A010672 -id:A010672 - cp's OEIS frontend

A033627 0-additive sequence: not the sum of any previous pair.

1, 2, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172, 175

Offset: 1

Jud McCranie

Conjecture: a(n+1) is the number of distinct numbers of steps required for the last n digits of integers to repeat themselves by iterating the map m -> m^2 + 1. - Ya-Ping Lu, Oct 19 2021

R. K. Guy, Unsolved Problems in Number Theory, C4

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
I. Dolinka, J. East and R. D. Gray, Motzkin monoids and partial Brauer monoids, arXiv preprint arXiv:1512.02279 [math.GR], 2015 (A sequence in Table 5 appears to match this. - _N. J. A. Sloane_, Sep 17 2016)
Eric Weisstein's World of Mathematics, Stöhr Sequence
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A002858, A010672, A016777.

See A244151 for another version.

import Data.List ((\\))
a033627 n = a033627_list !! (n-1)
a033627_list = f [1..] [] where
   f (x:xs) ys = x : f (xs \\ (map (+ x) ys)) (x:ys)
-- Reinhard Zumkeller, Jan 11 2012

Join[{1,2},Range[4,200,3]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2012 *)
f[s_List] := Block[{k = s[[-1]] + 1, ss = Union[ Plus @@@ Subsets[s, {2}]]}, While[ MemberQ[ ss, k], k++]; Append[ s, k]]; Nest[f, {1}, 70] (* Robert G. Wilson v, Jun 23 2014 *)
CoefficientList[Series[x(1+x^2+x^3)/(1-x)^2 , {x, 0, 70}], x] (* Stefano Spezia, Oct 04 2018 *)

a(n)=if(n>2,3*n-5,n) \\ Charles R Greathouse IV, Sep 01 2016

def a(n): return 3*n-5 if n > 2 else n
print([a(n) for n in range(1, 61)]) # Michael S. Branicky, Jun 09 2025

2 together with numbers of form 3k+1 (A016777).
From Gary W. Adamson, May 10 2008: (Start)
Equals binomial transform of [1, 1, 1, 0, -1, 2, -3, 4, -5, 6, -7, ...].
Equals sum of antidiagonal terms of the following arithmetic array: 1, 1, 1, 1, 1, ... 1, 2, 3, 4, 5, ... 1, 3, 5, 7, 9, ... . (End)
From Colin Barker, Sep 19 2012: (Start)
a(n) = 3*n - 5, for n > 2.
a(n) = 2*a(n-1) - a(n-2), for n > 4;
G.f.: x*(1+x^2+x^3)/(1-x)^2. (End)
E.g.f.: 5 + 3*x + x^2/2 + exp(x)*(3*x - 5). - Stefano Spezia, Apr 15 2023

A025582 A B_2 sequence: a(n) is the least value such that sequence increases and pairwise sums of elements are all distinct.

Original entry on oeis.org

0, 1, 3, 7, 12, 20, 30, 44, 65, 80, 96, 122, 147, 181, 203, 251, 289, 360, 400, 474, 564, 592, 661, 774, 821, 915, 969, 1015, 1158, 1311, 1394, 1522, 1571, 1820, 1895, 2028, 2253, 2378, 2509, 2779, 2924, 3154, 3353, 3590, 3796, 3997, 4296, 4432, 4778, 4850

Offset: 1

Dan Hoey

nonn

a(n) is also the least value such that sequence increases and pairwise differences of distinct elements are all distinct.

			After 0, 1, a(3) cannot be 2 because 2+0 = 1+1, so a(3) = 3.

David W. Wilson, Table of n, a(n) for n = 1..1000
Alon Amit, What are some interesting proofs using transfinite induction?, Quora, Nov 15 2014.
LiJun Zhang, Bing Li, and LeeTang Cheng, Constructions of QC LDPC codes based on integer sequences, Science China Information Sciences, June 2014, Volume 57, Issue 6, pp 1-14.
Eric Weisstein's World of Mathematics, B2 Sequence.
Index entries for B_2 sequences.

Row 2 of A365515.
See A011185 for more information.
A010672 is a similar sequence, but there the pairwise sums of distinct elements are all distinct.

from itertools import count, islice
def A025582_gen(): # generator of terms
    aset1, aset2, alist = set(), set(), []
    for k in count(0):
        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
A025582_list = list(islice(A025582_gen(),20)) # Chai Wah Wu, Sep 01 2023

def A025582_list(n):
    a = [0]
    psums = set([0])
    while len(a) < n:
        a += [next(k for k in IntegerRange(a[-1]+1, infinity) if not any(i+k in psums for i in a+[k]))]
        psums.update(set(i+a[-1] for i in a))
    return a[:n]
print(A025582_list(20))
# D. S. McNeil, Feb 20 2011

a(n) = A005282(n) - 1. - Tyler Busby, Mar 16 2024

A011185 A B_2 sequence: a(n) = least value such that sequence increases and pairwise sums of distinct elements are all distinct.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 21, 30, 39, 53, 74, 95, 128, 152, 182, 212, 258, 316, 374, 413, 476, 531, 546, 608, 717, 798, 862, 965, 1060, 1161, 1307, 1386, 1435, 1556, 1722, 1834, 1934, 2058, 2261, 2497, 2699, 2874, 3061, 3197, 3332, 3629, 3712, 3868, 4140, 4447, 4640

Offset: 1

Dan Hoey

nonn

a(n) = least positive integer > a(n-1) and not equal to a(i)+a(j)-a(k) for distinct i and j with 1 <= i,j,k <= n-1. [Comment corrected by Jean-Paul Delahaye, Oct 02 2020.]

Klaus Brockhaus and Dan Hoey, Table of n, a(n) for n = 1..2839
Index entries for B_2 sequences.

Cf. A010672.
Cf. A033627, A003278, A026471, A066720, A075122, A075123, A133097.
Cf. A036241, A349777.

from itertools import islice
def agen(): # generator of terms
    aset, sset, k = set(), set(), 0
    while True:
        k += 1
        while any(k+an in sset for an in aset): k += 1
        yield k; sset.update(k+an for an in aset); aset.add(k)
print(list(islice(agen(), 51))) # Michael S. Branicky, Feb 05 2023

a(n) = A010672(n-1)+1.

A062295 A B_2 sequence: a(n) is the smallest square such that pairwise sums of not necessarily distinct elements are all distinct.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jul 02 2001

nonn

			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.

Klaus Brockhaus, Table of n, a(n) for n = 1..4944
Eric Weisstein's World of Mathematics, B2-Sequence
Index entries for B_2 sequences

Cf. A000290, A011185, A010672, A025582, A005282, A062292, A133743, A133744.

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
A062295_list = list(islice(A062295_gen(),30)) # Chai Wah Wu, Sep 05 2023

Edited, corrected and extended by Klaus Brockhaus, Sep 24 2007

A062294 A B_2 sequence: a(n) is the smallest prime such that the pairwise sums of distinct elements are all distinct.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 29, 47, 67, 83, 131, 163, 233, 307, 397, 443, 617, 727, 809, 941, 1063, 1217, 1399, 1487, 1579, 1931, 2029, 2137, 2237, 2659, 2777, 3187, 3659, 3917, 4549, 4877, 5197, 5471, 5981, 6733, 7207, 7349, 8039, 8291, 8543, 9283, 9689, 10037

Offset: 1

Labos Elemer, Jul 02 2001

nonn

Klaus Brockhaus, Table of n, a(n) for n = 1..3600
Eric Weisstein's World of Mathematics, B2-Sequence
Index entries for B_2 sequences

Cf. A011185, A010672, A025582, A005282, A061781, A061784, A062292, A079848, A133096.

from itertools import islice
from sympy import nextprime
def A062294_gen(): # generator of terms
    aset2, alist, k = set(), [], 0
    while (k:=nextprime(k)):
        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)
A062294_list = list(islice(A062294_gen(),30)) # Chai Wah Wu, Sep 11 2023

Edited, corrected and extended by Klaus Brockhaus, Sep 17 2007

A133097 a(n) = A005282(n) - A011185(n-1).

Original entry on oeis.org

0, 0, 1, 3, 5, 8, 10, 15, 27, 28, 23, 28, 20, 30, 22, 40, 32, 45, 27, 62, 89, 62, 116, 167, 105, 118, 108, 51, 99, 151, 88, 137, 137, 265, 174, 195, 320, 321, 249, 283, 226, 281, 293, 394, 465, 369, 585, 565, 639, 404, 483, 221, 233, 428, 384, 370, 527, 431, 818

Offset: 1

Klaus Brockhaus, Sep 17 2007

sign

Also A025582(n) - A010672(n-1).
A005282 is the sequence of smallest numbers such that the pairwise sums of not necessarily distinct elements are all distinct, whereas A011185 is the sequence of smallest numbers such that the pairwise sums of distinct elements are all distinct.
Sequence has negative terms; the first one is a(65) = -130.

			a(6) = A005282(6) - A011185(6) = 21 - 13 = 8.

Klaus Brockhaus, Table of n, a(n) for n = 1..2400

Cf. A005282, A011185, A025582, A010672, A133096.

from itertools import count, islice
from collections import deque
def A133097_gen(): # generator of terms
    aset2, alist, bset2, blist, aqueue, bqueue = set(), [], set(), [], deque(), deque()
    for k 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()
A133097_list = list(islice(A133097_gen(),30)) # Chai Wah Wu, Sep 11 2023

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

Labos Elemer, Jul 02 2001

nonn

A Mian-Chowla sequence consisting only of cubes.

			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.

Cf. A011185, A010672, A025582, A005282, A062294.

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

A062559 A B2 sequence consisting of powers of primes but not primes: a(n) = least value from A025475 such that sequence increases and pairwise sums of distinct elements are all distinct.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 49, 64, 121, 243, 256, 343, 512, 729, 961, 1024, 1331, 1369, 1849, 2048, 2187, 2197, 2401, 3125, 3481, 4096, 4913, 5329, 6561, 6859, 6889, 8192, 10201, 12769, 14641, 15625, 16384, 16807, 19683, 22201, 22801, 27889, 28561, 29791, 32768

Offset: 0

Labos Elemer, Jul 03 2001

nonn

32 is the first prime power > 1 not in this sequence.

Cf. A000961, A025475, A010672, A011185, A025582, A005282.

from itertools import count, islice
from sympy import factorint
def A062559_gen(): # generator of terms
    aset2, alist = set(), []
    for k in count(0):
        if len(f:=factorint(k).values()) == 1 and max(f) > 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)
A062559_list = list(islice(A062559_gen(),30)) # Chai Wah Wu, Sep 11 2023

More terms from Sean A. Irvine, Apr 03 2023

cp's OEIS Frontend

A033627 0-additive sequence: not the sum of any previous pair.

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A025582 A B_2 sequence: a(n) is the least value such that sequence increases and pairwise sums of elements are all distinct.

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A011185 A B_2 sequence: a(n) = least value such that sequence increases and pairwise sums of distinct elements are all distinct.

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A062295 A B_2 sequence: a(n) is the smallest square such that pairwise sums of not necessarily distinct elements are all distinct.

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A062294 A B_2 sequence: a(n) is the smallest prime such that the pairwise sums of distinct elements are all distinct.

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A133097 a(n) = A005282(n) - A011185(n-1).

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A062292 A B_2 sequence: a(n) is the smallest cube such that the pairwise sums of {a(1)...a(n)} are all distinct.

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A062559 A B2 sequence consisting of powers of primes but not primes: a(n) = least value from A025475 such that sequence increases and pairwise sums of distinct elements are all distinct.

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