A004978 -id:A004978 - cp's OEIS frontend

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.

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.

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

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

A001856 A self-generating sequence: every positive integer occurs as a(i)-a(j) for a unique pair i,j.

Original entry on oeis.org

1, 2, 4, 8, 16, 21, 42, 51, 102, 112, 224, 235, 470, 486, 972, 990, 1980, 2002, 4004, 4027, 8054, 8078, 16156, 16181, 32362, 32389, 64778, 64806, 129612, 129641, 259282, 259313, 518626, 518658, 1037316, 1037349, 2074698, 2074734, 4149468

Offset: 1

N. J. A. Sloane

This is a B_2 sequence. More economical recursion: a(1)=1, a(2n)=2a(2n-1), a(2n+1)=a(2n)+r(n), where r(n) is the smallest positive integer not of the form a(j)-a(i) with 1<=iA247556. - Thomas Ordowski, Sep 28 2014

R. K. Guy, Unsolved Problems in Number Theory, E25.
W. Sierpiński, Elementary Theory of Numbers. Państ. Wydaw. Nauk., Warsaw, 1964, p. 444.
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).

T. D. Noe, Table of n, a(n) for n = 1..1000
R. L. Graham, Problem E1910, Amer. Math. Monthly, 73 (1966), 775.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
M. Hall, Cyclic projective planes, Duke Math. J., 4 (1947), 1079-1090.
C. B. A. Peck, Remark on Problem E1910, Amer. Math. Monthly, 75 (1968), 80-81.
W. Sierpiński, Elementary Theory of Numbers, Warszawa 1964.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)

Cf. A054540, A004978, A247556.

a[1] = 1; a[2] = 2; a[n_?OddQ] := a[n] = 2*a[n-1]; a[n_?EvenQ] := a[n] = a[n-1] + r[(n-2)/2]; r[n_] := ( diff = Table[a[j] - a[i], {i, 1, 2*n+1}, {j, i+1, 2*n+1}] // Flatten // Union; max = diff // Last; notDiff = Complement[Range[max], diff]; If[notDiff == {}, max+1, notDiff // First]); Table[a[n], {n, 1, 39}] (* Jean-François Alcover, Dec 31 2012 *)

a(1)=1, a(2)=2, a(2n+1) = 2a(2n), a(2n+2) = a(2n+1) + r(n), where r(n) = smallest positive number not of form a(j) - a(i) with 1 <= i < j <= 2n+1.

More terms from Larry Reeves (larryr(AT)acm.org), Sep 14 2000

A048204 a(n) = T(n+1,n), array T given by A048201.

Original entry on oeis.org

2, 6, 11, 19, 29, 43, 58, 74, 95, 117, 142, 168, 196, 229, 263, 298, 334, 372, 412, 454, 500, 548, 597, 647, 700, 757, 817, 879, 943, 1008, 1078, 1155, 1235, 1316, 1399, 1484, 1570, 1660, 1751, 1843, 1943, 2047, 2154, 2262, 2378, 2497, 2621, 2748, 2880, 3013

Offset: 1

Clark Kimberling

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A004978, A048201.

from itertools import count, accumulate, islice
from collections import deque
def A048204_gen(): # generator of terms
    aset, alist, c = {1}, deque([1]), 0
    for k in count(2):
        if k in aset:
            aset.remove(k)
        else:
            yield (c:=c+k)
            aset |= set(k+d for d in accumulate(alist))
            alist.appendleft(k)
A048204_list = list(islice(A048204_gen(),50)) # Chai Wah Wu, Sep 01 2025

a(n) = A004978(n+2) - 2. - Sean A. Irvine, Jun 05 2021

A048205 a(n)=T(n+2,n), array T given by A048201.

Original entry on oeis.org

4, 9, 17, 27, 41, 56, 72, 93, 115, 140, 166, 194, 227, 261, 296, 332, 370, 410, 452, 498, 546, 595, 645, 698, 755, 815, 877, 941, 1006, 1076, 1153, 1233, 1314, 1397, 1482, 1568, 1658, 1749, 1841, 1941, 2045, 2152, 2260, 2376

Offset: 1

Clark Kimberling

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A004978.

from itertools import count, accumulate, islice
from collections import deque
def A048205_gen(): # generator of terms
    aset, alist, c = {1,3}, deque([2,1]), 0
    for k in count(3):
        if k in aset:
            aset.remove(k)
        else:
            yield (c:=c+k)
            aset |= set(k+d for d in accumulate(alist))
            alist.appendleft(k)
A048205_list = list(islice(A048205_gen(),50)) # Chai Wah Wu, Sep 02 2025

a(n) = A004978(n+3) - 4. - Sean A. Irvine, Jun 05 2021

A048208 a(n) = T(2n-1,n), array T given by A048201.

Original entry on oeis.org

1, 6, 17, 37, 63, 98, 139, 186, 240, 298, 359, 431, 505, 589, 683, 779, 892, 1018, 1150, 1288, 1431, 1593, 1762, 1949, 2151, 2366, 2591, 2823, 3061, 3304, 3557, 3820, 4095, 4367, 4645, 4930, 5222, 5530, 5851, 6176, 6517, 6868, 7222, 7582, 7952, 8331, 8710

Offset: 1

Clark Kimberling

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A004978, A048201.

from itertools import count, accumulate, islice
from collections import deque
def A048208_gen(): # generator of terms
    aset, alist, b, blist, c = set(), deque(), 1, [], 0
    for k in count(1):
        if k in aset:
            aset.remove(k)
        else:
            if c&1:
                yield b-blist[c>>1]
            aset |= set(k+d for d in accumulate(alist))
            alist.appendleft(k)
            blist.append(b)
            b += k
            c += 1
A048208_list = list(islice(A048208_gen(),47)) # Chai Wah Wu, Sep 02 2025

a(n) = A004978(2*n) - A004978(n). - Sean A. Irvine, Jun 05 2021

More terms from Sean A. Irvine, Jun 05 2021

A048210 T(n,1) + T(n,n), array T given by A048201.

Original entry on oeis.org

2, 5, 11, 17, 28, 40, 58, 74, 91, 117, 140, 168, 195, 225, 263, 298, 334, 371, 411, 453, 497, 547, 597, 647, 698, 754, 815, 878, 942, 1008, 1074, 1149, 1233, 1316, 1398, 1483, 1570, 1657, 1751, 1843, 1936, 2044, 2152, 2262, 2371

Offset: 1

Clark Kimberling

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A048201, A004978.

from itertools import count, accumulate, islice
from collections import deque
def A048210_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)+k
            aset |= set(k+d for d in accumulate(alist))
            alist.appendleft(k)
A048210_list = list(islice(A048210_gen(),45)) # Chai Wah Wu, Sep 02 2025

a(n) = 2*A004978(n+1) - A004978(n) - 1. - Sean A. Irvine, Jun 05 2021

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Crossrefs

Formula

A002048 Segmented numbers, or prime numbers of measurement.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

C

Haskell

Maple

Mathematica

Python

Formula

Extensions

A002049 Prime numbers of measurement.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

Python

Formula

A001856 A self-generating sequence: every positive integer occurs as a(i)-a(j) for a unique pair i,j.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A048204 a(n) = T(n+1,n), array T given by A048201.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

Formula

A048205 a(n)=T(n+2,n), array T given by A048201.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

Formula

A048208 a(n) = T(2n-1,n), array T given by A048201.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

Formula

Extensions

A048210 T(n,1) + T(n,n), array T given by A048201.

Views