

Extended by Ray Chandler and Robert G. Wilson v, Aug 25 2005

Edited by Ray Chandler, Mar 20 2007

A079149 Primes p such that either p-1 or p+1 has at most 2 prime factors, counted with multiplicity; i.e., primes p such that either bigomega(p-1) <= 2 or bigomega(p+1) <= 2, where bigomega(n) = A001222(n).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 23, 37, 47, 59, 61, 73, 83, 107, 157, 167, 179, 193, 227, 263, 277, 313, 347, 359, 383, 397, 421, 457, 467, 479, 503, 541, 563, 587, 613, 661, 673, 719, 733, 757, 839, 863, 877, 887, 983, 997, 1019, 1093, 1153, 1187, 1201, 1213, 1237

Offset: 1

Cino Hilliard, Dec 27 2002

There are only 2 primes such that both p-1 and p+1 have at most 2 prime factors - 3 and 5. Proof: If p > 5 then whichever of p-1 and p+1 is divisible by 4 has at least 3 prime factors.

Primes which are not the sum of two consecutive composite numbers. - Juri-Stepan Gerasimov, Nov 15 2009

Union of A079147 and A079148. Cf. A060254, A079152.

Select[Prime[Range[500]],MemberQ[PrimeOmega[{#-1,#+1}],2]&] (* Harvey P. Dale, Sep 04 2011 *)

s(n) = {sr=0; ct=0; forprime(x=2,n, if(bigomega(x-1) < 3 || bigomega(x+1) < 3, print1(x" "); sr+=1.0/x; ct+=1; ); ); print(); print(ct" "sr); } \\ Lists primes p<=n such that p+-1 has at most 2 prime factors.

A079151 Primes p such that p-1 has at most 3 prime factors, counted with multiplicity; i.e., primes p such that bigomega(p-1) = A001222(p-1) <= 3.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 43, 47, 53, 59, 67, 71, 79, 83, 103, 107, 131, 139, 149, 167, 173, 179, 191, 223, 227, 239, 263, 269, 283, 293, 311, 317, 347, 359, 367, 383, 389, 419, 431, 439, 443, 467, 479, 499, 503, 509, 557, 563, 587, 599, 607, 619, 643

Offset: 1

Cino Hilliard, Dec 27 2002

Can it be proved that this is a subsequence of A301590, except for a(5) = 13? (Checked up to A301591(10^4) = 427421.) - M. F. Hasler, Aug 14 2021

			149 is in the sequence because 149 - 1 = 2*2*37 has 3 prime factors.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A079148, A079150, A079152, A079153.

s[n_] := Reap[sr = 0; ct = 0; For[x = 2, x <= n, x = NextPrime[x], If[PrimeOmega[x - 1] < 4, Sow[x]; sr += 1.0/x; ct += 1]]][[2, 1]]; s[700] (* Jean-François Alcover, Jun 08 2013, translated and adapted from Pari *)
Select[Prime[Range[120]],PrimeOmega[#-1]<4&] (* Harvey P. Dale, Oct 02 2017 *)

s(n)=sr=0; ct=0; forprime(x=2,n, if(bigomega(x-1) < 4, print1(x" "); sr+=1.0/x; ct+=1; ); ); print(); print(ct" "sr); \\ Lists primes p<=n such that p-1 has at most 3 prime factors.

list(lim)=my(v=List([2,3]),t); forprime(p=2,(lim-1)\2, if(isprime(t=2*p+1), listput(v,t))); forprime(p=2,(lim-1)\4, forprime(q=2,min(p,(lim-1)\2\p), if(isprime(t=2*p*q+1), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Mar 06 2017

Typos in definition corrected by Harvey P. Dale, Oct 02 2017

A109067 3-almost primes of the form semiprime + 1.

Original entry on oeis.org

27, 50, 52, 63, 66, 70, 75, 78, 92, 116, 124, 130, 147, 170, 186, 188, 195, 207, 222, 236, 238, 255, 266, 268, 275, 279, 290, 292, 310, 322, 356, 363, 366, 387, 399, 404, 412, 418, 423, 428, 438, 452, 455, 470, 474, 483, 494, 498, 506, 518, 530, 534, 539, 555

Offset: 1

Jonathan Vos Post, Aug 24 2005

			a(1) = 27 because (2*13+1)=(3^3) = 27.
a(2) = 50 because (7*7+1)=(2*5^2) = 50.
a(3) = 52 because (3*17+1)=(2^2*13) = 52.
a(4) = 63 because (2*31+1)=(3^2*7) = 63.
a(5) = 66 because (5*13+1)=(2*3*11) = 66.
a(6) = 70 because (3*23+1)=(2*5*7) = 70.
a(7) = 75 because (2*37+1)=(3*5^2) = 75.
a(8) = 78 because (7*11+1)=(2*3*13) = 78.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Primes are in A000040. Semiprimes are in A001358. 3-almost primes are in A014612.
Primes of the form semiprime + 1 are in A005385 (safe primes).
Semiprimes of the form semiprime + 1 are in A109373.
3-almost primes of the form semiprime + 1 are in this sequence.
4-almost primes of the form semiprime + 1 are in A109287.
5-almost primes of the form semiprime + 1 are in A109383.
Least n-almost prime of the form semiprime + 1 are in A128665.
Cf. A077065, A079148, A092307.

f[n_] := Plus @@ Last /@ FactorInteger[n];Select[Range[600], f[ # ] == 3 && f[ # - 1] == 2 &] (* Ray Chandler, Mar 20 2007 *)
Select[Select[Range[600],PrimeOmega[#]==2&]+1,PrimeOmega[#]==3&] (* Harvey P. Dale, Nov 24 2013 *)

list(lim)=my(v=List(),t); forprime(p=2,lim, forprime(q=2,min(p,lim\p), if(bigomega(t=p*q+1)==3, listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

a(n) is in this sequence iff a(n) is in A014612 and a(n)-1 is in A001358.

Edited and extended by Ray Chandler, Mar 20 2007

A079147 Primes p such that p+1 has at most 2 prime factors, counted with multiplicity; i.e., primes p such that bigomega(p+1) = A001222(p+1) <= 2.

Original entry on oeis.org

2, 3, 5, 13, 37, 61, 73, 157, 193, 277, 313, 397, 421, 457, 541, 613, 661, 673, 733, 757, 877, 997, 1093, 1153, 1201, 1213, 1237, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1933, 1993, 2017, 2137, 2341, 2473, 2557, 2593, 2797, 2857, 2917, 3061, 3217, 3253

Offset: 1

Cino Hilliard, Dec 27 2002

Sum of reciprocals ~ 1.266

			157 is in the sequence because 157 + 1 = 2*79 has 2 prime factors.

Except for 2, this is identical to A005383. Cf. A079148, A079149, A079150.

Select[Prime[Range[500]],PrimeOmega[#+1]<3&] (* Harvey P. Dale, May 17 2018 *)

s(n) = {sr=0; forprime(x=2,n, if(bigomega(x+1) < 3, print1(x" "); sr+=1.0/x; ); ); print(); print(sr); } \\ Lists primes p<=n such that p+1 has at most 2 prime factors.

A109383 5-almost primes of the form semiprime + 1.

Original entry on oeis.org

112, 120, 162, 300, 304, 378, 392, 396, 408, 520, 552, 567, 592, 612, 630, 656, 675, 680, 688, 696, 700, 750, 780, 918, 924, 944, 952, 980, 990, 1044, 1100, 1116, 1136, 1140, 1160, 1168, 1170, 1242, 1264, 1272, 1300, 1323, 1352, 1372, 1380, 1386, 1416, 1470

Offset: 1

Jonathan Vos Post, Aug 25 2005

			a(1) = 112 because (3*37)+1 = (2^4) * 7 = 112.
a(2) = 120 because (7*17)+1 = (2^3) * 3 * 5 = 120.
a(3) = 162 because (7*23)+1 = 2 * (3^4) = 162.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Primes are in A000040. Semiprimes are in A001358. 5-almost primes are in A014614.
Primes of the form semiprime + 1 are in A005385 (safe primes).
Semiprimes of the form semiprime + 1 are in A109373.
3-almost primes of the form semiprime + 1 are in A109067.
4-almost primes of the form semiprime + 1 are in A109287.
5-almost primes of the form semiprime + 1 are in this sequence.
Least n-almost prime of the form semiprime + 1 are in A128665.
Cf. A077065, A079148, A092307.

f[n_] := Plus @@ Last /@ FactorInteger[n];Select[Range[1500], f[ # ] == 5 && f[ # - 1] == 2 &] (* Ray Chandler, Mar 20 2007 *)

v=vector(10000);i=0; for(n=1,9e99, if(issemi(n)&bigomega(n+1)==5, v[i++]=n+1;if(i==#v, return))); v \\ Charles R Greathouse IV, Feb 14 2011

a(n) is in this sequence iff a(n) is in A014614 and (a(n)-1) is in A001358.

Extended by Ray Chandler, Mar 20 2007

A194593 Semiprimes s such that phi(s)/2 is prime.

Original entry on oeis.org

9, 10, 14, 22, 46, 94, 118, 166, 214, 334, 358, 454, 526, 694, 718, 766, 934, 958, 1006, 1126, 1174, 1438, 1678, 1726, 1774, 1966, 2038, 2374, 2566, 2614, 2638, 2734, 2878, 2974, 3046, 3238, 3646, 3814, 4054, 4078, 4126, 4198, 4414, 4894, 4918, 5158, 5638, 5758, 5806, 5926, 5998

Offset: 1

Juri-Stepan Gerasimov, Aug 30 2011

nonn

For n > 2, A001221(a(n)) = A001221(A000010(a(n))) = 2, and A008683(a(n)) = A008683(A000010(a(n))) = 1. - Torlach Rush, Aug 23 2018

For n > 1, A000010(a(n)) = A077065(n-1). - Torlach Rush, Sep 11 2018

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

Cf. A000010, A001358, A005384, A005385, A079148, A065966.

[9] cat [2*p: p in PrimesUpTo(3000) | IsPrime((p - 1) div 2)]; // Vincenzo Librandi, Aug 25 2018

9, 10, op(select(s -> isprime(s/2) and isprime((s-2)/4), [seq(s,s=6..10000,8)])); # Robert Israel, Apr 06 2016

Select[Range@ 6000, PrimeOmega@ # == 2 && PrimeQ[EulerPhi[#]/2] &] (* Michael De Vlieger, Apr 06 2016 *)

isok(n) = (bigomega(n)== 2) && isprime(eulerphi(n)/2); \\ Michel Marcus, Apr 06 2016

a(n) = 2*A005385(n-1), n>1.

a(n) = 4*A005384(n-1) + 2, n > 1. - Michel Marcus, Apr 02 2020

Corrected by R. J. Mathar, Oct 13 2011

A087656 Let f be defined on the rationals by f(p/q) =(p+1)/(q+1)=p_{1}/q_{1} where (p_{1},q_{1})=1. Let f^k(p/q)=p_{k}/q_{k} where (p_{k},q_{k})=1. Sequence gives least k such that p_{k}-q_{k} = 1 starting at n.

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 3, 4, 5, 10, 4, 12, 7, 6, 4, 16, 5, 18, 6, 8, 11, 22, 5, 8, 13, 6, 8, 28, 7, 30, 5, 12, 17, 10, 6, 36, 19, 14, 7, 40, 9, 42, 12, 8, 23, 46, 6, 12, 9, 18, 14, 52, 7, 14, 9, 20, 29, 58, 8, 60, 31, 10, 6, 16, 13, 66, 18, 24, 11, 70, 7, 72, 37, 10, 20, 16, 15, 78, 8, 8, 41, 82

Offset: 3

Benoit Cloitre, Oct 04 2003

nonn

Proof that this is the same as A059975 except for offset, from Joseph Myers, Feb 21 2004. Claim: a(n+1) = A059975(n). If p is the least prime factor of n then the rule here gives (n+1)/1 -> (n+2)/2 -> ... -> (n+p)/p = (n/p + 1)/1 so a(n+1) = a(n/p + 1) + (p-1) and clearly A059975(n) = A059975(n/p) + (p-1). The natural start for the induction is A059975(1) = a(2) = 0 (one place before the currently listed sequences start).

			6 -> (6+1)/(1+1) = 7/2 -> (7+1)/(2+1) = 8/3 -> (8+1)/(3+1) = 9/4 -> (9+1)/(4+1) = 2/1 and 2-1 = 1 hence a(6) = 4.

Same as A059975 apart from offset.

a(x)=if(x<0, 0, c=0; while(abs(numerator(x)-denominator(x)-1)>0, x=(numerator(x)+1)/(denominator(x)+1); c++); c)

If p is prime a(p+1)=p-1; it appears that a(n)=(n-1)/2 iff n is in A079148 or in A053177.

cp's OEIS Frontend

A005385 Safe primes p: (p-1)/2 is also prime.

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A109287 4-almost primes equal to p*q + 1, where p and q are (not necessarily distinct) primes.

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A109373 Semiprimes of the form semiprime + 1.

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A079149 Primes p such that either p-1 or p+1 has at most 2 prime factors, counted with multiplicity; i.e., primes p such that either bigomega(p-1) <= 2 or bigomega(p+1) <= 2, where bigomega(n) = A001222(n).

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A079151 Primes p such that p-1 has at most 3 prime factors, counted with multiplicity; i.e., primes p such that bigomega(p-1) = A001222(p-1) <= 3.

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A109067 3-almost primes of the form semiprime + 1.

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A079147 Primes p such that p+1 has at most 2 prime factors, counted with multiplicity; i.e., primes p such that bigomega(p+1) = A001222(p+1) <= 2.