A033627 -id:A033627 - cp's OEIS frontend

A171173 Triangle read by rows in which row n lists A033627(n) together with the first 2n-1 positive integers.

2, 1, 4, 1, 2, 3, 7, 1, 2, 3, 4, 5, 10, 1, 2, 3, 4, 5, 6, 7, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 22, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 25, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 28, 1

Offset: 1

Omar E. Pol, Feb 23 2010

The same as A171172 except the initial term.
Also, a(n) is the length of each component of the n-th L-toothpick added to the structure of A171165.
See also A171175, a permutation of the natural numbers.

			Triangle begins:
2,1,
4,1,2,3,
7,1,2,3,4,5,
10,1,2,3,4,5,6,7,
13,1,2,3,4,5,6,7,8,9,
16,1,2,3,4,5,6,7,8,9,10,11,
19,1,2,3,4,5,6,7,8,9,10,11,12,13,
22,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,
25,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,

Harvey P. Dale, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000027, A033627, A172310, A171164, A161165, A161166, A171172, A171174, A171175, A171176, A171177, A171178.

Join[{2,1},Flatten[Table[Flatten[{3n+1,Range[2n+1]}],{n,10}]]] (* Harvey P. Dale, Nov 24 2011 *)

A171174 Triangle read by rows in which row n lists A033627(n) together with the first 2n-1 numbers <> 0 of A038608.

Original entry on oeis.org

2, -1, 4, -1, 2, -3, 7, -1, 2, -3, 4, -5, 10, -1, 2, -3, 4, -5, 6, -7, 13, -1, 2, -3, 4, -5, 6, -7, 8, -9, 16, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 19, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 22, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 25, -1, 2, -3, 4

Offset: 1

Omar E. Pol, Feb 23 2010

Absolute values give A171173.

Note that the partial sums of this sequence gives A171175, a permutation of the natural numbers.

			Triangle begins:
2, -1,
4, -1,2,-3,
7, -1,2,-3,4,-5,
10,-1,2,-3,4,-5,6,-7,
13,-1,2,-3,4,-5,6,-7,8,-9,
16,-1,2,-3,4,-5,6,-7,8,-9,10,-11,
19,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,
22,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,14,-15,
25,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,14,-15,16,-17,

Cf. A033627, A038608, A172310, A171164, A161165, A161166, A171172, A171173, A171175, A171176, A171177, A171178.

A033628 Numbers that are in both the 1-additive and 0-additive sequences (A002858 and A033627).

Original entry on oeis.org

1, 2, 4, 13, 16, 28, 82, 97, 106, 145, 148, 175, 238, 241, 253, 316, 319, 358, 370, 382, 400, 409, 412, 451, 502, 544, 568, 607, 646, 673, 685, 688, 739, 751, 781, 820, 847, 949, 991, 1018, 1030, 1081, 1186, 1252, 1360, 1387, 1462, 1465, 1489, 1492

Offset: 1

Jud McCranie

nonn

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

A227393 a(n) = concatenation of first 3n terms of A033627.

Original entry on oeis.org

1, 124, 12471013, 12471013161922, 12471013161922252831, 124710131619222252831343740, 124710131619222252831343740434649, 124710131619222252831343740434649525558

Offset: 1

V. T. Jayabalaji, Jul 15 2013

All entries are == 1 (mod 3). V. T. Jayabalaji, Jul 13 2013

			124 is the concatenation of 1,2 and 4.
12471013 is the concatenation of 1, 2, 4, 7 10 and 13.

Cf. A033627

Edited by N. J. A. Sloane, Jul 16 2013

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.

A026474 a(n) = least positive integer > a(n-1) and not equal to a(i)+a(j) or a(i)+a(j)+a(k) for 1<=i

Original entry on oeis.org

1, 2, 4, 8, 15, 22, 29, 36, 43, 50, 57, 64, 71, 78, 85, 92, 99, 106, 113, 120, 127, 134, 141, 148, 155, 162, 169, 176, 183, 190, 197, 204, 211, 218, 225, 232, 239, 246, 253, 260, 267, 274, 281, 288, 295, 302, 309, 316, 323, 330, 337, 344, 351, 358, 365, 372

Offset: 1

Clark Kimberling

All h-Stohr sequences have formula: h terms 1,2,..,2^(n-1),..,2^(h-1) and then continue (2^h-1)(n-h)+1. - Henry Bottomley, Feb 04 2000

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Stoehr Sequence.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A026472, A026476, A033627, A051039, A051040.

[1,2,4] cat [7*n+1: n in [1..60]]; // Vincenzo Librandi, Oct 18 2013

Join[{1,2,4,8},Range[15,500,7]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2012 *)
CoefficientList[Series[(1 + x^2 + 2 x^3 + 3 x^4)/(1 - x)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Oct 18 2013 *)
LinearRecurrence[{2,-1},{1,2,4,8,15},60] (* Harvey P. Dale, May 14 2018 *)

a(n)=if(n>3,7*n-20,2^(n-1)) \\ Charles R Greathouse IV, Sep 17 2015

Starts 1, 2, 4 then the numbers 7*(n-3)+1.

a(n) = 7*n-20 for n>3. a(n) = 2*a(n-1)-a(n-2) for n>5. G.f.: x*(1+x^2+2*x^3+3*x^4)/(1-x)^2. - Colin Barker, Sep 19 2012

More terms from Eric W. Weisstein

A002048 Segmented numbers, or prime numbers of measurement.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 14, 15, 16, 21, 22, 25, 26, 28, 33, 34, 35, 36, 38, 40, 42, 46, 48, 49, 50, 53, 57, 60, 62, 64, 65, 70, 77, 80, 81, 83, 85, 86, 90, 91, 92, 100, 104, 107, 108, 116, 119, 124, 127, 132, 133, 137, 141, 144, 145, 148, 150, 151, 154, 158, 159, 163, 165

Offset: 1

N. J. A. Sloane

The segmented numbers are the positive integers excluding those equal to the sum of two or more consecutive smaller terms. The prime numbers of measurement are their partial sums, cf. A002049. - M. F. Hasler, Jun 26 2019

Without the requirement that the smaller terms be consecutive, the sequence becomes the sequence of powers of 2 (A000079). - Alonso del Arte, Jan 25 2020

			Although 5 is the sum of the terms 1 and 4, those prior terms are not consecutive, and therefore 5 is in the sequence.
6 is not in the sequence because it is the sum of consecutive prior terms 2 and 4.
7 is not in the sequence either because it is also the sum of consecutive prior terms, in this case 1, 2, 4.
8 is in the sequence because no sum whatsoever of distinct prior terms adds up to 8.

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..7836 from R. J. Mathar)
G. E. Andrews, MacMahon's prime numbers of measurement, Amer. Math. Monthly, 82 (1975), 922-923.
Thomas Bloom, Problem 359, Erdős Problems.
R. L. Graham and C. B. A. Peck, Problem E1910, Amer. Math. Monthly, 75 (1968), 80-81.
R. K. Guy, Letter to G. E. Andrews, Apr 14 1975
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.
Samuel B. Reid, C program for A002048
Eric Weisstein's World of Mathematics, Prime Number of Measurement.

Cf. A002049 (partial sums), A004978, A005242, A033627.

// See Links section for C program by Samuel B. Reid, Jan 26 2020

import Data.List ((\\))
a002048 n = a002048_list !! (n-1)
a002048_list = f [1..] [] where
   f (x:xs) ys = x : f (xs \\ scanl (+) x ys) (x : ys)
-- Reinhard Zumkeller, May 23 2013

A002048 := proc(anmax::integer,printlist::boolean)
local a, asum,su,i,piv,j;
a := [];
for i from 1 to anmax do
a := [op(a),i];
od:
if printlist then
printf("%d %d\n",1,a[1]);
printf("%d %d\n",2,a[2]);
fi;
asum := [a[1]+a[2],a[2]];
for i from 3 to anmax do
asum := [op(asum),0];
od:
piv := 3;
while piv <= nops(a) do
for i from 1 to piv-2 do
a := remove(has,a, asum[i]);
od:
if printlist then
printf("%a %a\n",piv,a[piv]);
fi;
for i from 1 to piv do
asum := subsop(i=asum[i]+a[piv], asum);
od:
piv := piv+1;
od;
RETURN(a);
end:
A002048(40000,true);
# R. J. Mathar, Jun 04 2006

A002048[anmax_] := (a = {}; Do[AppendTo[a, i], {i, 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, piv}]; piv = piv + 1;]; a); A002048[63] (* Jean-François Alcover, Jul 28 2011, converted from R. J. Mathar's Maple prog. *)
searchMax = 200; segmNums = {1}; curr = 2; While[curr < searchMax, If[Not[MemberQ[Apply[Plus, Subsequences[segmNums], 1], curr]], AppendTo[segmNums, curr], ];  curr = curr + 1]; segmNums (* Alonso del Arte, Jan 25 2020 *)

from itertools import count, accumulate, islice
from collections import deque
def A002048_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))
            alist.appendleft(k)
A002048_list = list(islice(A002048_gen(),20)) # Chai Wah Wu, Sep 01 2025

Andrews conjectures that lim_{n -> oo} n log n / (a(n) loglog n) = 1. - N. J. A. Sloane, Dec 01 2013

More terms from R. J. Mathar, May 31 2006

A051039 4-Stohr sequence.

Original entry on oeis.org

1, 2, 4, 8, 16, 31, 46, 61, 76, 91, 106, 121, 136, 151, 166, 181, 196, 211, 226, 241, 256, 271, 286, 301, 316, 331, 346, 361, 376, 391, 406, 421, 436, 451, 466, 481, 496, 511, 526, 541, 556, 571, 586, 601, 616, 631, 646, 661, 676, 691, 706, 721, 736, 751

Offset: 1

Eric W. Weisstein

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Stoehr Sequence.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A033627, A026474, A051040.

Join[{1,2,4,8},Range[16,3001,15]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2012 *)
LinearRecurrence[{2,-1},{1,2,4,8,16,31},60] (* Harvey P. Dale, Dec 25 2016 *)

a(n)=if(n>4,15*n-59,2^(n-1)) \\ Charles R Greathouse IV, Jun 20 2024

a(n) = 15*n-59 for n>4. a(n) = 2*a(n-1)-a(n-2) for n>6. G.f.: x*(7*x^5+4*x^4+2*x^3+x^2+1)/(x-1)^2. - Colin Barker, Sep 19 2012

A026471 a(n) = least positive integer > a(n-1) and not of the form a(i) + a(j) + a(k) for 1 <= i < j < k <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 13, 14, 15, 25, 26, 27, 37, 38, 48, 49, 50, 60, 61, 71, 72, 73, 83, 84, 94, 95, 96, 106, 107, 117, 118, 119, 129, 130, 140, 141, 142, 152, 153, 163, 164, 165, 175, 176, 186, 187, 188, 198, 199, 209, 210, 211, 221, 222, 232, 233, 234, 244, 245, 255

Offset: 1

Clark Kimberling

Wieb Bosma, Rene Bruin, Robbert Fokkink, Jonathan Grube, Anniek Reuijl, and Thian Tromp, Using Walnut to solve problems from the OEIS, arXiv:2503.04122 [math.NT], 2025.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A003278, A033627, A075122, A075123.

{1, 5, 13} union {n congruent 2, 3, 4, 14, 15 mod 23}, proved by Matthew Akeran. - Ralf Stephan, Nov 15 2004

G.f.: (9*x^11-7*x^10+9*x^8+7*x^5+x^4+x^3+x^2+x+1)*x/(x^6-x^5-x+1). - Alois P. Heinz, Aug 06 2018

Edited by Floor van Lamoen, Sep 02 2002

A060469 Smallest positive a(n) such that number of solutions to a(n) = a(j)+a(k) j

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 23, 25, 28, 30, 35, 37, 40, 42, 47, 49, 52, 54, 59, 61, 64, 66, 71, 73, 76, 78, 83, 85, 88, 90, 95, 97, 100, 102, 107, 109, 112, 114, 119, 121, 124, 126, 131, 133, 136, 138, 143, 145, 148, 150, 155, 157, 160, 162, 167, 169, 172, 174

Offset: 1

Henry Bottomley, Mar 15 2001

Numbers {1,4,6,11} mod 12 plus {2,3,8}.

			11 is in the sequence since it is 3+8 but no other sum of two distinct terms.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A002858, A033627, A056594, A060469, A060470, A060471, A060472.

Virtually identical to A003662.

LinearRecurrence[{1, 0, 0, 1, -1}, {1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 23}, 100] (* Paolo Xausa, Mar 04 2024 *)

Vec(x*(2*x^10+x^8+x^7+2*x^6+x^5+x^4+x^3+x^2+x+1)/((x-1)^2*(x+1)*(x^2+1)) + O(x^100)) \\ Colin Barker, Feb 27 2015

a(n) = a(n-1)+a(n-4)-a(n-5) for n>9. - Colin Barker, Feb 27 2015
G.f.: x*(2*x^10+x^8+x^7+2*x^6+x^5+x^4+x^3+x^2+x+1) / ((x-1)^2*(x+1)*(x^2+1)). - Colin Barker, Feb 27 2015
a(n) = (6*n - 22 - (-1)^n + A056594(n) - A056594(n+1))/2 for n > 6. - Stefano Spezia, Mar 11 2025

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A171173 Triangle read by rows in which row n lists A033627(n) together with the first 2n-1 positive integers.

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A171174 Triangle read by rows in which row n lists A033627(n) together with the first 2n-1 numbers <> 0 of A038608.

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A033628 Numbers that are in both the 1-additive and 0-additive sequences (A002858 and A033627).

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A227393 a(n) = concatenation of first 3n terms of A033627.

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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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A026474 a(n) = least positive integer > a(n-1) and not equal to a(i)+a(j) or a(i)+a(j)+a(k) for 1<=i

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A002048 Segmented numbers, or prime numbers of measurement.

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A051039 4-Stohr sequence.

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