A062295 -id:A062295 - cp's OEIS frontend

A133744 a(n) = A062295(n) - A133743(n).

0, 0, 0, 0, 0, 0, 15, -19, -44, 0, 31, 33, 80, 43, 92, 0, 112, 305, 140, -77, 336, 261, 0, -103, -228, 129, 131, 268, 429, 292, -153, -805, -352, 189, 985, 2040, 1260, 440, -693, -468, 239, -2367, -1365, -285, 885, 596, 3531, 2608, 3360, 2752, -2196, 0, 2709, 4367, 4411, 2105

Offset: 1

Klaus Brockhaus, Sep 24 2007

sign

A062295 is the sequence of smallest squares such that the pairwise sums of not necessarily distinct elements are all distinct, whereas A133743 is the sequence of smallest squares such that the pairwise sums of distinct elements are all distinct.

			a(7) = A062295(7) - A133743(7) = 64 - 49 = 15.

Klaus Brockhaus, Table of n, a(n) for n = 1..4944

Cf. A062295, A133743, A133745.

from collections import deque
from itertools import count, islice
def A133744_gen(): # generator of terms
    aset2, alist, bset2, blist, aqueue, bqueue = set(), [], set(), [], deque(), deque()
    for k in (n**2 for n in count(1)):
        cset2 = {k<<1}
        if (k<<1) not in aset2:
            for a in alist:
                if (m:=a+k) in aset2:
                    break
                cset2.add(m)
            else:
                aqueue.append(k)
                alist.append(k)
                aset2.update(cset2)
        cset2 = set()
        for b in blist:
            if (m:=b+k) in bset2:
                break
            cset2.add(m)
        else:
            bqueue.append(k)
            blist.append(k)
            bset2.update(cset2)
        if len(aqueue) > 0 and len(bqueue) > 0:
            yield aqueue.popleft()-bqueue.popleft()
A133744_list = list(islice(A133744_gen(),30)) # Chai Wah Wu, Sep 11 2023

A133743 a(n) is the smallest positive square such that pairwise sums of distinct elements are all distinct.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Sep 24 2007

nonn

			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.

Klaus Brockhaus, Table of n, a(n) for n = 1..4948
Eric Weisstein's World of Mathematics, B2-Sequence
Index entries for B_2 sequences

Cf. A000290, A062295, A133744, A133745.

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)
A133743_list = list(islice(A133743_gen(),30)) # Chai Wah Wu, Sep 11 2023

A133745 Numbers n such that A133744(n) = 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 16, 23, 52, 71, 137, 224, 260, 361, 668, 695, 699, 1518, 1775, 1776, 3285, 7030, 36261

Offset: 1

Klaus Brockhaus, Sep 24 2007

Conjecture: sequence is infinite.

Cf. A062295, A133743, A133744.

a(23)-a(24) from Chai Wah Wu, Sep 11 2023

cp's OEIS Frontend

A133744 a(n) = A062295(n) - A133743(n).

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A133743 a(n) is the smallest positive square such that pairwise sums of distinct elements are all distinct.

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Python

A133745 Numbers n such that A133744(n) = 0.

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