A086173 -id:A086173 - cp's OEIS frontend

A086172 Numbers n such that n*prime(n)+1 is prime.

1, 2, 4, 6, 24, 30, 36, 42, 72, 84, 98, 100, 116, 126, 128, 144, 162, 174, 206, 212, 228, 288, 312, 318, 324, 336, 350, 360, 406, 408, 416, 418, 452, 458, 466, 498, 510, 522, 528, 530, 536, 546, 548, 556, 604, 654, 660, 666, 682, 684, 690, 708, 710, 728, 738

Offset: 1

Zak Seidov, Jul 11 2003

See also A086173, A086174, A086175, A086176, A086177.

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

Cf. A086173, 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]&]

is(n,p=prime(n))=isprime(n*p+1) \\ Charles R Greathouse IV, Feb 17 2017

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 *)

A096065 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

5, 151, 443, 601, 1097, 3389, 13553, 20921, 25847, 32719, 41669, 46153, 56813, 66109, 84691, 119087, 129449, 150559, 188857, 201847, 206273, 240173, 266863, 271109, 310577, 318751, 346553, 395749, 430819, 446951, 451933, 507079, 645527

Offset: 0

Alonso del Arte, Jul 20 2004

nonn

			a(3)=443 because the twelfth prime is 37 and 12 * 37 - 1 = 443.

Cf. A086173, A096064.

Select[Table[Prime[n]*n - 1, {n, 320}], PrimeQ]

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}]

A288491 Numbers k such that k^k*prime(k) - 1 is prime.

Original entry on oeis.org

2, 6, 40, 56, 954

Offset: 1

Vincenzo Librandi, Jun 10 2017

a(6) > 22000. - Giovanni Resta, Jun 12 2017

a(6) > 40000. - Michael S. Branicky, Jan 05 2025

			a(1) = 2 because 3*2^2-1 = 11 a prime.
a(2) = 6 because 13*6^6-1 = 606527 a prime.

Cf. A086173.

[n: n in [1..1000] | IsPrime(NthPrime(n)*n^n-1)];

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

cp's OEIS Frontend

A086172 Numbers n such that n*prime(n)+1 is prime.

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

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

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

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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

Mathematica

A096065 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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A232442 a(n) = |{0 < k < n: m*prime(m) - 1 and m*prime(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.

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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.