A304978 Numbers that can be expressed in more than one way as 6xy + x + y with x >= y > 0.

106, 155, 197, 204, 253, 288, 302, 351, 379, 400, 421, 449, 470, 498, 504, 535, 547, 554, 561, 596, 645, 652, 687, 694, 704, 729, 743, 779, 782, 792, 820, 834, 841, 873, 890, 904, 925, 939, 953, 988, 1016, 1029, 1037, 1042, 1054, 1079, 1086, 1107, 1121, 1135, 1184, 1198, 1204, 1211, 1219, 1233, 1254, 1276, 1282, 1289, 1329

Offset

1,1

Comments

Is it possible to find a closed form formula for this sequence?

Numbers k such that 6*k+1 has at least 5 divisors == 1 (mod 6). - Robert Israel, Jan 20 2019

Links

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

Example

106 is in this sequence because 106 can be expressed in two different ways as 6xy + x + y: 6*8*2 + 8 + 2 and 6*15*1 + 15 + 1.

Maple

filter:= proc(n) nops(select(t -> t mod 6 =1, numtheory:-divisors(6*n+1)))>= 5 end proc:
select(filter, [$1..2000]); # Robert Israel, Jan 20 2019

Mathematica

Select[Range[1329], 2 == Length@ FindInstance[ 6*x*y+x+y == # && x >= y > 0, {x, y}, Integers, 2] &] (* Giovanni Resta, May 29 2018 *)

Prog

(Python)
# Finding duplicates in matrix A where {a_ij}=6*i*j+i+j
def cuenta(a):
    repa=[]
    dupo=[]
    k=0
    sumo=0
    while k<len(a)-1:
        j=k+1
        dupa = 1
        while j<len(a) and a[j]==a[k]:
            dupa +=1
            j+=1
        if dupa>1:
            repa.append(a[k])
            dupo.append(dupa)
            sumo +=(dupa-1)
        k +=dupa
    return repa, dupo, sumo
kuno = []
for k in range(1, a + 1):
    for j in range(1, k + 1):
        m=6 * k * j + k + j
        if m <= NN // 6: kuno.append(m)
kuno=sorted(kuno)
repkuno, dupkuno, suno= cuenta(kuno)
print(repkuno)
#
#END
(PARI) is(n) = my(i=0); for(x=1, n, for(y=1, x, if(n==6*x*y+x+y, i++; if(i==2, return(1))))); 0 \\ Felix Fröhlich, May 29 2018

Crossrefs

Subsequence of A067611.

Sequence in context: A163625 A070796 A045093 * A078776 A188006 A160725

Adjacent sequences: A304975 A304976 A304977 * A304979 A304980 A304981

Keyword

nonn

Author

Pedro Caceres, May 22 2018

Status

approved