Paul F. Marrero Romero - cp's OEIS frontend

A376895 Primes of the form 3^k*k^3 + 2.

2, 5, 14348909, 3502727633, 150094635296999123, 269211745384444720788843377, 2075640621314051693456929619860436129299430333182575810508680776710025092954370975575949

Offset: 1

Paul F. Marrero Romero, Oct 08 2024

nonn

A366997 are the corresponding integers k to obtain the terms of this sequence.
The next term a(8)=~2.20847*10^143 is too large to include.
The last known integer k in A366997 is 20803 and corresponds to a(13)=~3.21988*10^9938.

Cf. A366997.

Select[Table[3^k*k^3+2,{k,0,1000}],PrimeQ]

a(n) = 3^A366997(n)*A366997(n)^3 + 2.

A377045 Number of partitions of cuban primes.

Original entry on oeis.org

15, 490, 21637, 1121505, 3913864295, 1131238503938606, 78801255302666615, 5589233202595404488, 29349508915133986374841, 2163909235608484556362424, 913865816485680423486405066750, 191623400974625892978847721669762887224010

Offset: 1

Paul F. Marrero Romero, Oct 14 2024

nonn

Number of partitions of prime numbers that are the difference of two consecutive cubes.

Number of partitions of primes p such that p=(3*k^2 + 1)/4 for some integer k (A121259).

Robert Israel, Table of n, a(n) for n = 1..153

Cf. A000041, A002407, A121259.

R:= NULL: count:= 0:
for i from 1 while count < 30 do
  p:= (i+1)^3 - i^3;
  if isprime(p) then count:= count+1; v:= combinat:-numbpart(p); R:= R,v; fi
od:
R; # Robert Israel, Nov 14 2024

PartitionsP[Select[Table[(3k^2 + 1)/4,{k,50}],PrimeQ]]

a(n) = A000041(A002407(n)).

a(n) = A000041((3*A121259(n)^2 + 1)/4).

A363743 a(n) = floor(sqrt(log_10(n!))).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 0

Paul F. Marrero Romero, Aug 17 2023

nonn

Every nonnegative integer occurs at least 4 times.
Each integer k > 14 appears fewer than k times.
The only integers k that occur exactly k times are 11, 13 and 14.

Cf. A000196, A034886.

Mathematica
```
Array[Floor@ Sqrt[Log10[#!]] &, 93, 0]
```

a(n) = sqrtint(log(n!)/log(10)); \\ Michel Marcus, Sep 27 2023

a(n) = floor(sqrt(A034886(n) - 1)).

a(n) = A000196(A034886(n) - 1).

A348960 a(n) = floor(log_10(Pi*n!)).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 21, 22, 24, 25, 27, 28, 29, 31, 32, 34, 35, 37, 38, 40, 42, 43, 45, 46, 48, 50, 51, 53, 54, 56, 58, 59, 61, 63, 64, 66, 68, 70, 71, 73, 75, 77, 78, 80, 82, 84, 85, 87, 89, 91, 93, 95, 96, 98, 100

Offset: 0

Paul F. Marrero Romero, Nov 05 2021

nonn

Cf. A004216, A034886, A053511.

Array[Floor@ Log10[Pi*#!] &, 71, 0] (* Michael De Vlieger, Nov 05 2021 *)

a(n) = floor(log_10(Pi*n!)).

a(n) = floor(A053511 + log_10(n!)).

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User: Paul F. Marrero Romero

A376895 Primes of the form 3^k*k^3 + 2.

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A377045 Number of partitions of cuban primes.

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A363743 a(n) = floor(sqrt(log_10(n!))).

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A348960 a(n) = floor(log_10(Pi*n!)).

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