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A276525 Composite numbers m for which L(p(i)/p(j)) = 1 for all i, j, where p(k) are the prime factors of m and L(x/y) is the Legendre symbol of x and y, defined to be 1 if x is a quadratic residue (mod y) and -1 if x is a quadratic non-residue (mod y). 2
14, 28, 34, 39, 46, 55, 56, 62, 68, 82, 92, 94, 95, 98, 111, 112, 117, 124, 136, 142, 145, 146, 155, 158, 164, 178, 183, 184, 188, 194, 196, 203, 205, 206, 219, 221, 224, 226, 248, 254, 259, 272, 274, 275, 284, 291, 292, 295, 299, 302, 305, 316, 323, 327, 328 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

L(x/y) = L(y/x) for primes x, y, and either x = 4r+1 for some r, or y = 4r+1 for some r.

We observe pairs of the form (a(n), a(n)+1) = (55, 56), (94, 95), (111, 112), (145, 146), (183, 184), (205, 206), (274, 275), (291, 292), (327, 328), ..., .

From Robert G. Wilson v, Nov 16 2016: (Start)

We observe triples of the form (a(n), a(n)+1, a(n)+2) with a(n): 542, 543, 655, 1262, 1411, 1646, 1982, 2305, 2306, 2942, 3025, ..., .

We observe quadruples of the form (a(n), a(n)+1, a(n)+2, a(n)+3) with a(n): 542, 2305, 7022, 10081, 19981, 35821, 41372, 50011, 50941, ..., .

We observe quintuples of the form (a(n), a(n)+1, a(n)+2, a(n)+3, a(n)+4) with a(n): 85631, ..., .

Number of terms less than 10^k: 0, 14, 168, 1591, 14175, 127791, ...

Number of twin terms less than 10^k: 0, 2, 31, 250, 1737, 13604, ...

(End)

LINKS

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Legendre Symbol

EXAMPLE

14 is in the sequence because the prime factors are 2 and 7 => L(2,7)= L(7,2)= 1.

137678 is in the sequence because the prime factors are 2, 23, 41 and 73 => L(2,23) = L(2,41) = L(2,73) = L(23,41) = L(23,73) = L(41,73) = L(73,41) = L(73,23) = L(73,2) = L(41,23) = L(41,2) = L(23,2) = 1.

MAPLE

with(numtheory): nn:=400:

for n from 1 to nn do:

x:=factorset(n):n0:=nops(x):

  if n0>1

   then

   ii:=0:

    for i from 1 to n0-1 while(ii=0) do:

     for j from 2 to n0 while(ii=0) do:

       p:=legendre(x[i], x[j]):q:=legendre(x[j], x[i]):

        if p=-1 or q=-1 then ii:=1:

        else fi:

     od:od:

     if ii=0 then printf(`%d, `, n):

     else fi:fi:

od:

MATHEMATICA

fQ[n_] := If[ CompositeQ@ n, Block[{pf = Transpose[ FactorInteger[n]][[1]]}, lng = Length@ pf; Union[ Flatten[ Table[ JacobiSymbol[pf[[i]], pf[[j]]], {i, lng}, {j, lng}]]] == {0, 1}], False]; Select[ Range@ 330, fQ] (* Robert G. Wilson v, Nov 16 2016 *)

CROSSREFS

Cf. A002144, A002145.

Sequence in context: A297282 A095782 A143204 * A118904 A164722 A162020

Adjacent sequences:  A276522 A276523 A276524 * A276526 A276527 A276528

KEYWORD

nonn

AUTHOR

Michel Lagneau, Nov 16 2016

STATUS

approved

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Last modified August 24 03:51 EDT 2019. Contains 326260 sequences. (Running on oeis4.)