A064370 -id:A064370 - cp's OEIS frontend

A232861 Numbers k with k - 1, k + 1, prime(k) - k, prime(k) + k, kprime(k) - 1, kprime(k) + 1 all prime.

22110, 23742, 128238, 275592, 346560, 1061910, 1281522, 1339002, 1378188, 1461600, 1850130, 2064150, 2354952, 2478270, 2523708, 2689260, 2694300, 3916638, 4422618, 4933530, 6179082, 6541080, 6641562, 6740478, 6759030, 7315812, 8484798, 8711010, 9133308, 9687720

Offset: 1

Zhi-Wei Sun, Dec 01 2013

nonn

Obviously, each term of the sequence is a multiple of 6.
Conjecture: (i) This sequence contains infinitely many terms.
(ii) Let P(x) be a non-constant integer-valued polynomial with positive leading coefficient. Then, there are infinitely many positive integers k with prime(k) - k in the range P(Z) = {P(m): m is an integer}, if and only if the degree of P(x) is at most 3. We may also replace prime(k) - k by prime(k) + k.

			a(1) = 22110 with the six numbers 22110 - 1 = 22109, 22110 + 1 = 22111, prime(22110) - 22110 = 228841, prime(22110) + 22110 = 273061, 22110*prime(22110) - 1 = 5548526609, 22110*prime(22110) + 1 = 5548526611 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000
Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2017.

Cf. A000040, A014574, A014689, A064269, A064270, A064370, A105645, A232443, A232463, A232548.

n=0
Do[If[PrimeQ[k-1]&&PrimeQ[k+1]&&PrimeQ[Prime[k]-k]&& PrimeQ[Prime[k]+k]&& PrimeQ[k*Prime[k]-1]&& PrimeQ[k*Prime[k]+1],n=n+1;Print[n," ",k]],{k,1,9700000}]

A115883 The n-th prime minus n gives a triangular number.

Original entry on oeis.org

1, 2, 4, 5, 7, 13, 34, 37, 46, 62, 104, 111, 210, 259, 274, 296, 306, 439, 488, 502, 513, 751, 763, 817, 969, 998, 1132, 1231, 1405, 1586, 1849, 1982, 2107, 2488, 2578, 2695, 2732, 2752, 2989, 3008, 3079, 3322, 3958, 4201, 4628, 5035, 5594, 5722, 5929

Offset: 1

Giovanni Resta, Feb 06 2006

nonn

			prime(13)-13 = 41-13 = 28 = T(7).

Harvey P. Dale and T. D. Noe, Table of n, a(n) for n = 1..1000 (first 500 terms from Harvey P. Dale)

Cf. A064370, A115882, A115887.

Flatten[Position[Table[Prime[n]-n,{n,6000}],?(IntegerQ[(Sqrt[ 8#+1]- 1)/2]&)]] (* _Harvey P. Dale, Oct 08 2012 *)
pntQ[{a_,b_}]:=OddQ[Sqrt[8(a-b)+1]]; Module[{nn=6000},Select[Thread[ {Prime[ Range[nn]],Range[nn]}],pntQ]][[All,2]] (* Harvey P. Dale, Dec 20 2019 *)

isok(n) = ispolygonal(prime(n) - n, 3); \\ Michel Marcus, Jan 25 2014

A104269 Prime numbers p such that primepi(p) + p is a square.

Original entry on oeis.org

11, 37, 443, 571, 1049, 1307, 1451, 1523, 2837, 3593, 5233, 8539, 9257, 9439, 10391, 10987, 17579, 21881, 23321, 23909, 25117, 30557, 30893, 31231, 42239, 47123, 64811, 65789, 83089, 91631, 92219, 95747, 97549, 99971, 101197, 101807, 110603, 114487, 120431

Offset: 1

Zak Seidov, Feb 26 2005

nonn

A064371(p) + A000040(A064371(p)) = A086968(p)^2.
p^2 is prime + its index A086968; p + p-th prime is a square A064371.
Equals the prime terms of A073945. - Bill McEachen, Oct 26 2021

			37 is a term because 37 is 12th prime and 37 + 12 = 49 = 7^2.

Cf. A000040, A000720, A064371, A086968, A064370, A073945.

q:= n-> isprime(n) and issqr(n+numtheory[pi](n)):
select(q, [$0..150000])[];  # Alois P. Heinz, Oct 27 2021

Select[Prime@Range[10^4],IntegerQ@Sqrt[PrimePi@#+#]&] (* Giorgos Kalogeropoulos, Oct 26 2021 *)

isok(n) = isprime(n) && issquare(n + primepi(n)); \\ Michel Marcus, Oct 05 2013

a(n) = A086968(n)^2 - pi(a(n)).

Definition corrected by Michel Marcus, Oct 05 2013

A105645 Numbers n such that the n-th prime - n is a cube.

Original entry on oeis.org

1, 2, 38, 152, 542, 746, 1632, 2243, 5317, 7520, 15006, 33156, 39925, 101946, 130340, 136572, 331757, 397252, 560017, 722898, 1037524, 1197551, 1737710, 1754109, 2160356, 2217439, 2559702, 2820804, 5173565, 6197364, 7014969, 7597461

Offset: 1

Zak Seidov, May 03 2005

nonn

Corresponding cubes are: {1,1,5,9,15,17,23,26,36,41,53,71,76}^3. Cf. A064370: numbers n such that the n-th prime - n is a square.

Cf. A064370.

Transpose[Select[Table[{Prime[n],n},{n,76*10^5}],IntegerQ[Surd[#[[1]]- #[[2]],3]]&]][[2]] (* Harvey P. Dale, Jun 22 2016 *)

isok(n) = ispower(prime(n)-n, 3); \\ Michel Marcus, Oct 05 2013

a(14)-a(32) from Donovan Johnson, Dec 02 2009

A113410 Numbers n such that n^2 is of the form k-th prime - k for some k.

Original entry on oeis.org

1, 5, 21, 23, 27, 31, 33, 37, 45, 53, 67, 70, 96, 101, 102, 128, 135, 144, 167, 178, 186, 188, 196, 197, 199, 202, 216, 219, 246, 247, 252, 255, 264, 299, 300, 341, 356, 363, 369, 381, 382, 407, 410, 426, 427, 494, 503, 506, 520, 528, 538, 550, 562, 573, 607

Offset: 1

Zak Seidov, Oct 28 2005

nonn

Values of k: A064370. Numbers n such that n^2 is of the form k-th prime + k for some k A086968.

Cf. A064370, A086968.

Sqrt[#]&/@Select[Table[Prime[k]-k,{k,2,40000}],IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Aug 04 2019 *)

A245061 Prime numbers p such that p - primepi(p) is a square, where primepi is the prime counting function.

Original entry on oeis.org

2, 3, 37, 541, 647, 881, 1151, 1301, 1627, 2377, 3271, 5179, 5641, 10501, 11597, 11821, 18503, 20543, 23339, 31259, 35461, 38669, 39499, 42901, 43331, 44201, 45523, 51973, 53407, 67213, 67757, 70489, 72169, 77291, 98893, 99551, 128291, 139721, 145207, 150011

Offset: 1

Chai Wah Wu, Jul 10 2014

nonn

			37 is in the sequence because primepi(37) = 12, and 37 - 12 = 5^2.
541 is in the sequence because primepi(541) = 100, and 541 - 100 = 21^2.
547 is not in the sequence because primepi(547) = 101, and 547 - 101 = 446, which is not a perfect square.

Chai Wah Wu, Table of n, a(n) for n = 1..500

Cf. A104269, A113410, A064370.

with(numtheory): A245061:=n->`if`(type(sqrt(n-pi(n)),integer) and type(n,prime), n, NULL): seq(A245061(n), n=2..10^5); # Wesley Ivan Hurt, Jul 10 2014

Select[Prime[Range[200]], IntegerQ[Sqrt[# - PrimePi[#]]] &] (* Alonso del Arte, Jul 11 2014 *)

select(p->issquare(p-primepi(p)), primes(15000)) \\ Michel Marcus, Jul 11 2014

import sympy,gmpy2
[sympy.prime(n) for n in range(1,10**6) if gmpy2.is_square(sympy.prime(n)-n)] # Chai Wah Wu, Jul 11 2014

a(n) = prime(A064370(n+1)). - Michel Marcus, Jul 11 2014

More terms from Michel Marcus, Jul 11 2014

A114067 The n-th prime minus n gives a fourth power.

Original entry on oeis.org

1, 2, 2603, 4485, 2894125, 8422110, 16832512, 17322137, 20485427, 32550001, 34980568, 49197457, 58325415

Offset: 1

Giovanni Resta, Feb 13 2006

nonn

			p(32550001)-32550001 = 236421376 = 592240896 = 156^4.

Subsequence of A064370 (gives a square).

isok(n) = ispower(prime(n) - n, 4); \\ Michel Marcus, Jan 22 2014

cp's OEIS Frontend

A232861 Numbers k with k - 1, k + 1, prime(k) - k, prime(k) + k, k*prime(k) - 1, k*prime(k) + 1 all prime.

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A115883 The n-th prime minus n gives a triangular number.

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A104269 Prime numbers p such that primepi(p) + p is a square.

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A105645 Numbers n such that the n-th prime - n is a cube.

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A113410 Numbers n such that n^2 is of the form k-th prime - k for some k.

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A245061 Prime numbers p such that p - primepi(p) is a square, where primepi is the prime counting function.

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A114067 The n-th prime minus n gives a fourth power.

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A232861 Numbers k with k - 1, k + 1, prime(k) - k, prime(k) + k, kprime(k) - 1, kprime(k) + 1 all prime.