Mark E. Shoulson - cp's OEIS frontend

User: Mark E. Shoulson

Mark E. Shoulson has authored 3 sequences.

A246419 Numbers n such that n^6 - 2 is not squarefree.

10, 22, 50, 143, 204, 267, 311, 312, 423, 455, 461, 479, 506, 556, 579, 600, 649, 818, 845, 889, 987, 1008, 1104, 1108, 1134, 1178, 1258, 1273, 1333, 1343, 1416, 1423, 1467, 1537, 1610, 1637, 1712, 1756, 1779, 2001, 2005, 2045, 2065, 2066, 2104, 2166, 2205

Offset: 1

Mark E. Shoulson, Aug 25 2014

nonn

			10^6 - 2 = 999998, which is divisible by 16129 = 127^2, so 10 is in the sequence.
11^6 - 2 = 1771559, which is prime, and 12^6 - 2 = 2985982 = 2 * 17 * 31 * 2833 (all prime), so neither of these numbers is in the sequence.

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

Cf. A005117, A013928.

[n: n in [0..3000] | not IsSquarefree(n^6-2)]; // Vincenzo Librandi, Oct 16 2014

remove(t -> numtheory:-issqrfree(t^6-2), [$1..3000]); # Robert Israel, Aug 25 2014

Select[Range[2000], MoebiusMu[#^6 - 2] == 0 &] (* Alonso del Arte, Aug 25 2014 *)

isok(n) = ! issquarefree(n^6-2); \\ Michel Marcus, Oct 11 2014

from sympy import factorint
[i for i in range(1, 500) if any(v>1 for v in factorint(i**6-2).values())]

A247025 Lengths of prefixes of the infinite string of digits repeat(1379) that are prime.

Original entry on oeis.org

2, 3, 7, 81, 223, 250, 255, 537, 543, 1042, 2103, 4285, 25015, 35361, 43525

Offset: 1

Mark E. Shoulson, Sep 09 2014

Every prime > 5 in base 10 ends in 1, 3, 7, or 9.  If those digits are repeated, in order, some prefixes of that string are prime.
n such that floor(1379/9999 * 10^n) is prime. - Robert Israel, Sep 09 2014
a(13) > 15500. - Daniel Starodubtsev, Mar 16 2021

			1 and 3 are the first two digits of the string, and 13 is prime. 13 has length 2, so 2 is a term.
137 is prime and three digits long, so 3 is a term.
1379137 is prime and seven digits long, so 7 is a term.

Cf. A000040, A007652.

[n: n in [0..300] | IsPrime(Floor(1379/9999 * 10^n))]; // Vincenzo Librandi, Oct 17 2014

Select[Range[4300],PrimeQ[FromDigits[PadRight[{},#,{1,3,7,9}]]]&] (* The program generates the first 12 terms of the sequence. *) (* Harvey P. Dale, Jun 11 2024 *)

lista(nn) = {s = 0; digs = [1,3,7,9]; id = 1; for (n=1, nn, s = 10*s + digs[id]; if (isprime(s), print1(n, ", ")); id++; if (id==5, id = 1););} \\ Michel Marcus, Oct 11 2014

from sympy import isprime
from itertools import cycle
it=cycle([1,3,7,9])
c=0
a=0
for i in it:
    c+=1
    a*=10
    a+=i
    if isprime(a):
        print(c)

Edited. Name specified. Example reformulated. a(12) added (using R. Israel's formula). Keyword less and Crossreferences added. - Wolfdieter Lang, Nov 03 2014
a(13)-a(14) from Michael S. Branicky, May 29 2023
a(15) from Michael S. Branicky, Jun 13 2024

A242098 Numbers n such that the residue of n modulo floor(sqrt(n)) is prime.

Original entry on oeis.org

11, 14, 18, 19, 22, 23, 27, 28, 32, 33, 38, 39, 41, 44, 45, 47, 51, 52, 54, 58, 59, 61, 66, 67, 69, 71, 74, 75, 77, 79, 83, 84, 86, 88, 92, 93, 95, 97, 102, 103, 105, 107, 112, 113, 115, 117, 123, 124, 126, 128, 134, 135, 137, 139, 146, 147, 149

Offset: 1

Mark E. Shoulson, Aug 14 2014

Also, i^2+p(1), i^2+p(2),..., i^2+p(k), i^2+i+p(1), i^2+i+p(2),..., i^2+i+p(k), for i>=3, where p(n) is the n-th prime and p(k) is the largest prime strictly less than i.

			floor(sqrt(28)) = 5. 28 modulo 5 = 3, which is prime, so 28 is in the sequence.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Select[Range[200],PrimeQ[ Mod[#,Floor[Sqrt[#]]]]&] (* Harvey P. Dale, May 31 2019 *)

for(n=1, 10^3, if(isprime(n%sqrtint(n)), print1(n", "))) \\ Jens Kruse Andersen, Aug 16 2014

from sympy import isprime
from math import sqrt, floor
from itertools import count
sequence = ( for  in count(1) if isprime( % floor(sqrt())))
print([next(sequence) for i in range(50)])

Added alternative formulation in comment.

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User: Mark E. Shoulson

A246419 Numbers n such that n^6 - 2 is not squarefree.

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A247025 Lengths of prefixes of the infinite string of digits repeat(1379) that are prime.

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Magma

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A242098 Numbers n such that the residue of n modulo floor(sqrt(n)) is prime.

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Author

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Mathematica

PARI

Python

Extensions