A104082 -id:A104082 - cp's OEIS frontend

A013602 a(n) = nextprime(4^n)-4^n.

1, 1, 1, 3, 1, 7, 3, 27, 1, 3, 7, 15, 43, 15, 3, 3, 15, 25, 31, 7, 15, 15, 7, 15, 21, 55, 21, 159, 81, 69, 33, 135, 13, 9, 33, 25, 15, 37, 15, 7, 13, 9, 3, 27, 7, 133, 25, 129, 61, 7, 277, 267, 111, 99, 33, 27, 25, 43, 33, 25, 451, 277, 67, 7, 51, 169, 67, 27, 85, 87

Offset: 0

James Kilfiger (mapdn(AT)csv.warwick.ac.uk)

nonn

Daniel Starodubtsev, Table of n, a(n) for n = 0..1000

Cf. A000302, A104082.

Maple
```
seq(nextprime(4^i)-4^i,i=0..100);
```

np4[n_]:=Module[{c=4^n},NextPrime[c]-c]; Array[np4,70,0] (* Harvey P. Dale, Jan 23 2012 *)

a(n) = nextprime(4^n)-4^n; \\ Michel Marcus, Aug 13 2019

a(n) = A104082(n) - A000302(n). - Michel Marcus, Aug 13 2019

a(n) = A013597(2*n), n >= 0. - A.H.M. Smeets, Aug 13 2019

A104089 Largest prime <= 4^n.

Original entry on oeis.org

3, 13, 61, 251, 1021, 4093, 16381, 65521, 262139, 1048573, 4194301, 16777213, 67108859, 268435399, 1073741789, 4294967291, 17179869143, 68719476731, 274877906899, 1099511627689, 4398046511093, 17592186044399, 70368744177643

Offset: 1

Cino Hilliard, Mar 03 2005

Cf. A014234, A104082, A104088, A104090-A104094, A003618, A104096.

Cf. A000040, A000302, A007917.

NextPrime[4^Range[30], -1] (* Paolo Xausa, Oct 28 2024 *)

g(n,b) = for(x=0,n,print1(precprime(b^x)","))

a(n) = A007917(A000302(n)). - Paolo Xausa, Oct 28 2024

A340959 Table read by antidiagonals of the smallest prime >= n^k, n >= 1 and k >= 0.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 3, 5, 2, 2, 5, 11, 11, 2, 2, 5, 17, 29, 17, 2, 2, 7, 29, 67, 83, 37, 2, 2, 7, 37, 127, 257, 251, 67, 2, 2, 11, 53, 223, 631, 1031, 733, 131, 2, 2, 11, 67, 347, 1297, 3137, 4099, 2203, 257, 2, 2, 11, 83, 521, 2411, 7789, 15629, 16411, 6563

Offset: 1

Donald S. McDonald, Jan 31 2021

			Table begins:
  2, 2,  2,   2,   2,    2, ...
  2, 2,  5,  11,  17,   37, ...
  2, 3, 11,  29,  83,  251, ...
  2, 5, 17,  67, 257, 1031, ...
  2, 5, 29, 127, 631, 3137, ...
  ...;
yielding the triangle:
  2;
  2, 2;
  2, 2,  2;
  2, 3,  5,  2;
  2, 5, 11, 11,  2;
  2, 5, 17, 29, 17, 2;
  ...

Cf. A104080 (n=2), A104081 (n=3), A104082 (n=4), A104083 (n=5), A104084 (n=7).

T[n_,k_]:=NextPrime[n^k-1];Flatten[Table[T[n-k,k],{n,11},{k,0,n-1}]] (* Stefano Spezia, Feb 01 2021 *)

T(n,k) = nextprime(n^k); \\ Michel Marcus, Feb 01 2021

T(n,k) = next_prime(n^k-1).

A361173 Numbers k such that, in base 4, the greatest prime less than 4^k and the least prime greater than 4^k have no common digit.

Original entry on oeis.org

1, 4, 28, 83, 1816

Offset: 1

Lewis Baxter, Mar 02 2023

In base 4 all consecutive primes with no common digit are of this form, except for 2 and 3.
It is unknown whether this sequence is infinite.
Base 2 and base 3 have no such primes.

			k=4 is a term: the consecutive primes are 251 and 257. In base 4 their representations are 3323 and 10001, which have no common digit.

Mathematics StackExchange, Are there an infinity of disjoint consecutive primes?

Cf. A104082, A104089, A068802, A156981.

Select[Range[100], ! IntersectingQ @@ IntegerDigits[NextPrime[4^#, {-1, 1}], 4] &] (* Amiram Eldar, Mar 03 2023 *)

isok(k) = #setintersect(Set(digits(precprime(4^k), 4)), Set(digits(nextprime(4^k), 4))) == 0; \\ Michel Marcus, Mar 03 2023

from sympy.ntheory import digits, nextprime, prevprime
def ok(n):
    p, q = prevprime(4**n), nextprime(4**n)
    return set(digits(p, 4)[1:]) & set(digits(q, 4)[1:]) == set()
print([k for k in range(1, 99) if ok(k)]) # Michael S. Branicky, Mar 03 2023

cp's OEIS Frontend

A013602 a(n) = nextprime(4^n)-4^n.

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A104089 Largest prime <= 4^n.

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A340959 Table read by antidiagonals of the smallest prime >= n^k, n >= 1 and k >= 0.

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Mathematica

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A361173 Numbers k such that, in base 4, the greatest prime less than 4^k and the least prime greater than 4^k have no common digit.

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Mathematica

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Python