A002048 -id:A002048 - cp's OEIS frontend

A004978 a(n) = least integer m > a(n-1) such that m - a(n-1) != a(j) - a(k) for all j, k less than n; a(1) = 1, a(2) = 2.

1, 2, 4, 8, 13, 21, 31, 45, 60, 76, 97, 119, 144, 170, 198, 231, 265, 300, 336, 374, 414, 456, 502, 550, 599, 649, 702, 759, 819, 881, 945, 1010, 1080, 1157, 1237, 1318, 1401, 1486, 1572, 1662, 1753, 1845, 1945, 2049, 2156, 2264, 2380, 2499, 2623, 2750, 2882

Offset: 1

N. J. A. Sloane. This was in the 1973 "Handbook", but was then dropped from the database. Resubmitted by Clark Kimberling. Entry revised by N. J. A. Sloane, Jun 10 2012

Equivalently, if S(n) = { a(j) - a(k); n > j > k > 0 }, then a(n) = a(n-1) + M where M = min( {1, 2, 3, ...} \ S(n) ) is the smallest positive integer not in S(n). - M. F. Hasler, Jun 26 2019

			From _M. F. Hasler_, Jun 26 2019: (Start)
After a(1) = 1, a(2) = 2, we have a(3) = least m > a(2) such that m - a(2) = m - 2 is not in { a(j) - a(k); 1 <= k < j < 3 } = { a(2) - a(1) } = { 1 }. Thus we must have m - 2 = 2, whence m = 4.
The next term a(4) is the least m > a(3) such that m - a(3) = m - 4 is not in { a(j) - a(k); 1 <= k < j < 4 } = { 1, 4 - 2 = 2, 4 - 1 = 3 }, i.e., m = 4 + 4 = 8.
The next term a(5) is the least m > a(4) such that m - a(4) = m - 8 is not in { a(j) - a(k); 1 <= k < j < 5 } = { 1, 2, 3, 8 - 4 = 4, 8 - 2 = 6, 8 - 1 = 7 }, i.e., m = 5 + 8 = 13. (End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Nathaniel Johnston)
G. E. Andrews, MacMahon's prime numbers of measurement, Amer. Math. Monthly, 82 (1975), 922-923.

Differences give A002048, see also A048201.
See also A001856.
For n>2, a(n) equals A002049(n-1)+1 and A048204(n-2)+2.

s=1:2000^2;d(1)=1;A004978(1)=1;A004978(2)=2;
for n=3:2000
  A004978(n)=A004978(n-1)+find([d,0]~=s(1:max(size(d))+1),1);
  d(end+1:end+n-1)=A004978(n)-A004978(1:n-1);
  d=sort(unique(d));
end
% Nathaniel Johnston, Feb 09 2011

A004978_vec(N,a=[1..N],S=[1])={for(n=3,N,a[n]=a[n-1]+S[1]+1;S=setunion(S,select(t->t>S[1],vector(n-1,k,a[n]-a[n-k])));for(k=1,#S-1, if(S[k+1]-S[k]>1, S=S[k..-1];next(2)));S[#S]==#S&&S=[#S]);a} \\ M. F. Hasler, Jun 26 2019

from itertools import count, accumulate, islice
from collections import deque
def A004978_gen(): # generator of terms
    aset, alist, c = set(), deque(), 1
    for k in count(1):
        if k in aset:
            aset.remove(k)
        else:
            yield c
            aset |= set(k+d for d in accumulate(alist))
            alist.appendleft(k)
            c += k
A004978_list = list(islice(A004978_gen(),20)) # Chai Wah Wu, Sep 01 2025

Definition corrected by Bryan S. Robinson, Mar 16 2006

Name edited by M. F. Hasler, Jun 26 2019

A048201 Triangular array T read by rows: T(i,j)=b(i+1)-b(i+1-j); j=1,2,...,i; i=1,2,3,...; b=A004978.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 5, 9, 11, 12, 8, 13, 17, 19, 20, 10, 18, 23, 27, 29, 30, 14, 24, 32, 37, 41, 43, 44, 15, 29, 39, 47, 52, 56, 58, 59, 16, 31, 45, 55, 63, 68, 72, 74, 75, 21, 37, 52, 66, 76, 84, 89, 93, 95, 96, 22, 43, 59, 74, 88, 98, 106, 111

Offset: 1

Clark Kimberling

			Rows: {1}, {2,3},{4,6,7}, ...

T(n, 1)=c(n), where c=A002048.

T(n, n)=d(n), where d=A002049.

T(n, 1)=least number not in any preceding row and T(n, k)=T(n, 1)+T(n-1, 1)+...+T(n-k+1, 1) for k >= 2.

A002049 Prime numbers of measurement.

Original entry on oeis.org

1, 3, 7, 12, 20, 30, 44, 59, 75, 96, 118, 143, 169, 197, 230, 264, 299, 335, 373, 413, 455, 501, 549, 598, 648, 701, 758, 818, 880, 944, 1009, 1079, 1156, 1236, 1317, 1400, 1485, 1571, 1661, 1752, 1844, 1944, 2048, 2155, 2263, 2379, 2498, 2622, 2749, 2881

Offset: 1

N. J. A. Sloane

Partial sums of A002048. - Reinhard Zumkeller, May 23 2013

R. K. Guy, Unsolved Problems in Number Theory, E30.
Š. Porubský, On MacMahon's segmented numbers and related sequences. Nieuw Arch. Wisk. (3) 25 (1977), no. 3, 403--408. MR0485763 (58 #5575)
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..3000 from Reinhard Zumkeller)
G. E. Andrews, MacMahon's prime numbers of measurement, Amer. Math. Monthly, 82 (1975), 922-923.
R. L. Graham and C. B. A. Peck, Problem E1910, Amer. Math. Monthly, 75 (1968), 80-81.
P. A. MacMahon, The prime numbers of measurement on a scale, Proc. Camb. Phil. Soc. 21 (1923), 651-654; reprinted in Coll. Papers I, pp. 797-800.

Cf. A002048.

a(n) = A004978(n+1)-1 = A048204(n-1)+1.

import Data.List ((\\))
a002049 n = a002049_list !! (n-1)
a002049_list = g [1..] [] where
   g (x:xs) ys = (last zs) : g (xs \\ zs) (x : ys) where
     zs = scanl (+) x ys
-- Reinhard Zumkeller, May 23 2013

A002048[anmax_] := (a = {}; Do[AppendTo[a, i], {i, 1, anmax}]; asum = {a[[1]] + a[[2]], a[[2]]}; Do[AppendTo[asum, 0], {i, 3, anmax}]; piv = 3; While[piv <= Length[a], Do[a = DeleteCases[a, asum[[i]]], {i, 1, piv - 2}]; Do[asum[[i]] += a[[piv]], {i, 1, piv}]; piv = piv + 1;]; a); A002048[200] // Accumulate (* Jean-François Alcover, Oct 05 2016, adapted from R. J. Mathar's Maple code in A002048. *)

from itertools import count, accumulate, islice
from collections import deque
def A002049_gen(): # generator of terms
    aset, alist, c = set(), deque(), 0
    for k in count(1):
        if k in aset:
            aset.remove(k)
        else:
            yield (c:=c+k)
            aset |= set(k+d for d in accumulate(alist))
            alist.appendleft(k)
A002049_list = list(islice(A002049_gen(),20)) # Chai Wah Wu, Sep 01 2025

Andrews conjectures that a(n) ~ (1/2) n^2 log n / loglog n. - N. J. A. Sloane, Dec 01 2013

A005242 A self-generating sequence.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 12, 14, 15, 16, 19, 20, 21, 24, 25, 27, 28, 32, 33, 34, 37, 38, 40, 42, 43, 44, 46, 47, 48, 51, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 66, 68, 69, 72, 73, 74, 77, 79, 81, 83, 84, 86, 88, 89, 91, 92, 94, 96, 97, 98, 100

Offset: 1

N. J. A. Sloane

nonn

Presumably this is the lexicographically earliest sequence of distinct positive numbers with the property that sums of two or three consecutive earlier terms are excluded. - N. J. A. Sloane, Jan 07 2021

R. K. Guy, Unsolved Problems in Number Theory, E30.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Number of Measurement.

Cf. A002048, A033627.

import Data.List ((\\))
a005242 n = a005242_list !! (n-1)
a005242_list = f [1..] 0 0 where
   f (x:xs) y z = x : f (xs \\ [x + y, x + y + z]) x y
-- Reinhard Zumkeller, May 23 2013

More terms from Jud McCranie, Feb 15 1997

A145198 a(n) is the least number not already in the sequence and not the product of consecutive terms in the sequence.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Gerald Hillier, Oct 04 2008

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A002048, A145278 (complement).

{m=72; v=vector(m); z=2*m; u=vectorsmall(z); k=1; for(n=1, m, while(u[k], k++); v[n]=k; u[k]=1; j=n-1; p=k; while(j>0&&(p=p*v[j])<=z, u[p]=1; j--)); for(i=1, m, print1(v[i], ","))} \\ Klaus Brockhaus, Oct 06 2008
(HP 49G)
HP 49G calculator program from Gerald Hillier, Oct 24 2008
<< 0 OVER NDUPN
->LIST DUPDUP + 1 0
0 -> M V U K J P
<< 1 M
FOR N K
WHILE U OVER
GET
REPEAT 1 +
END 'K' STO V
N K PUT 'V' STO U K
1 PUT N 1 - 'J' STO
K 'P' STO
WHILE J 0 >
IF
THEN P V J
GET * DUP 'P' STO M
2 * <=
ELSE 0
END
REPEAT P 1
PUT J 1 - 'J' STO
END 'U' STO
NEXT V
>>
>>

from operator import mul
from itertools import count, accumulate, islice
from collections import deque
def A145198_gen(): # generator of terms
    aset, alist = set(), deque()
    for k in count(1):
        if k in aset:
            aset.remove(k)
        else:
            yield k
            aset |= set(k*d for d in accumulate(alist,mul))
            alist.appendleft(k)
A145198_list = list(islice(A145198_gen(),50)) # Chai Wah Wu, Sep 01 2025

Extended by Klaus Brockhaus, Oct 06 2008

A362040 a(n) is the number of distinct sums of one or more contiguous terms in the sequence thus far.

Original entry on oeis.org

0, 1, 2, 4, 7, 10, 15, 21, 26, 34, 42, 52, 63, 75, 86, 96, 109, 125, 142, 160, 179, 197, 216, 238, 259, 281, 306, 332, 359, 387, 416, 442, 473, 505, 536, 567, 600, 636, 669, 707, 746, 784, 823, 865, 906, 948, 992, 1036, 1083, 1129, 1172, 1222, 1269, 1321, 1374, 1428

Offset: 1

Neal Gersh Tolunsky, Apr 15 2023

nonn

			At n=1, there are no contiguous subsequences, so a(1)=0.
At n=2, there is one contiguous subsequence: [0], so a(2)=1.
At n=3, there are three contiguous subsequences: [0], [1] and [0, 1], but only two distinct sums (0 and 1), so a(3)=2.

Winston de Greef, Table of n, a(n) for n = 1..10000

Cf. A361798 (number of sums).

Cf. A000217, A002048, A002049.

from itertools import islice
def gen_a():
    seen = set()
    sums = []
    new = 0
    while True:
        for v in sums: seen.add(v + new)
        sums = [v + new for v in sums]
        sums.append(0)
        new = len(seen)
        yield new
print(list(islice(gen_a(), 60))) # Winston de Greef, Apr 15 2023

a(n) <= A000217(n).

a(13)-a(15) corrected and more terms from Winston de Greef, Apr 15 2023

A048202 a(n)=T(n,2), array T given by A048201.

Original entry on oeis.org

3, 6, 9, 13, 18, 24, 29, 31, 37, 43, 47, 51, 54, 61, 67, 69, 71, 74, 78, 82, 88, 94, 97, 99, 103, 110, 117, 122, 126, 129, 135, 147, 157, 161, 164, 168, 171, 176, 181, 183, 192, 204, 211, 215, 224, 235, 243, 251, 259, 265, 270, 278

Offset: 2

Clark Kimberling

Chai Wah Wu, Table of n, a(n) for n = 2..10000

Cf. A048201, A002048.

from itertools import count, accumulate, islice
from collections import deque
def A048202_gen(): # generator of terms
    aset, alist = {1}, deque([1])
    for k in count(2):
        if k in aset:
            aset.remove(k)
        else:
            yield k+alist[0]
            aset |= set(k+d for d in accumulate(alist))
            alist.appendleft(k)
A048202_list = (list(islice(A048202_gen(),52))) # Chai Wah Wu, Sep 02 2025

a(n) = A002048(n) + A002048(n+1). [Gerald Hillier, Oct 04 2008]

A048206 a(n)=least m such that row m of T contains n, array T given by A048201.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 3, 5, 4, 6, 4, 4, 5, 7, 8, 9, 5, 6, 5, 5, 10, 11, 6, 7, 12, 13, 6, 14, 6, 6, 9, 7, 15, 16, 17, 18, 7, 19, 8, 20, 7, 21, 7, 7, 9, 22, 8, 23, 24, 25, 13, 8, 26, 14, 9, 8, 27, 8, 8, 28, 15, 29, 9, 30, 31, 10, 16, 9, 17, 32, 18, 9, 13, 9, 9

Offset: 1

Clark Kimberling

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A048206

Cf. A002048, A048201.

PARI
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See Links section.
```

A338789 Lexicographically earliest sequence of positive integers such that the sum of any number of consecutive terms can only repeat as sum of other consecutive terms after two or more terms between them.

Original entry on oeis.org

1, 2, 4, 1, 9, 2, 1, 21, 2, 3, 1, 8, 2, 23, 18, 2, 8, 7, 9, 2, 1, 20, 16, 1, 21, 36, 3, 32, 2, 3, 4, 11, 13, 8, 2, 13, 27, 4, 18, 3, 7, 5, 8, 133, 3, 22, 31, 46, 19, 8, 47, 14, 3, 2, 14, 10, 44, 3, 5, 1, 10, 3, 9, 6, 19, 73, 39, 22, 36, 6, 1, 60, 17, 32, 227, 2, 134, 9, 45, 11, 33, 3, 37, 1, 8, 12, 14, 8, 1, 67

Offset: 1

S. Brunner, Nov 09 2020

			The solution for a(5):
We look through the numbers step by step and if groups with the same sum are less than 2 terms apart they are put in brackets:
   1,2,4,[1],[1?] - not possible
   [1,2],4,[1,2?] - not possible
   1,2,[4],[1,3?] - not possible
   1,2,[4],1,[4?] - not possible
   1,2,[4,1],[5?] - not possible
   1,[2,4],1,[6?] - not possible
   1,[2,4,1],[7?] - not possible
   [1,2,4,1],[8?] - not possible
   1,2,4,1,9?
There are no 2 sums which contradict the definition of this sequence with a(5) = 9, so this is the next term. In this case we knew it must be the solution because the upper bound of a(n) is always the sum of all previous terms + 1.
Another example for a(8) = 21:
  1,2,4,1,9,2,[1],[1?]   ;  1,2,4,1,9,[2],1,[2?]   ;  1,2,4,1,9,[2,1],[3?]
  1,[2,4,1],9,[2,1,4?]   ;  [1,2,4,1],9,[2,1,5?]   ;  1,2,4,1,[9],[2,1,6?]
  1,2,4,[1,9],[2,1,7?]   ;  1,2,4,1,[9],2,[1,8?]   ;  1,2,4,[1,9],2,[1,9?]
  1,2,4,1,[9,2],[1,10?]  ;  1,2,4,1,[9,2],1,[11?]  ;  1,2,4,1,[9,2,1],[12?]
  1,2,4,[1,9,2,1],[13?]  ;  [1,2,4,1,9],[2,1,14?]  ;  1,2,[4,1,9,2],[1,15?]
  1,2,[4,1,9,2],1,[16?]  ;  1,2,[4,1,9,2,1],[17?]  ;  1,[2,4,1,9,2],1,[18?]
  1,[2,4,1,9,2,1],[19?]  ;  [1,2,4,1,9,2,1],[20?]  ;  1,2,4,1,9,2,1,21?
  -> a(8) = 21.

S. Brunner, Table of n, a(n) for n = 1..10000

Cf. A002048.

def A(lastn):
    n,a,chk,nchk=1,[],[],[]
    while n<=lastn:
        i=1
        while i in chk: i+=1
        for x, v in enumerate(chk): chk[x]=v-i
        chk.extend(nchk)
        for x, v in enumerate(nchk): nchk[x]=v+i
        nchk.append(i)
        chk.extend(nchk)
        chk=[x for x in chk if x>0]
        chk=list(set(chk))
        a.append(i)
        print(i)
        n += 1
    return a

A344841 a(n) is the least positive number not of the form a(k) XOR ... XOR a(m) with 1 <= k <= m < n (where XOR denotes the bitwise XOR operator).

Original entry on oeis.org

1, 2, 4, 5, 8, 12, 14, 16, 17, 19, 20, 21, 23, 32, 33, 35, 48, 49, 51, 56, 57, 59, 64, 65, 67, 68, 69, 71, 76, 77, 79, 80, 81, 83, 84, 85, 87, 92, 93, 95, 128, 129, 131, 132, 133, 135, 140, 141, 143, 192, 193, 195, 196, 197, 199, 204, 205, 207, 224, 225, 227

Offset: 1

Rémy Sigrist, May 29 2021

This sequence has similarities with A002048; here we use XOR, there addition.

All powers of 2 appear.

			  n   a(n)  a(k) XOR ... XOR a(n) for k=1..n
  --  ----  -------------------------------------
   1     1  1
   2     2  3, 2
   3     4  7, 6, 4
   4     5  2, 3, 1, 5
   5     8  10, 11, 9, 13, 8
   6    12  6, 7, 5, 1, 4, 12
   7    14  8, 9, 11, 15, 10, 2, 14
   8    16  24, 25, 27, 31, 26, 18, 30, 16
   9    17  9, 8, 10, 14, 11, 3, 15, 1, 17
  10    19  26, 27, 25, 29, 24, 16, 28, 18, 2, 19

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A002048, A246360.

s=2^0; for (n=1, #a=vector(61), print1 (a[n]=valuation(s+1, 2)", "); z=0; forstep (k=n, 1, -1, s=bitor(s, 2^z=bitxor(z,a[k]))))

from operator import xor
from itertools import count, accumulate, islice
from collections import deque
def A344841_gen(): # generator of terms
    aset, alist = set(), deque()
    for k in count(1):
        if k in aset:
            aset.remove(k)
        else:
            yield k
            aset |= set(k^d for d in accumulate(alist,xor))
            alist.appendleft(k)
A344841_list = list(islice(A344841_gen(),60)) # Chai Wah Wu, Sep 01 2025

Apparently, a(A246360(k)) = 2^(k-1) for any k > 0.

cp's OEIS Frontend

A004978 a(n) = least integer m > a(n-1) such that m - a(n-1) != a(j) - a(k) for all j, k less than n; a(1) = 1, a(2) = 2.

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A048201 Triangular array T read by rows: T(i,j)=b(i+1)-b(i+1-j); j=1,2,...,i; i=1,2,3,...; b=A004978.

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A002049 Prime numbers of measurement.

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A005242 A self-generating sequence.

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A145198 a(n) is the least number not already in the sequence and not the product of consecutive terms in the sequence.

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PARI

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A362040 a(n) is the number of distinct sums of one or more contiguous terms in the sequence thus far.

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Python

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A048202 a(n)=T(n,2), array T given by A048201.

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A048206 a(n)=least m such that row m of T contains n, array T given by A048201.

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