A235538
Earliest infinite sequence of natural numbers such that the members of this sequence as well as the absolute values of the members of the k-th differences of this sequence, for all k>0, are all distinct.
Original entry on oeis.org
1, 3, 9, 26, 5, 13, 31, 15, 27, 81, 22, 45, 92, 20, 50, 145, 46, 89, 32, 71, 151, 40, 75, 163, 73, 124, 60, 126, 244, 97, 219, 63, 132, 306, 68, 144, 297, 79, 166, 354, 83, 187, 394, 94, 203, 419, 108, 220, 460, 127, 260, 110, 247, 513, 161, 340, 117, 252
Offset: 1
For n=1:
- 1 is admissible; hence a(1)=1.
For n=2:
- 1 is not admissible (as it already appears in the sequence),
- 2 is not admissible (as a(1) would appear in the first differences),
- 3 is admissible; hence a(2)=3.
For n=3:
- 1 is not admissible (as it already appears in the sequence),
- 2 is not admissible (as it already appears in the first differences),
- 3 is not admissible (as it already appears in the sequence),
- 4 is not admissible (as a(1) would appear in the first differences),
- 5 is not admissible (as 2 would appear twice in the first differences),
- 6 is not admissible (as a(2) would appear in the first differences),
- 7 is not admissible (as 2 would appear in the first and second differences),
- 8 is not admissible (as a(2) would appear in the second differences),
- 9 is admissible; hence a(3)=9.
-
a[1] = 1; diffs0 = {1} (* flattened array of successive differences *);
a[n_] := a[n] = Module[{}, aa = Array[a, n-1]; m0 = 1; While[ MemberQ[ diffs0, m0], m0++]; For[m = m0, True, m++, am = Append[aa, m]; td = Table[Differences[am, k], {k, 0, n-1}]; diffs = Abs[Flatten[td]]; If[ Length[diffs] == Length[Union[diffs]], diffs0 = diffs//Sort; Return[m]]] ];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 31 2018 *)
-
See Link section.
A062292
A B_2 sequence: a(n) is the smallest cube such that the pairwise sums of {a(1)...a(n)} are all distinct.
Original entry on oeis.org
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 2197, 2744, 3375, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 15625, 17576, 19683, 21952, 24389, 27000, 29791, 35937, 42875, 50653, 54872, 59319, 64000, 68921, 74088, 79507, 85184, 91125, 97336
Offset: 1
During recursive construction of this set, for n=1-50, the cubes of 12,18,24,32,34,36,48 are left out to keep all sums of distinct cubes distinct from each other.
-
from itertools import count, islice
def A062292_gen(): # generator of terms
aset1, aset2, alist = set(), set(), []
for k in (n**3 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.update(bset2)
A062292_list = list(islice(A062292_gen(),30)) # Chai Wah Wu, Sep 05 2023
A080932
Non-occurring pairwise differences between the elements of the Mian-Chowla sequence variant A058335.
Original entry on oeis.org
26, 33, 50, 55, 68, 93, 94, 98, 107, 108, 109, 115, 122, 138, 144, 150, 155, 163, 178, 181, 182, 183, 185, 186, 193, 196, 200, 202, 204, 208, 210, 212, 223, 227, 232, 235, 239, 242, 245, 250, 253, 257, 263, 264, 268
Offset: 1
A096772
A B3-sequence: a(1) = 1; for n>1, a(n) = smallest number > a(n-1) such that the sums of any three terms are all distinct.
Original entry on oeis.org
1, 2, 5, 14, 33, 72, 125, 219, 376, 573, 745, 1209, 1557, 2442, 3098, 4048, 5298, 6704, 7839, 10987, 12332, 15465, 19144, 24546, 28974, 34406, 37769, 45864, 50877, 61372, 68303, 77918, 88545, 101917, 122032, 131625, 148575, 171237, 197815, 201454
Offset: 1
-
from itertools import count, islice
def A096772_gen(): # generator of terms
aset1, aset2, aset3, alist = set(), set(), set(), []
for k in count(1):
bset2, bset3 = {k<<1}, {3*k}
if 3*k not in aset3:
for d in aset1:
if (m:=d+(k<<1)) in aset3:
break
bset2.add(d+k)
bset3.add(m)
else:
for d in aset2:
if (m:=d+k) in aset3:
break
bset3.add(m)
else:
yield k
alist.append(k)
aset1.add(k)
aset2 |= bset2
aset3 |= bset3
A096772_list = list(islice(A096772_gen(),30)) # Chai Wah Wu, Sep 05 2023
A133605
Elements of A011185 that are also the sum of a pair of distinct elements of A011185.
Original entry on oeis.org
3, 5, 8, 13, 21, 74, 95, 182, 212, 413, 862, 1060, 1435, 1934, 4447, 5323, 7588, 19934, 23725, 24970, 29558, 43344, 45425, 48622, 55240, 63835, 91336, 98178, 177387, 180356, 206088, 333837, 400924, 418503, 429115, 598604, 776150, 990510, 993089
Offset: 1
A011185(1) = 1, A011185(2) = 2. 1 + 2 = 3 = A011185(3), hence 3 is in the sequence.
A011185(16) = 212, A011185(35) = 1722. 212 + 1722 = 1934 = A011185(37), hence 1934 is in the sequence.
-
from itertools import count, islice
def A133605_gen(): # generator of terms
aset2, alist = set(), []
for k in count(1):
bset2 = set()
for a in alist:
if (b:=a+k) in aset2:
break
bset2.add(b)
else:
if k in aset2:
yield k
alist.append(k)
aset2.update(bset2)
A133605_list = list(islice(A133605_gen(),30)) # Chai Wah Wu, Sep 11 2023
A247556
Exact differential base (a B_2 sequence) constructed as follows: Start with a(0)=0. For n>=1, let S be the set of all differences a(j)-a(i) for 0 <= i < j <= n-1, and let d be the smallest positive integer not in S. If, for every i in 1..n-1, a(n-1) + d - a(i) is not in S, then a(n) = a(n-1) + d. Otherwise, let r be the smallest positive integer such that, for every i in 1..n-1, neither a(n-1) + r - a(i) nor a(n-1) + r + d - a(i) is in S; then a(n) = a(n-1) + r and a(n+1) = a(n) + d.
Original entry on oeis.org
0, 1, 3, 7, 12, 20, 30, 44, 65, 80, 96, 143, 165, 199, 224, 306, 332, 415, 443, 591, 624, 678, 716, 934, 973, 1134, 1174, 1449, 1491, 1674, 1720, 2113, 2161, 2468, 2517, 2855, 2906, 2961, 3245, 3302, 3711, 3772, 4081, 4148, 4603, 4673, 5557, 5628, 5917, 5989
Offset: 0
Given a(0)=0, a(1)=1, a(2)=3, a(3)=7, the differences used are 1,2,3,4,6,7, so d=5, and we can use a(4) = a(3)+d = 7+5 = 12 because appending a(4)=12 to the sequence will result in the differences 12-0=12, 12-1=11, 12-3=9, 12-7=5, none of which had already been used.
Similarly, given a(0)..a(4) = 0,1,3,7,12, the differences used are 1..7,9,11,12, so d=8, and we can use a(5) = a(4)+d = 12+8 = 20 because the resulting differences will be 20, 19, 17, 13, 8, none of which had already been used.
Proceeding as above, we get a(6)=30 and a(7)=44.
Given a(0)..a(7) = 0,1,3,7,12,20,30,44, the differences used are 1..14,17..20,23..24,27,29..30,32,37,41,43..44, so d=15, but we cannot use a(8) = a(7)+d = 44+15 = 59 because the difference 29 would be repeated: 59-30 = 30-1. Thus, we must find the smallest r such that using both a(8) = a(7)+r and a(9) = a(8)+d will not repeat any differences. The smallest such r is 21, so a(8) = a(7)+r = 44+21 = 65 and a(9) = a(8)+d = 65+15 = 80.
- Jerzy Browkin, Rozwiązanie pewnego zagadnienia A. Schinzla (Polish) [The solution of a certain problem of A. Schinzel], Roczniki Polskiego Towarzystwa Matematycznego [Annals Polish Mathematical Society], Seria I, Prace Matematyczne III (1959).
Cf.
A001856, where a(1)=1, a(2)=2, a(2n+1)=2*a(2n), a(2n+2) = a(2n+1) + d.
A284916
Lexicographically earliest sequence of positive integers such that the same (Euclidean) distance does not occur twice between any two distinct pairs of points ((n, a(n)), (k, a(k))).
Original entry on oeis.org
1, 1, 2, 5, 9, 14, 7, 19, 25, 2, 33, 43, 54, 67, 27, 47, 64, 78, 94, 118, 17, 129, 144, 103, 156, 174, 199, 37, 114, 199, 78, 183, 220, 168, 239, 70, 272, 302, 258, 292, 311, 350, 376, 409, 431, 458, 479, 324, 504, 550, 281, 424, 563, 527, 489, 591, 129, 636
Offset: 1
Let p_n = (n, a(n)).
For n = 4, a(4) = 5 because
d(p_3, p_4) = sqrt(2) = d(p_2, p_3) if a(4) = 1,
d(p_3, p_4) = 1 = d(p_1, p_2) if a(4) = 2,
d(p_3, p_4) = sqrt(2) = d(p_2, p_3) if a(4) = 3,
d(p_3, p_4) = sqrt(5) = d(p_1, p_3) if a(4) = 4, therefore
a(4) = 5, the least value that does not create a contradiction.
-
dq[p_, q_] := Total[(p - q)^2]; good[w_] := Catch[ Do[ If[ MemberQ[di, dq[w, P[[i]]]], Throw@False], {i, Length@ P}]; True];P = di = {}; n = 0; While[n < 58, n++; k = 1; While[! good[{n, k}], k++]; di = Join[di, dq[{n, k}, #] & /@ P]; AppendTo[P, {n, k}]]; Last /@ P (* Giovanni Resta, Apr 06 2017 *)
A284917
Lexicographically earliest infinite sequence of positive integers such that the Taxicab distance is unique to each pair of distinct points ((n, a(n)), (k, a(k))).
Original entry on oeis.org
1, 1, 2, 5, 9, 16, 25, 38, 58, 72, 87, 112, 136, 169, 190, 237, 274, 344, 383, 456, 545, 572, 640, 752, 798, 891, 944, 989, 1131, 1283, 1365, 1492, 1540, 1788, 1862, 1994, 2218, 2342, 2472, 2741, 2885, 3114, 3312, 3548, 3753, 3953, 4251, 4386, 4731, 4802, 5073
Offset: 1
Let p_n = (n, a(n)).
For n = 4, a(4) = 5 because
d(p_1, p_4) = 3 = d(p_1, p_3) if a(4) = 1,
d(p_3, p_4) = 1 = d(p_1, p_2) if a(4) = 2,
d(p_3, p_4) = 2 = d(p_2, p_3) if a(4) = 3,
d(p_3, p_4) = 3 = d(p_1, p_3) if a(4) = 4, therefore
a(4) = 5, the least value that does not create a contradiction.
-
dq[p_, q_] := Total@Abs[p - q]; good[w_] := Catch[Do[If[ MemberQ[di, dq[w, P[[i]]]], Throw@False], {i, Length@P}]; True]; di = P = {}; n = 0; While[n < 51, n++; k = 1; While[! good[{n, k}], k++]; di = Join[di, dq[{n, k}, #] & /@ P]; AppendTo[P, {n, k}]]; Last /@ P (* Giovanni Resta, Apr 06 2017 *)
A293542
a(1)=1; for n>1, a(n) = least integer greater than a(n-1) such that the sums of the divisors of the pairwise sums of a(1),...,a(n) are all distinct.
Original entry on oeis.org
1, 2, 4, 8, 19, 28, 56, 72, 101, 144, 202, 240, 261, 448, 511, 602, 772, 806, 1152, 1296, 1541, 1602, 2016, 2256, 2900, 3322, 3362, 3978, 4376, 5887, 6416, 7702, 8228, 8578, 11341, 11382, 13376, 13692, 16083, 16380, 16544, 17382, 22726, 24944, 26302, 27508, 30580, 33184, 34020, 37474
Offset: 1
Let s(n) be the sum of the divisors of n. a(3)!=3 because s(1+3)=s(2+2)=7. a(3)=4 because s(1+1)=3, s(1+2)=4, s(1+4)=6, s(2+2)=7, s(2+4)=12, and s(4,4)=15 are all distinct.
A333866
Lexicographically earliest sequence of distinct positive integers, which when mapped onto a square spiral, gives a set without four distinct points forming a square.
Original entry on oeis.org
1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 18, 20, 21, 26, 27, 30, 31, 37, 38, 41, 43, 44, 46, 50, 52, 55, 57, 58, 65, 66, 70, 73, 79, 82, 83, 88, 91, 92, 101, 102, 104, 110, 111, 116, 122, 124, 127, 132, 133, 141, 143, 145, 146, 152, 156, 157, 167, 170, 171, 180
Offset: 1
The first terms, mapped onto a square spiral, are:
65---*---*---*---*---*---*--58--57
| |
66 37---*---*---*---*---*--31 *
| | | |
* 38 17---*---*--14--13 30 55
| | | | | |
* * 18 5---*---3 * * * <-- As the sequence contains 2, 13
| | | | | | | | and 27, it cannot contain 54.
* * * 6 1---2 11 * *
| | | | | | |
70 41 20 7---*---*--10 27 52
| | | | |
* * 21---*---*---*---*--26 *
| | |
* 43--44---*--46---*---*---*--50
|
73---*---*---*---*---*--79---*---*
See
A333825 for a similar sequences.
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