A086172 -id:A086172 - cp's OEIS frontend

A086173 Numbers k such that k*prime(k)-1 is prime.

2, 8, 12, 14, 18, 30, 54, 66, 72, 80, 90, 94, 102, 110, 124, 144, 150, 160, 178, 184, 186, 198, 208, 210, 222, 224, 234, 250, 260, 264, 266, 280, 312, 336, 342, 370, 390, 400, 414, 432, 450, 462, 468, 470, 472, 476, 510, 564, 570, 596, 598, 600, 616, 652, 690

Offset: 1

Zak Seidov, Jul 11 2003

See also A086172, A086174, A086175, A086176, A086177.

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

Cf. A086172, A086174, A086175, A086176, A086177.

[n: n in [1..800] | IsPrime(n*NthPrime(n) - 1)]; // Vincenzo Librandi, Oct 05 2012

Select[Range[1000], PrimeQ[ # Prime[ # ]-1]&]

A086174 Numbers n such that n*prime(n)+2 is a prime.

Original entry on oeis.org

3, 29, 33, 45, 65, 81, 91, 93, 95, 101, 103, 105, 109, 123, 153, 155, 189, 201, 225, 251, 253, 273, 283, 291, 305, 321, 363, 367, 371, 375, 387, 429, 431, 469, 475, 501, 515, 517, 525, 541, 567, 601, 613, 627, 633, 643, 669, 675, 701, 715, 717, 723, 729, 735

Offset: 1

Zak Seidov, Jul 11 2003

See also A086172, A086173, A086175, A086176, A086177.

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

Cf. A086172, A086173, A086175, A086176, A086177.

[n: n in [1..800] | IsPrime(n*NthPrime(n) + 2)]; // Vincenzo Librandi, Oct 05 2012

Select[Range[1000], PrimeQ[ # Prime[ # ]+2]&]

A086175 Numbers n such that n*prime(n)-2 is prime.

Original entry on oeis.org

3, 5, 21, 23, 25, 33, 37, 45, 57, 81, 83, 85, 93, 121, 123, 133, 137, 173, 183, 187, 193, 195, 215, 219, 225, 231, 245, 247, 285, 289, 295, 301, 315, 317, 327, 329, 353, 357, 359, 391, 395, 403, 419, 423, 429, 435, 447, 477, 479, 503, 513, 549, 561, 567, 571

Offset: 1

Zak Seidov, Jul 11 2003

See also A086172, A086173, A086174, A086176, A086177.

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

Cf. A086172, A086173, A086174, A086176, A086177.

[n: n in [1..800] | IsPrime(n*NthPrime(n) - 2)]; // Vincenzo Librandi, Oct 05 2012

Select[Range[1000], PrimeQ[ # Prime[ # ]-2]&]

A086176 Numbers n such that n*prime(n)+3 is prime.

Original entry on oeis.org

1, 4, 10, 20, 22, 28, 34, 38, 46, 56, 62, 92, 98, 112, 146, 148, 154, 166, 170, 176, 178, 200, 208, 254, 256, 260, 262, 266, 284, 340, 346, 352, 364, 394, 406, 412, 418, 460, 476, 500, 514, 524, 548, 550, 560, 574, 584, 586, 590, 610, 614, 620, 656, 664, 698

Offset: 1

Zak Seidov, Jul 11 2003

See also A086172, A086173, A086174, A086175, A086177.

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

Cf. A086172, A086173, A086174, A086175, A086177.

[n: n in [1..800] | IsPrime(n*NthPrime(n) + 3)]; // Vincenzo Librandi, Oct 05 2012

Select[Range[1000], PrimeQ[ # Prime[ # ]+3]&]

A086177 Numbers n such that n*prime(n)-3 is prime.

Original entry on oeis.org

2, 8, 14, 34, 40, 46, 50, 80, 82, 88, 110, 116, 118, 130, 142, 200, 224, 226, 238, 256, 274, 278, 280, 286, 292, 302, 322, 328, 332, 350, 352, 362, 380, 398, 412, 430, 436, 464, 496, 544, 572, 586, 616, 620, 622, 634, 638, 646, 650, 662, 676, 688, 700, 734

Offset: 1

Zak Seidov, Jul 11 2003

See also A086172, A086173, A086174, A086175, A086176.

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

Cf. A086172, A086173, A086174, A086175, A086176.

[n: n in [1..800] | IsPrime(n*NthPrime(n) - 3)]; // Vincenzo Librandi, Oct 05 2012

Select[Range[1000], PrimeQ[ # Prime[ # ]-3]&]

A085637 Numbers k such that k*prime(k) -+ 1 are twin primes.

Original entry on oeis.org

2, 30, 72, 144, 312, 336, 510, 690, 990, 1122, 1254, 1272, 1410, 2082, 2376, 2508, 2586, 2664, 2802, 3060, 3096, 3180, 3432, 3510, 3684, 4062, 4506, 5526, 5790, 6174, 7224, 8064, 8388, 9078, 9390, 9504, 10698, 10794, 10884, 10992, 11046, 11334

Offset: 1

Zak Seidov, Jul 11 2003

Intersection of A086172 and A086173. See also A086174, A086175, A086176, A086177.

			k=30 is a term because 30*prime(30) +- 1 = 3390 +- 1 are twin primes.

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

Cf. A086172, A086173, A086174, A086175, A086176, A086177.

[n: n in [1..11500] | IsPrime(n*NthPrime(n) - 1) and IsPrime(n*NthPrime(n) + 1) ]; // Vincenzo Librandi, Oct 05 2012

Select[Range[15000], PrimeQ[ # Prime[ # ] - 1] && PrimeQ[ # Prime[ # ] + 1] &]
Select[Range[12000],AllTrue[#*Prime[#]+{1,-1},PrimeQ]&] (* Harvey P. Dale, Mar 21 2025 *)

A096064 Let p(k) = k-th prime; sequence gives primes q of the form q = k*p(k) + 1 for some k.

Original entry on oeis.org

3, 7, 29, 79, 2137, 3391, 5437, 7603, 25849, 36373, 51059, 54101, 74357, 88327, 92033, 119089, 154387, 179743, 263063, 275813, 328093, 540577, 645529, 671299, 694333, 761713, 824951, 872281, 1133147, 1142809, 1190177, 1206767, 1442333

Offset: 1

Alonso del Arte, Jul 20 2004

nonn

			29 is in the sequence because the fourth prime is 7 and 4 * 7 + 1 = 29.

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

Cf. A086172, A096065.

[a: n in [0..500] | IsPrime(a) where a is NthPrime(n)*n + 1 ]; // Vincenzo Librandi, Oct 05 2012

Select[Table[Prime[n]*n + 1, {n, 455}], PrimeQ]

Offset changed from 0 to 1 by Vincenzo Librandi, Oct 05 2012

A192611 Primes prime(k) such that prime(k)*k+1 is also prime.

Original entry on oeis.org

2, 3, 7, 13, 89, 113, 151, 181, 359, 433, 521, 541, 641, 701, 719, 827, 953, 1033, 1277, 1301, 1439, 1877, 2069, 2111, 2143, 2267, 2357, 2423, 2791, 2801, 2861, 2887, 3191, 3251, 3313, 3557, 3643, 3739, 3797, 3821, 3863, 3931, 3947, 4021, 4447

Offset: 1

Andrea Raffetti, Jul 05 2011

nonn

Primes p such that p*pi(p)+1 is prime (see Crossrefs).

			13 is in the list because, being the 6th prime, 13*6+1=79 is prime.

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

Cf. A096064, A086172.
Cf. A000040 The prime numbers.
Cf. A000720 pi(n), the number of primes <= n.

[ NthPrime(n): n in [1..650] | IsPrime(NthPrime(n)*n+1) ]; // Bruno Berselli, Jul 05 2011

Select[Prime[Range[700]],PrimeQ[# PrimePi[#]+1]&] (* Harvey P. Dale, May 28 2012 *)

k=0;forprime(p=2,1e4,if(isprime(k++*p+1),print1(p", "))) \\ Charles R Greathouse IV, Jul 05 2011

A232442 a(n) = |{0 < k < n: mprime(m) - 1 and mprime(m) + 1 are both prime with m = sigma(k) + phi(n-k)}|, where sigma(k) is the sum of all positive divisors of k and phi(.) is Euler's totient function.

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 2, 1, 1, 0, 0, 1, 1, 6, 1, 2, 2, 0, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 4, 1, 1, 0, 1, 2, 2, 2, 4, 0, 0, 1, 2, 0, 3, 3, 3, 2, 0, 1, 1, 2, 1, 2, 0, 1, 1, 14, 3, 2, 2, 2, 2, 3, 4, 5, 3, 2, 3, 1, 3, 3, 4, 6, 3, 0, 5, 3, 1, 0, 5, 2, 0, 3, 6, 1

Offset: 1

Zhi-Wei Sun, Jan 14 2014

nonn

Conjecture: a(n) > 0 for all n > 214.
This implies that there are infinitely many twin prime pairs of the special form {m*prime(m) - 1, m*prime(m) + 1}.
We have verified the conjecture for n up to 10^5.

			a(25) = 1 since sigma(6) + phi(19) = 12 + 18 = 30 with {30*prime(30) - 1, 30*prime(30) + 1} = {3389, 3391} a twin prime pair.
a(100) = 1 since sigma(75) + phi(25) = 124 + 20 = 144 with {144*prime(144) - 1, 144*prime(144) + 1} = {119087, 119089} a twin prime pair.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A000203, A085637, A086172, A086173, A232270, A232861, A233544, A233539.

sigma[n_]:=DivisorSigma[1,n]
q[n_]:=PrimeQ[n*Prime[n]-1]&&PrimeQ[n*Prime[n]+1]
f[n_,k_]:=sigma[k]+EulerPhi[n-k]
a[n_]:=Sum[If[q[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A277124 Numbers k such that (k+1)*prime(k) + k is a prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 11, 13, 14, 20, 21, 26, 29, 30, 33, 35, 36, 39, 41, 43, 49, 54, 55, 62, 68, 69, 75, 76, 79, 81, 89, 90, 105, 110, 113, 117, 119, 126, 134, 141, 146, 154, 162, 174, 176, 178, 179, 186, 191, 195, 207, 209, 215, 216, 222, 225, 227, 230, 231, 234, 237

Offset: 1

Alex Ratushnyak, Sep 30 2016

nonn

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

Cf. A000040, A086172.

p:= 1: count:= 0: Res:= NULL:
for n from 1 while count < 100 do
  p:= nextprime(p);
  if isprime(n*p+n+p) then
    count:= count+1; Res:= Res, n
  fi
od:
Res; # Robert Israel, Oct 25 2018

npQ[n_]:=Module[{p=Prime[n]},PrimeQ[n*p+n+p]]; Select[Range[250],npQ] (* Harvey P. Dale, Mar 27 2022 *)

is(n) = isprime((n + 1) * (prime(n) + 1) - 1); \\ Altug Alkan, Oct 01 2016

cp's OEIS Frontend

A086173 Numbers k such that k*prime(k)-1 is prime.

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Magma

Mathematica

A086174 Numbers n such that n*prime(n)+2 is a prime.

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Magma

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A086175 Numbers n such that n*prime(n)-2 is prime.

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Magma

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A086176 Numbers n such that n*prime(n)+3 is prime.

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Magma

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A086177 Numbers n such that n*prime(n)-3 is prime.

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Magma

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A085637 Numbers k such that k*prime(k) -+ 1 are twin primes.

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Magma

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A096064 Let p(k) = k-th prime; sequence gives primes q of the form q = k*p(k) + 1 for some k.

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Magma

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A192611 Primes prime(k) such that prime(k)*k+1 is also prime.

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A232442 a(n) = |{0 < k < n: mprime(m) - 1 and mprime(m) + 1 are both prime with m = sigma(k) + phi(n-k)}|, where sigma(k) is the sum of all positive divisors of k and phi(.) is Euler's totient function.