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Numbers n such that A133744(n) = 0.
+20
2
1, 2, 3, 4, 5, 6, 10, 16, 23, 52, 71, 137, 224, 260, 361, 668, 695, 699, 1518, 1775, 1776, 3285, 7030, 36261
COMMENTS
Conjecture: sequence is infinite.
A B_2 sequence: a(n) is the smallest square such that pairwise sums of not necessarily distinct elements are all distinct.
+10
4
1, 4, 9, 16, 25, 36, 64, 81, 100, 169, 256, 289, 441, 484, 576, 625, 841, 1089, 1296, 1444, 1936, 2025, 2401, 2601, 3136, 4225, 4356, 4624, 5329, 5476, 5776, 6084, 7569, 9025, 10201, 11449, 11664, 12321, 12996, 13456, 14400, 16129, 17956, 20164, 22201
EXAMPLE
36 is in the sequence since the pairwise sums of {1, 4, 9, 16, 25, 36} are all distinct: 2, 5, 8, 10, 13, 17, 18, 20, 25, 26, 29, 32, 34, 37, 40, 41, 45, 50, 52, 61, 72.
49 is not in the sequence since 1 + 49 = 25 + 25.
PROG
(Python)
from itertools import count, islice
def A062295_gen(): # generator of terms aset1, aset2, alist = set(), set(), []
for k in (n**2 for n in count(1)):
bset2 = {k<<1}
if (k<<1) not in aset2:
for d in aset1:
if (m:=d+k) in aset2:
break
bset2.add(m)
else:
yield k
alist.append(k)
aset1.add(k)
aset2 |= bset2
a(n) is the smallest positive square such that pairwise sums of distinct elements are all distinct.
+10
4
1, 4, 9, 16, 25, 36, 49, 100, 144, 169, 225, 256, 361, 441, 484, 625, 729, 784, 1156, 1521, 1600, 1764, 2401, 2704, 3364, 4096, 4225, 4356, 4900, 5184, 5929, 6889, 7921, 8836, 9216, 9409, 10404, 11881, 13689, 13924, 14161, 18496, 19321, 20449, 21316
EXAMPLE
49 is in the sequence since the pairwise sums of distinct elements of {1, 4, 9, 16, 25, 36, 49} are all distinct: 5, 10, 13, 17, 20, 25, 26, 29, 34, 37, 40, 41, 45, 50, 52, 53, 58, 61, 65, 74, 85.
64 is not in the sequence since 1 + 64 = 16 + 49.
PROG
(Python)
from itertools import count, islice
def A133743_gen(): # generator of terms aset2, alist = set(), []
for k in map(lambda x:x**2, count(1)):
bset2 = set()
for a in alist:
if (b:=a+k) in aset2:
break
bset2.add(b)
else:
yield k
alist.append(k)
aset2.update(bset2)
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