A218562 -id:A218562 - cp's OEIS frontend

A218563 Numbers n such that n^2 + 1 is divisible by a 4th power.

182, 239, 443, 807, 1068, 1432, 1693, 2057, 2318, 2682, 2943, 3307, 3568, 3932, 4193, 4557, 4818, 5182, 5443, 5807, 6068, 6432, 6693, 7057, 7318, 7682, 7943, 8307, 8568, 8932, 9193, 9557, 9818, 10182, 10443, 10807, 11068, 11432, 11693, 12057, 12318, 12682

Offset: 1

Michel Lagneau, Nov 02 2012

nonn

Includes all n == 182 or 443 (mod 625). In particular, the sequence has positive asymptotic density. # Robert Israel, Oct 06 2016

			239 is in the sequence because 239^2+1 = 57122 = 2*13^4;
27493 is in the sequence because 27493^2+1 = 755865050 = 2*5^2*17^4*181.

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

Cf. A002522, A049532, A034939, A218562.

N:= 100000: # to get all terms <= N
res:= {}:
p:= 2;
while p^4 <= N^2+1 do
  for v in map(t -> subs(t,n), [msolve(n^2+1, p^4)]) do
    res:= res union {seq(k*p^4+v, k = 0 .. (N-v)/p^4)}
  od;
  p:= nextprime(p);
od:
sort(convert(res,list)); # Robert Israel, Oct 06 2016

Select[Range[2,13000],Max[Transpose[FactorInteger[#^2+1]][[2]]]>3&]

A218564 Numbers n such that n^2 + 1 is divisible by a 5th power.

Original entry on oeis.org

1068, 2057, 4193, 5182, 7318, 8307, 10443, 11432, 13568, 14557, 16693, 17682, 19818, 20807, 22943, 23932, 26068, 27057, 29193, 30182, 32318, 33307, 35443, 36432, 38568, 39557, 41693, 42682, 44818, 45807, 47943, 48932, 51068, 52057, 54193, 55182, 57318, 58307

Offset: 1

Michel Lagneau, Nov 02 2012

nonn

For each prime p == 1 (mod 4), there are two values of x (mod p^5) that solve x^2 + 1 == 0 (mod p^5), and then x + k*p^5 is in the sequence for every k. Thus the asymptotic density of this sequence should be 1 - Product_p (1 - 2/p^5), where the product is over all primes p == 1 (mod 4). - Robert Israel, Sep 04 2018

			1068 is in the sequence because 1068^2+1 = 1140625 = 5^6*73;
143044 is in the sequence because 143044^2+1 = 20461585937 = 13^5*55109;
390112 is in the sequence because 390112^2+1 = 152187372545 = 5*13*17^6*97.

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

Cf. A002522, A049532, A034939, A218562, A218563.

N:= 10^5: # to get all terms <= N
P:= select(isprime,[seq(i,i=5..floor((N^2+1)^(1/5)),4)]):
g:= proc(x,r,N) local t; t:= rhs(op(x)); seq(t+r*k,k=0..(N-t)/r) end proc:
R:= `union`(seq(map(g, {msolve(n^2+1,p^5)},p^5,N),p=P)):
sort(convert(R,list)); # Robert Israel, Sep 04 2018

Select[Range[2,20000],Max[Transpose[FactorInteger[#^2+1]][[2]]]>4&]

isok(n) = vecmax(factor(n^2+1)[,2]) >= 5; \\ Michel Marcus, Sep 04 2018

A218565 Numbers k such that k^2 + 1 is divisible by a 6th power.

Original entry on oeis.org

1068, 14557, 16693, 30182, 32318, 45807, 47943, 61432, 63568, 77057, 79193, 92682, 94818, 108307, 110443, 123932, 126068, 139557, 141693, 155182, 157318, 170807, 172943, 186432, 188568, 202057, 204193, 217682, 219818, 233307, 235443, 248932, 251068, 264557

Offset: 1

Michel Lagneau, Nov 02 2012

nonn

			1068 is in the sequence because 1068^2 + 1 = 5^6 * 73.
390112 is in the sequence because 390112^2 + 1 = 5 * 13 * 17 ^ 6 * 97.
1999509 is in the sequence because 1999509^2 + 1 = 2 * 13 ^ 6 * 29 * 14281.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002522, A049532, A034939, A218562, A218563, A218564.

Cf. A001014 (6th powers).

Select[Range[2,27000],Max[Transpose[FactorInteger[#^2+1]][[2]]]>5&]

A218574 Numbers k such that k^2 + 1 is divisible by a 7th power.

Original entry on oeis.org

32318, 45807, 110443, 123932, 188568, 202057, 266693, 280182, 344818, 358307, 422943, 436432, 501068, 514557, 579193, 592682, 657318, 670807, 735443, 748932, 813568, 827057, 891693, 905182, 969818, 983307

Offset: 1

Michel Lagneau, Nov 02 2012

nonn

			32318 is in the sequence because 32318^2 + 1 =  5 ^ 7 * 29 * 461.
6826318 is in the sequence because 6826318^2 + 1 = 5 ^ 3 * 13 ^ 8 * 457.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002522, A049532, A034939, A218562, A218563, A218564, A218565.

Cf. A001015 (7th powers).

Select[Range[2,1500000],Max[Transpose[FactorInteger[#^2+1]][[2]]]>6&]

cp's OEIS Frontend

A218563 Numbers n such that n^2 + 1 is divisible by a 4th power.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A218564 Numbers n such that n^2 + 1 is divisible by a 5th power.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A218565 Numbers k such that k^2 + 1 is divisible by a 6th power.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A218574 Numbers k such that k^2 + 1 is divisible by a 7th power.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica