A011757 -id:A011757 - cp's OEIS frontend

A347000 The (m^n)-th prime, written as square array T(n,m) read by falling antidiagonals.

2, 3, 2, 5, 7, 2, 7, 23, 19, 2, 11, 53, 103, 53, 2, 13, 97, 311, 419, 131, 2, 17, 151, 691, 1619, 1543, 311, 2, 19, 227, 1321, 4637, 8161, 5519, 719, 2, 23, 311, 2309, 10627, 28687, 38873, 19289, 1619, 2, 29, 419, 3671, 21391, 79349, 171529, 180503, 65687, 3671, 2

Offset: 1

Hugo Pfoertner, Aug 10 2021

			The array begins
  2   3     5      7     11      13       17 ...
  2   7    23     53     97     151      227 ...
  2  19   103    311    691    1321     2309 ...
  2  53   419   1619   4637   10627    21391 ...
  2 131  1543   8161  28687   79349   185707 ...
  2 311  5519  38873 171529  567871  1549817 ...
  2 719 19289 180503 994837 3950183 12579617 ...

Hugo Pfoertner, Table of k, a(k) for k = 1..351, antidiagonals for m+n<=26, flattened.

Cf. A000040, A033844, A038833, A119772, A011757, A055875, A109791, A062448.

Cf. A003320, A051129.

T[n_,m_]:=Prime[m^n];Flatten[Table[Reverse[Table[T[n-m+1,m],{m,n}]],{n,10}]] (* Stefano Spezia, Aug 10 2021 *)

A067853 Numbers k such that k divides prime(k^2)+1.

Original entry on oeis.org

1, 2, 3, 8, 12, 37, 72, 73, 95, 472, 582, 735, 745, 1293, 1295, 1354, 2886, 6088, 7418, 7650, 8922, 9348, 9584, 14592, 25212, 34997, 36360, 102226, 114713, 160044, 190402, 376024

Offset: 1

Joseph L. Pe, Feb 15 2002

nonn

a(32) > 3*10^5. [Donovan Johnson, Nov 15 2009]

			8 divides prime(8^2)+1 = 312 = 39*8, so 8 is a term of the sequence.

Cf. A011757.

Select[Range[10^4], Mod[Prime[ #^2] + 1, # ] == 0 &]

a(24)-a(31) from Donovan Johnson, Nov 15 2009

a(32) from Amiram Eldar, Jul 23 2021

A109801 Cumulative sum of squares of primes indexed by squares.

Original entry on oeis.org

4, 53, 582, 3391, 12800, 35601, 87130, 183851, 359412, 652093, 1089014, 1772943, 2791024, 4214273, 6250602, 8871763, 12402404, 16994853, 22933822, 30446903, 39951792, 51930313, 66393122, 84125643, 105627412, 131140013, 161599374

Offset: 1

Jonathan Vos Post, Aug 15 2005

nonn

Related to Prime(1^2) + prime(2^2) + ... + prime(n^2) (A109724).

			a(1) = 4 because (prime[1^2])^2 = (prime[1])^2 = 2^2.
a(2) = 53 because (prime[1^2])^2 + (prime[2^2])^2 = 2^2 + 7^2 = 4 + 49 = 53 (which is prime).
a(3) = 582 because (prime[1^2])^2 + (prime[2^2])^2 + (prime[3^2])^2 = 2^2 + 7^2 + 23^2 = 582.
a(4) = 582 because (prime[1^2])^2 + (prime[2^2])^2 + (prime[3^2])^2 + (prime[4^2])^2 = 2^2 + 7^2 + 23^2 + 53^2 = 3391 (which is prime).
a(32) = a(31) + (prime[32^2])^2 = 345995122 + 8161^2 = 412597043 (which is prime).
a(34) = a(33) + (prime[34^2])^2 = 488932212 + 9341^2 = 576186493 (which is prime).

Cf. A000040, A000290, A000583, A011757, A109724, A109770.

Accumulate[Prime[Range[30]^2]^2] (* Harvey P. Dale, Mar 28 2012 *)

(Prime[1^2])^2 + (prime[2^2])^2 + ... + (prime[n^2])^2. a(n+1) = a(n) + (A011757(n+1))^2.

A145317 Numbers n such that prime[(n + 1)^2] - prime[n^2] is a perfect cube.

Original entry on oeis.org

79, 10288, 16181, 306998, 394021

Offset: 1

Zak Seidov, Oct 07 2008

nonn

n such that A011757(n + 1) - A011757(n) is a perfect cube

			p1=prime[(n + 1)^2], p= prime[n^2], p1-p2=q^3: n = 79, p1 = 63809, p = 62081, q = 12; n = 10288, p1 = 2163941687, p = 2163502711, q = 76; n = 16181, p1 = 5602364903, p = 5601635903, q = 90; n = 306998, p1=2596139184841, p = 2596121608841, q = 260. n = 394021, p1 = 4357072887373, p = 4357049981069, q = 284. Next n, if any, is > 500000.

A145290 prime[(n + 1)^2] - prime[n^2] is a perfect square, A011757 prime(n^2).

Select[Range[400000],IntegerQ[(Prime[(#+1)^2]-Prime[#^2])^(1/3)]&] (* Harvey P. Dale, Dec 13 2012 *)

A254955 Prime numbers indexed by oblong numbers.

Original entry on oeis.org

3, 13, 37, 71, 113, 181, 263, 359, 463, 601, 743, 911, 1091, 1291, 1511, 1747, 2017, 2297, 2617, 2903, 3271, 3617, 4003, 4409, 4831, 5297, 5743, 6247, 6761, 7297, 7853, 8443, 9029, 9631, 10271, 10973, 11717, 12413, 13109, 13879, 14717, 15461, 16301, 17191, 18059

Offset: 1

Waldemar Puszkarz, Feb 11 2015

nonn

			a(1) = prime(1 + 1^2) = prime(2) = 3.
a(2) = prime(2 + 2^2) = prime(6) = 13.

Cf. A000040, A002378 (n*(n+1)), A011756 (prime(n(n+1)/2)), A011757 (prime(n^2)).

[NthPrime(n+n^2): n in [1..50]]; // Vincenzo Librandi, Feb 24 2015

Table[Prime[n + n^2], {n, 100}] (* Puszkarz *)
Prime[2Accumulate[Range[40]]] (* Alonso del Arte, Feb 11 2015 *)

vector(80, n, prime(n+n^2)) \\ Michel Marcus, Feb 12 2015

a(n) = prime(n + n^2) = A000040(A002378(n)).

A259648 a(n) = floor( prime(n)^3 / (n*log(n)) ).

Original entry on oeis.org

19, 37, 61, 165, 204, 360, 412, 615, 1059, 1129, 1698, 2066, 2151, 2555, 3356, 4264, 4362, 5376, 5973, 6084, 7250, 7928, 9242, 11341, 12162, 12279, 13129, 13261, 14141, 19242, 20270, 22285, 22399, 26583, 26688, 28965, 31330, 32597, 35090, 37668, 37773, 43082

Offset: 2

Ilya Gutkovskiy, Jul 02 2015

nonn

Cf. A024702, A011757, A138672.

[Floor((NthPrime(n))^3/(n*Log(n))): n in [2..60]]; // Vincenzo Librandi, Jul 03 2015

Table[Floor[Prime[n]^3/(n Log[n])], {n, 2, 30}]

a(n) = floor(prime(n)^3/(n*log(n))); \\ Michel Marcus, Jul 02 2015

a(n) = floor( A030078(n) / (n*log(n))).

A299118 Squares s such that prime(s) + 2 is a square.

Original entry on oeis.org

1, 4, 9, 7745089

Offset: 1

Waldemar Puszkarz, Feb 02 2018

Primes corresponding to the first four squares are 2, 7, 23, and 136866599. The sequence may be finite.
There may be no square s such that prime(s) + 1 is square (none was found up to 10^9).
This is a Diophantine problem of the form f(n^2) + A = m^2, where f(x) = prime(x), and the simplest case of A = 1 has probably no solutions unlike the same case with f(x) = primepi(x) that may even have an infinite number of solutions.

			prime(4) + 2 = 7 + 2 = 9, and both 4 and 9 are squares.

Cf. A000290 (squares), A000040 (primes), A011757 (primes with square indices).

Select[Range[10^4]^2, IntegerQ@Sqrt[Prime[#] + 2] &]

for(n=1, 10^4, issquare(prime(n^2)+2)&&print1(n^2 ", "))

cp's OEIS Frontend

A347000 The (m^n)-th prime, written as square array T(n,m) read by falling antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A067853 Numbers k such that k divides prime(k^2)+1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A109801 Cumulative sum of squares of primes indexed by squares.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A145317 Numbers n such that prime[(n + 1)^2] - prime[n^2] is a perfect cube.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A254955 Prime numbers indexed by oblong numbers.

Views

Author

Keywords

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A259648 a(n) = floor( prime(n)^3 / (n*log(n)) ).

Views

Author

Keywords

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A299118 Squares s such that prime(s) + 2 is a square.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI