A157480 -id:A157480 - cp's OEIS frontend

A251859 First appearance of prime(n) in A157480.

1, 6, 5, 10, 3, 2, 8, 7, 13, 20, 30, 28, 27, 26, 35, 58, 45, 56, 55, 21, 96, 142, 53, 93, 262, 14, 139, 12, 195, 47, 87, 57, 214, 404, 133, 255, 81, 252, 37, 36, 187, 128, 127, 479, 75, 572, 477, 313, 70, 475, 179, 68, 241, 310, 19, 98, 115, 469, 762, 114, 234, 94, 302, 231, 1238, 229, 298, 376, 50, 161

Offset: 1

Zak Seidov, Dec 10 2014

nonn

Apparently all primes eventually appear in A157480. Note that this sequence is one-to-one map while A157480 not.

			a(3)=5 because A157480(5)=prime(3)=5 and prime(3)+prime(5)=5+11=16=4^2,
a(4)=10 because A157480(10)=prime(4)=7 and prime(4)+prime(10)=7+29=36=6^2.

Cf. A157480.

A157480(a(n)) = prime(n), or prime(a(n))+prime(n) is a square.

A259226 Numbers n such that A157480(n) = A259232(n).

Original entry on oeis.org

2, 3, 7, 12, 14, 19, 29, 36, 37, 38, 42, 46, 50, 67, 73, 74, 82, 84, 106, 110, 112, 125, 134, 143, 157, 168, 169, 177, 183, 202, 222, 226, 232, 249, 263, 275, 278, 282, 314, 327, 355, 369, 399, 433, 457, 464, 483, 557, 617, 685, 820, 826, 835, 838, 935, 937, 1031, 1059, 1080, 1087, 1112, 1205, 1276, 1296, 1316, 1349, 1421, 1503, 1505, 1713, 1827, 2017, 2085, 2092, 2263, 2337, 2357, 2498, 2591, 2594, 2634, 2676, 2771, 2984, 3128, 3776, 3777, 3928, 4382, 5037, 5319, 5448, 5621, 5839, 6137, 6183, 6521, 7373, 9867

Offset: 1

Zak Seidov, Jun 22 2015

Or numbers n such that A157480(n) > A000040(n).

It is a strong conjecture that the sequence is complete, and the last term is a(99)=9867.

Cf. A000040, A157480, A259232.

A259232 Smallest prime q > p such that q + p is a square, where p is the n-th prime.

Original entry on oeis.org

7, 13, 11, 29, 53, 23, 19, 557, 41, 71, 113, 107, 59, 101, 53, 271, 137, 83, 257, 73, 251, 821, 113, 107, 227, 223, 797, 149, 467, 211, 197, 193, 263, 761, 251, 173, 167, 1601, 233, 227, 397, 719, 293, 383, 379, 701, 1553, 353, 257, 347, 251, 337, 659, 773, 419, 313, 307, 1493, 1019, 503, 293

Offset: 1

Zak Seidov, Jun 22 2015

nonn

Corresponding squares a(n)+prime(n): 9,16,16,36,64,36,36,576,64,100.

Also, a(n) >= A157480(n).

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A000040, A157480.

Table[p=Prime[n];x=1+Floor[Sqrt[2*p]];While[!PrimeQ[q=x^2-p],x++];q,{n,100}]

a(n)=p = prime(n); k = nextprime(p+1); while(!issquare(p+k), k = nextprime(k+1)); k; \\ Michel Marcus, Jun 22 2015

a(n,p=prime(n))=my(s=sqrtint(2*p)); while(!isprime(s++^2-p),); s^2-p \\ Charles R Greathouse IV, May 06 2016

A259237 a(n) = least prime q such that q + prime(n) is a cube.

Original entry on oeis.org

727, 5, 3, 1721, 53, 499, 47, 197, 41, 971, 1697, 179, 23, 173, 17, 11, 5, 3, 149, 929, 439, 137, 4013, 127, 2647, 1627, 113, 109, 107, 103, 89, 1597, 79, 373, 67, 2593, 59, 53, 3929, 43, 37, 331, 809, 23, 19, 17, 5, 2521, 773, 283, 3863, 761, 271, 5581, 743, 3833

Offset: 1

Zak Seidov, Jun 22 2015

Corresponding values of (a(n)+prime(n))^(1/3): 9,2,2,12,4,8,4,6,4,10,12,6,4,6,4,4,4,4,6,10,8,6,16,6,14,12,6,6,6,6,6.

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

Cf. A157480, A259232.

f:= proc(n) local p,k;
  p:= ithprime(n);
  for k from ceil(p^(1/3)) do
    if isprime(k^3 - p) then return k^3 - p fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Oct 17 2023

Table[p=Prime[n];x=Ceiling[p^(1/3)];While[!PrimeQ[q=x^3-p],x++];q,{n,100}]

a(n) = {p = prime(n); k=2; while(!ispower(p+k,3), k = nextprime(k+1)); k;} \\ Michel Marcus, Jun 22 2015

cp's OEIS Frontend

A251859 First appearance of prime(n) in A157480.

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A259226 Numbers n such that A157480(n) = A259232(n).

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A259232 Smallest prime q > p such that q + p is a square, where p is the n-th prime.

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A259237 a(n) = least prime q such that q + prime(n) is a cube.

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