A240370 -id:A240370 - cp's OEIS frontend

A053443 x^2 + y^2 does not take on all possible values mod n.

4, 8, 9, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 45, 48, 49, 52, 54, 56, 60, 63, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 126, 128, 132, 135, 136, 140, 144, 147, 148, 152, 153, 156, 160, 162, 164, 168, 171

Offset: 1

David W. Wilson

nonn

Sequence gives values of n such there is not always a solution 1 < z < n to x^2 + y^2 = z (mod n). - Benoit Cloitre, Jan 04 2002; corrected by Carmine Suriano, Jun 19 2013

The asymptotic density of this sequence is 1- 3/(8*K^2) = 1 - (3/4) * A243379 = 0.35791..., where K is the Landau-Ramanujan constant (A064533). - Amiram Eldar, Dec 19 2020

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Complement of A240370.

Cf. A053444, A064533, A243379.

Select[Range[200], AnyTrue[FactorInteger[#], Mod[First[#1], 4] > 1 && Last[#1] > 1 &] &] (* Amiram Eldar, Dec 19 2020 *)

is(n)=my(v=vectorsmall(n,i,1));for(x=0,n\2, for(y=0,x, v[(x^2+y^2)%n+1]=0)); vecmax(v) \\ Charles R Greathouse IV, Jun 19 2013

is(n)=forprime(p=2,97,my(o=valuation(n,p));if(o,if(o>1&&p%4>1,return(1));n/=p^o));my(f=factor(n));for(i=1,#f[,1],if(f[i,2]>1&&f[i,1]%4>1,return(1)));0 \\ Charles R Greathouse IV, Jun 19 2013

n divisible by p^2 where p = 2 or prime p == 3 (mod 4).

A193304 Squarefree numbers multiplied by powers of 5.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119, 122, 123, 125, 127, 129, 130, 131, 133, 134, 137, 138, 139, 141, 142, 143, 145, 146, 149, 150, 151, 154, 155, 157, 158, 159, 161, 163, 165, 166, 167, 170

Offset: 1

José María Grau Ribas, Jul 21 2011

nonn

Numbers k such that A008683(A132739(k)) is not zero, where A008683 is the Moebius mu function. - Antti Karttunen, Jun 21 2014

Antti Karttunen, Table of n, a(n) for n = 1..10000

Differs from A240370 for the first time at n=109, where A240370(109)=169, while here it is missing, and a(109)=170.

Cf. A008683, A132739.

N:= 1000: # to get all terms <= N
sf:= select(numtheory:-issqrfree,{$1..N}):
map(t -> seq(t*5^i, i=0..floor(log[5](N/t))), sf);
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list)); # Robert Israel, Apr 16 2015

lim = 102; sf = Select[Range[lim], SquareFreeQ]; lim5 = 5^Range[0, Log[5, lim]]; Select[Union[Flatten[Outer[Times, sf, lim5]]], # <= lim &]

is(n)=issquarefree(n/5^valuation(n,5)) \\ Charles R Greathouse IV, Jul 31 2011

;; With Antti Karttunen's IntSeq-library.
(define A193304 (NONZERO-POS 1 1 (COMPOSE A008683 A132739)))
;; Reflecting essentially the above Pari-program, Antti Karttunen, Jun 21 2014

a(n) ~ kn with k = 10*Pi^2/63. - Charles R Greathouse IV, Apr 16 2015

cp's OEIS Frontend

A053443 x^2 + y^2 does not take on all possible values mod n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A193304 Squarefree numbers multiplied by powers of 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Scheme

Formula