Gionata Neri - cp's OEIS frontend

A334212 Least number k such that n^k + 1 is not squarefree.

3, 1, 5, 3, 7, 1, 1, 5, 11, 1, 10, 7, 3, 1, 17, 1, 2, 1, 3, 11, 10, 1, 1, 13, 1, 1, 10, 3, 31, 1, 2, 10, 5, 1, 37, 10, 2, 1, 5, 2, 10, 1, 1, 21, 47, 1, 1, 1, 3, 1, 10, 1, 5, 1, 3, 2, 10, 1, 14, 21, 1, 1, 5, 3, 21, 1, 2, 3, 2, 1, 10, 10, 1, 1, 7, 3, 10, 1

Offset: 2

Gionata Neri, Apr 18 2020

nonn

For n == 1 (mod 4) (n not 1), a(n) <= (n + 1)/2.
For n == 3 (mod 4), a(n) = 1.
For even n, a(n) <= n + 1.
Existence proof for n >= 2 and upper bounds use the binomial formula.

Cf. A013929, A334213, A334214.

for(n=2,79, for(k=1,n+1, !issquarefree(n^k+1)&!print1(k", ")&break))

A334213 Numbers m such that m^k + 1 is squarefree for all 0 <= k <= m.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 16, 30, 36, 46, 256

Offset: 1

Gionata Neri, Apr 18 2020

m = 2^i is a term iff k*i is not in A049096 with 0 < k < m + 1. Up to i = 128, there are no more terms of the form 2^i. a(12) > 10^7, if it exists. - Jinyuan Wang, May 01 2020

			4^0 + 1 = 2 is squarefree, 4^1 + 1 = 5 is squarefree, 4^2 + 1 = 17 is squarefree, 4^3 + 1 = 5*13 is squarefree and 4^4 + 1 = 257 is squarefree, so 4 is in the sequence.

Cf. A049096, A334212, A334214.

Do[L=Length[a];a=Select[a=m^Range[0,m-1]+1,SquareFreeQ[#]&];If[L==m-1,Print[m-1]],{m,0,1000}] (* Metin Sariyar, Apr 21 2020 *)

isOK(m) = k=0; until(k>m, if(!issquarefree(m^k+1), return(0)); k++); 1;
for(m=0, 99, if(isOK(m), print1(m, ", ")))

a(11) from Jinyuan Wang, May 01 2020

A334214 Odd numbers m such that m^k + 1 is squarefree for all 0 <= k <= (m - 1)/2.

Original entry on oeis.org

1, 5, 9, 13, 21, 25, 85, 105, 165

Offset: 1

Gionata Neri, Apr 18 2020

a(10) > 10000, if it exists. Most of the relevant factorizations of 165^k+1 were already present in the FactorDB database. The factorization of 165^79+1 has been completed with the software CADO-NFS. - Giovanni Resta, Apr 22 2020

Cf. A334212, A334213.

isOK(m) = k=0; until(k>(m-1)/2, if(!issquarefree(m^k+1), return(0)); k++); 1;
for(m=0, 140, if(isOK(m)&&m%2==1, print1(m, ", ")))

a(9) from Giovanni Resta, Apr 22 2020

A306878 Number of 0 < k < n such that n-k and n+k are both nonprimes.

Original entry on oeis.org

0, 0, 0, 0, 2, 1, 2, 3, 3, 3, 5, 5, 6, 6, 7, 6, 9, 10, 8, 10, 11, 10, 12, 13, 13, 13, 15, 14, 16, 18, 15, 18, 20, 16, 20, 21, 20, 21, 24, 21, 23, 26, 24, 24, 29, 25, 27, 30, 26, 30, 32, 29, 31, 33, 31, 33, 36, 33, 34, 41, 34, 36, 42, 35, 40, 42, 40, 40, 43, 42, 44, 48, 44, 44

Offset: 1

Gionata Neri, Mar 14 2019

nonn

Cf. A061357, A291564.

a(n)=my(s); for(c=1, n-1, s+=(bigomega(n-c)-1)*(bigomega(n+c)-1)!=0); s

a(n) = n - 1 - A061357(n) - A291564(n).

A319081 Number of primes of the form k^3 + n^2 for 0 <= k <= n.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 1, 0, 2, 4, 1, 2, 1, 3, 2, 4, 3, 3, 4, 3, 2, 2, 3, 5, 3, 4, 0, 3, 3, 2, 3, 7, 7, 1, 4, 4, 3, 2, 4, 9, 3, 5, 5, 3, 4, 7, 5, 5, 6, 4, 6, 4, 4, 11, 5, 5, 4, 6, 8, 6, 8, 4, 7, 0, 8, 8, 10, 7, 4, 3, 3, 12, 6, 13, 9, 3, 5, 5, 9, 11, 12, 7, 3, 8, 3, 8, 12, 11, 8, 4

Offset: 0

Gionata Neri, Sep 09 2018

nonn

Cf. A066649.

for(n=0,90,a=sum(k=0,n,isprime(k^3+n^2));print1(a", "))

a(n) = 0 if n is a cube other than 1.

A305629 Number of values of k, 0 < k < n, for which (2n^2 + 2k*n - k^2 - k)/2 is prime.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 0, 3, 3, 1, 3, 5, 2, 2, 4, 2, 1, 4, 3, 2, 4, 1, 4, 4, 2, 4, 4, 3, 1, 3, 2, 4, 12, 3, 2, 7, 1, 7, 15, 3, 4, 6, 3, 8, 8, 2, 4, 10, 6, 7, 7, 3, 6, 16, 3, 4, 6, 3, 8, 7, 4, 5, 11, 5, 4, 7, 5, 8, 12, 2, 12, 8, 4, 16, 9, 3, 8, 24, 6, 8, 11, 5, 8, 19, 3

Offset: 0

Gionata Neri, Jul 08 2018

nonn

It appears that a(A046174(n)) = 0.

Cf. A046174.

a(n) = sum(k=1, n-1, isprime((2*n^2+2*k*n-k^2-k)/2))

A300737 Numbers of the form (ki + 1)(k*j - 1) with i, j >= 1 and k >= 2.

Original entry on oeis.org

3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 35, 37, 38, 39, 41, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 77, 79, 80, 81, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 95, 97, 98, 99, 101

Offset: 1

Gionata Neri, Mar 11 2018

nonn

Cf. A056868.

limit=100; v=vector(limit); for(n=0,limit, for(k=2,floor(limit^(1/2)), for(i=1,limit/k, for(j=1,limit/k,if(n==(k*i+1)*(k*j-1),v[n]=1,))))); for(n=1,limit,if(v[n],print1(n", ")))

A299145 Primes of the form j^k + (j-1)^k + ... + 2^k, for j > 1 and k > 0.

Original entry on oeis.org

2, 5, 13, 29, 97, 139, 353, 4889, 72353, 353815699, 42065402653, 84998999651, 102769130749, 15622297824266188673, 28101527071305611527, 20896779938941631284493075599148668795944697935466419104293, 105312291668560568089831550410013687058921146068446092937783402353

Offset: 1

Gionata Neri, Feb 03 2018

nonn

Except for the terms 2, 5, 13, 29, 139, the exponent k satisfies k >= 4. More generally, if Q(j) = j^k + (j-1)^k + ... + 2^k is a term, then j-1 is a divisor of A064538(k). This is because (j-1) is a factor of Q(j) and thus Q(j) is prime only if j-1 is a divisor of the denominator of Q(j), i.e. A064538(k). Thus for each k there is only a finite number of values of j to check. This provides an efficient algorithm to find terms of this sequence by looking only for primes in the numbers H_{j,-k} - 1 = j^k + (j-1)^k + ... + 2^k for j-1 a divisor of A064538(k). - Chai Wah Wu, Mar 06 2018

			2 = 2^1;
5 = 3^1 + 2^1;
13 = 3^2 + 2^2;
29 = 4^2 + 3^2 + 2^2;
97 = 3^4 + 2^4;
139 = 7^2 + 6^2 + 5^2 + 4^2 + 3^2 + 2^2;
353 = 4^4 + 3^4 + 2^4;
4889 = 4^6 + 3^6 + 2^6;
72353 = 4^8 + 3^8 + 2^8;

Chai Wah Wu, Table of n, a(n) for n = 1..45 (all terms < 10^1000).

Cf. A007689, A074526, A064538.

With[{nn = 350}, Sort@ Flatten@ Map[Select[#, PrimeQ] &, Table[Total[Range[j, 1, -1]^k] - 1, {j, 2, nn}, {k, nn - j}]]] (* Michael De Vlieger, Feb 03 2018 *)

limit=100000; v=vector(limit); for(n=1, ceil((-1+(1+8*limit)^(1/2))/2), for(k=1, logint(limit, n+0^(n-1)), a=sum(i=1,n,i^k)-1;if(isprime(a)&&a

a(10)-a(15) from Michael De Vlieger, Feb 03 2018

a(16)-a(17) from Chai Wah Wu, Mar 07 2018

A293271 Numbers n such that n - p and n + p are both prime for some prime p.

Original entry on oeis.org

5, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 30, 32, 34, 36, 39, 40, 42, 44, 45, 46, 48, 50, 54, 56, 60, 64, 66, 69, 70, 72, 76, 78, 81, 84, 86, 90, 92, 96, 99, 100, 102, 104, 105, 106, 108, 110, 111, 114, 116, 120, 126, 129, 130, 132, 134, 138, 140, 142

Offset: 1

Gionata Neri, Oct 04 2017

nonn

Apart from a(1), all terms are composite.

Union of A087679 and 2*A063713. - Robert Israel, Oct 09 2017

Cf. A087679, A087695, A087696, A087697 (subsequences).

Cf. A063713.

filter:= proc(n) local k;
  k:= 1;
  while k < n do
    k:= nextprime(k);
    if isprime(n+k) and isprime(n-k) then return true fi
  od;
  false
end proc:
select(filter, [$1..1000]); # Robert Israel, Oct 09 2017

Select[Range@ 142, Function[n, AnyTrue[Prime@ Range@ PrimePi@ n, PrimeQ[n + {-#, #}] == {True, True} &]]] (* Michael De Vlieger, Oct 09 2017 *)

a(n) = forprime(p=1, n, i=n-p; j=n+p; if(isprime(i)&&isprime(j), n; break))

A292922 Triangle read by rows: T(n,k) is the number of numbers <= primorial(n) with k prime factors, counted without multiplicity.

Original entry on oeis.org

1, 4, 1, 16, 12, 1, 60, 116, 32, 1, 377, 1085, 745, 101, 1, 3323, 11172, 11534, 3735, 264, 1, 42518, 153752, 195801, 99914, 17808, 715, 1, 646580, 2464246, 3535748, 2314475, 667138, 69877, 1624, 1, 12285485, 48959467

Offset: 1

Gionata Neri, Sep 26 2017

			The triangle T(n, m) begins:
n\k   1     2     3     4     5     6
1:    1
2:    4     1
3:   16    12     1
4:   60   116    32     1
5:  377  1085   745   101     1
6: 3323 11172 11534  3735   264     1

Cf. A001221.

a(n) = r=prod(i=1,n,prime(i)); for(s=1,n,k=sum(t=2,r,omega(t)==s) ; k)

cp's OEIS Frontend

User: Gionata Neri

A334212 Least number k such that n^k + 1 is not squarefree.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A334213 Numbers m such that m^k + 1 is squarefree for all 0 <= k <= m.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A334214 Odd numbers m such that m^k + 1 is squarefree for all 0 <= k <= (m - 1)/2.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Extensions

A306878 Number of 0 < k < n such that n-k and n+k are both nonprimes.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A319081 Number of primes of the form k^3 + n^2 for 0 <= k <= n.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A305629 Number of values of k, 0 < k < n, for which (2*n^2 + 2*k*n - k^2 - k)/2 is prime.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A300737 Numbers of the form (k*i + 1)*(k*j - 1) with i, j >= 1 and k >= 2.

Views

Author

Keywords

Crossrefs

Programs

PARI

A299145 Primes of the form j^k + (j-1)^k + ... + 2^k, for j > 1 and k > 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A293271 Numbers n such that n - p and n + p are both prime for some prime p.

Views

Author

Keywords

Comments

Crossrefs

Programs

A305629 Number of values of k, 0 < k < n, for which (2n^2 + 2k*n - k^2 - k)/2 is prime.

A300737 Numbers of the form (ki + 1)(k*j - 1) with i, j >= 1 and k >= 2.