A104089 -id:A104089 - cp's OEIS frontend

A104090 Largest prime <= 5^n.

5, 23, 113, 619, 3121, 15619, 78121, 390581, 1953109, 9765619, 48828113, 244140613, 1220703073, 6103515623, 30517578121, 152587890563, 762939453109, 3814697265523, 19073486328109, 95367431640599, 476837158203071, 2384185791015571, 11920928955078089

Offset: 1

Cino Hilliard, Mar 03 2005

Cf. A014234, A054321, A104088, A104089, A104091-A104094, A003618, A104096.

Cf. A000040, A000351, A007917.

NextPrime[5^Range[30] + 1, -1] (* Paolo Xausa, Oct 29 2024 *)

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

a(n) = A007917(A000351(n)). - Paolo Xausa, Oct 29 2024

A013606 a(n) = 4^n - prevprime(4^n).

Original entry on oeis.org

1, 3, 3, 5, 3, 3, 3, 15, 5, 3, 3, 3, 5, 57, 35, 5, 41, 5, 45, 87, 11, 17, 21, 59, 27, 47, 33, 5, 27, 93, 57, 59, 5, 23, 35, 93, 35, 15, 11, 65, 57, 35, 35, 299, 33, 83, 3, 17, 51, 15, 33, 17, 117, 59, 21, 75, 11, 3, 5, 119, 3, 59, 137, 159, 5, 347, 45, 113, 105, 27, 111

Offset: 1

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

nonn

Daniel Starodubtsev, Table of n, a(n) for n = 1..1000

Cf. A000302, A104089.

Maple
```
seq(4^i-prevprime(4^i),i=1..100);
```

#-NextPrime[#,-1]&/@(4^Range[80]) (* Harvey P. Dale, Dec 13 2016 *)

a(n) = 4^n - precprime(4^n); \\ Michel Marcus, Sep 01 2019

a(n) = A000302(n) - A104089(n). - Michel Marcus, Sep 01 2019

A365265 Numbers k for which sqrt(k/2) divides k and the symmetric representation of sigma(k) consists of a single part and its width at the diagonal equals 1.

Original entry on oeis.org

2, 8, 18, 32, 128, 162, 200, 392, 512, 882, 968, 1352, 1458, 2048, 2178, 3042, 3872, 5000, 5202, 5408, 6498, 8192, 9248, 9522, 11552, 13122, 15138, 16928, 17298, 19208, 26912, 30752, 32768, 36992, 43218, 43808, 46208, 53792, 58482, 59168, 67712, 70688

Offset: 1

Hartmut F. W. Hoft, Aug 29 2023

nonn

Every number a(n) has the form 2^(2*i + 1) * s^2, i>= 0 and s odd, the single middle divisor of a(n) is sqrt(a(n)/2), and sqrt(2*a(n)) - 1 = floor((sqrt(8*n + 1) - 1)/2) = A003056(a(n)).
The least number in the sequence with 3 odd prime divisors is a(126) = 1630818 = 2^1 * 3^2 * 7^2 * 43^2.
Conjecture: Let a(n) = 2^(2i+1) * s^2, i>=0 and s odd, be a number in the sequence.
(1) For any odd prime divisor p of s, number a(n) * p^2 is in the sequence.
(2) For any odd prime p not a divisor of s, number a(n) * p^2 is in the sequence if p satisfies sqrt(2*a(n)) < p < 2*a(n).

			a(5) = 128 = 2^7  has 2^3 as its single middle divisor, and its symmetric representation of sigma consists of one part of width 1.
a(10) = 882 = 2 * 3^2 * 7^2 has 3 * 7 as its single middle divisor, its symmetric representation of sigma is the smallest in this sequence of maximum width 3, consists of one part, and has width 1 at the diagonal.
A table of ranges for the single odd prime factor p for numbers k in the sequence having the form 2^(2i+1) * p^(2j), i>=0 and j>0, indexed by exponent 2i+1 of 2 in number k. The lower bound is A014210(i+1) and the upper bound is A014234(2(i+1)) = A104089(i+1):
---------------------
  2i+1  /---- p ----/
---------------------
  1       3  ..    3
  3       5  ..   13
  5      11  ..   61
  7      17  ..  251
  9      37  .. 1021
...

Intersection of A361903 and A361905.
Also subsequence of the following sequences: A001105, A071562, A238443 = A174973, A319796, A320137.
The powers of 2 with an odd index (A004171) form a subsequence.
Cf. A003056, A014210, A014234, A104089, A235791, A237048, A237270, A237271, A237593, A249223, A250068.

(* a2[ ] and its support functions are defined in A249223 *)
a365265Q[n_] := Module[{list=If[Divisible[n, Sqrt[n/2]], a2[n], {0}]}, Last[list]==1&&AllTrue[list, #>0&]]
a365265[{m_, n_}] := Select[Range[m, n], a365265Q]
a365265[{1,75000}]

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

A104090 Largest prime <= 5^n.

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A013606 a(n) = 4^n - prevprime(4^n).

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Maple

Mathematica

PARI

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A365265 Numbers k for which sqrt(k/2) divides k and the symmetric representation of sigma(k) consists of a single part and its width at the diagonal equals 1.

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Author

Keywords

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Examples

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python