A002049 -id:A002049 - cp's OEIS frontend

A173702 Partial sums of prime numbers of measurement A002049.

1, 4, 11, 23, 43, 73, 117, 176, 251, 347, 465, 608, 777, 974, 1204, 1468, 1767, 2102, 2475, 2888, 3343, 3844, 4393, 4991, 5639, 6340, 7098, 7916, 8796, 9740, 10749, 11828, 12984, 14220, 15537, 16937

Offset: 1

Jonathan Vos Post, Nov 25 2010

Prime partial sums of prime numbers of measurement begin: 11, 23, 43, 73, 251, 347, 3343, 5639, 16937. The subsequence of squares begins: 1, 4, 3844 = 2^2 * 31^2.

			a(10) =  1 + 3 + 7 + 12 + 20 + 30 + 44 + 59 + 75 + 96 = 347 is prime.

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

Cf. A002049.

from itertools import count, accumulate, islice
from collections import deque
def A173702_gen(): # generator of terms
    aset, alist, a, b = set(), deque(), 0, 0
    for k in count(1):
        if k in aset:
            aset.remove(k)
        else:
            a += k
            yield (b:=a+b)
            aset |= set(k+d for d in accumulate(alist))
            alist.appendleft(k)
A173702_list = list(islice(A173702_gen(),50)) # Chai Wah Wu, Sep 02 2025

a(n) = Sum_{i=1..n} A002049(i).

A185846 Primes in A002049.

Original entry on oeis.org

3, 7, 59, 197, 373, 701, 1009, 1571, 2749, 3581, 5701, 5881, 6247, 8269, 9397, 14369, 15881, 31019, 32707, 41801, 47269, 53633, 70177, 81931, 84701, 93239, 118369, 131213, 133873, 138373, 147661, 149561, 161159, 191929, 194069, 203857, 221813, 252823, 290161, 298201

Offset: 1

Jonathan Vos Post, Feb 06 2011

			a(3) = A002049(8) = 59 = the 17th prime.

Chai Wah Wu, Table of n, a(n) for n = 1..8849

Cf. A002049.

from itertools import count, accumulate, islice
from collections import deque
from sympy import isprime
def A185846_gen(): # generator of terms
    aset, alist, c = set(), deque(), 0
    for k in count(1):
        if k in aset:
            aset.remove(k)
        else:
            c += k
            if isprime(c): yield c
            aset |= set(k+d for d in accumulate(alist))
            alist.appendleft(k)
A185846_list = list(islice(A185846_gen(),40)) # Chai Wah Wu, Sep 01 2025

More terms from Nathaniel Johnston, Feb 09 2011

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

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.

Original entry on oeis.org

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.

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

cp's OEIS Frontend

A173702 Partial sums of prime numbers of measurement A002049.

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A185846 Primes in A002049.

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A002048 Segmented numbers, or prime numbers of measurement.

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

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