A002033 -id:A002033 - cp's OEIS frontend

A143104 Infinite Redheffer matrix read by upwards antidiagonals.

1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1

Offset: 1

Mats Granvik, Roger L. Bagula and Gary W. Adamson, Jul 24 2008

Note that Redheffer's matrix (1977) is actually given by A077049: the first row starts with a single 1. We follow the nomenclature of Wilf, Dana, Vaughan and Weisstein, which uses the transpose and sets the first column to all-1. - R. J. Mathar, Jul 22 2017
The determinant of the n X n Redheffer matrix is given by Mertens's function A002321(n) [Barrett].
For n > 1, replacing a(n,n) with 0 in the Redheffer matrix and taking the determinant gives Moebius(n) = A008683(n). The number of permutations with nonzero contribution to this determinant is given by A002033. For first few n, these permutations are shown in the sequences A144193 to A144201. - Mats Granvik, Sep 14 2008
The determinant that is the Moebius function was discovered by reading the blog post "The Mobius function is strongly orthogonal to nilsequences" by Terence Tao. - Mats Granvik, Jan 24 2009

			1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0
1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0
1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0
1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

R. C. Vaughan, On the eigenvalues of Redheffer's matrix I, in: Number Theory with an Emphasis on the Markoff Spectrum (Provo, Utah, 1991), 283-296, Lecture Notes in Pure and Appl. Math., 147, Dekker, New-York, 1993.

Enrique Pérez Herrero, Rows n = 1..100 of triangle, flattened
W. B. Barret, R. W. Forcade and A. D. Pollington, On the spectral radius of a (0,1) matrix related to Mertens' Function, Lin. Alg. Applic. 107 (1988) 151-159.
Olivier Bordellès and Benoit Cloitre, A matrix inequality for Möbius functions, J. Inequal. Pure and Appl. Math., Volume 10 (2009), Issue 3, Article 62, 9 pp.
Will Dana, Eigenvalues of the Redheffer Matrix and their relation to the Mertens Function, (2015)
R. M. Redheffer, Eine explizit lösbare Optimierungsaufgabe, Internat. Schiftenreihe Numer. Math., 36 (1977), 213-216.
T. Tao, The Mobius function is strongly orthogonal to nilsequences
R. C. Vaughan, On the eigenvalues of Redheffer's matrix, II, J. Austral. Math. Soc. (Series A) 60 (1996), 260-273.
Eric Weisstein's World of Mathematics, Redheffer Matrix.
Herbert S. Wilf, The Redheffer matrix of a partially ordered set, arXiv:math/0408263 [math.CO], 2004.
Herbert S. Wilf, The Redheffer matrix of a partially ordered set, The Electronic Journal of Combinatorics 11(2) (2004), #R10.

Cf. A008683, A051731.

Cf. A002033, A144193 .. A144201, A143142. - Mats Granvik, Sep 14 2008

=if(mod(column();row())=0;1;if(column()=1;1;0)). Produces the Redheffer matrix.

A143104 := proc(i,j)
    if modp(j,i) =0 or j = 1 then
        1;
    else
        0;
    end if;
end proc:
for d from 2 to 10 do
    for m from d-1 to 1 by -1 do
        n := d-m ;
        printf("%d ",A143104(n,m)) ;
    end do:
end do: # R. J. Mathar, Jul 23 2017

Redheffer[i_, j_] := Boole[Divisible[i, j] || (i == 1)];
T[n_] := n*(n + 1)/2;
S[n_] := Floor[1/2 + Sqrt[2 n]];
j[n_] := 1 + T[S[n]] - n;
i[n_] := 1 + S[n] - j[n];
A143104[n_] := Redheffer[i[n], j[n]]; (* Enrique Pérez Herrero, Apr 13 2010 *)
a[i_, j_] := If[j == 1 || Divisible[j, i], 1, 0];
Table[a[i-j+1, j], {i, 1, 14}, {j, 1, i}] // Flatten (* Jean-François Alcover, Aug 07 2018 *)

PARI
```
{ a(i,j) = (j==1) || (j%i==0); }
```

a(i,j) = 1 if j=1 or i|j; 0 otherwise.

a(A000217(n)) = a(A000217(n)+1) = 1. - Enrique Pérez Herrero, Apr 16 2010

Edited and extended by Max Alekseyev, Oct 28 2008

A353867 Heinz numbers of integer partitions where every partial run (consecutive constant subsequence) has a different sum, and these sums include every integer from 0 to the greatest part.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 20, 30, 32, 56, 64, 90, 128, 140, 176, 210, 256, 416, 512, 616, 990, 1024, 1088, 1540, 2048, 2288, 2310, 2432, 2970, 4096, 4950, 5888, 7072, 7700, 8008, 8192, 11550, 12870, 14848, 16384, 20020, 20672, 30030, 31744, 32768, 38896, 50490, 55936

Offset: 1

Gus Wiseman, Jun 07 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
Related concepts:
- A partition whose submultiset sums cover an initial interval is said to be complete (A126796, ranked by A325781).
- In a knapsack partition (A108917, ranked by A299702), every submultiset has a different sum.
- A complete partition that is also knapsack is said to be perfect (A002033, ranked by A325780).
- A partition whose partial runs have all different sums is said to be rucksack (A353864, ranked by A353866, complement A354583).

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   16: {1,1,1,1}
   20: {1,1,3}
   30: {1,2,3}
   32: {1,1,1,1,1}
   56: {1,1,1,4}
   64: {1,1,1,1,1,1}
   90: {1,2,2,3}
  128: {1,1,1,1,1,1,1}
  140: {1,1,3,4}
  176: {1,1,1,1,5}
  210: {1,2,3,4}
  256: {1,1,1,1,1,1,1,1}

Knapsack partitions are counted by A108917, ranked by A299702.
Complete partitions are counted by A126796, ranked by A325781.
These partitions are counted by A353865.
This is a special case of A353866, counted by A353864, complement A354583.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A073093 counts prime-power divisors.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run-sums, nonprime A353834.
A353836 counts partitions by number of distinct run-sums.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353863 counts partitions whose weak run-sums cover an initial interval.
Cf. A018818, A181819, A182857, A304442, A316413, A325862, A353835, A353838, A353839, A353861, A353931.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
norqQ[m_]:=Sort[m]==Range[0,Max[m]];
msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
Select[Range[1000],norqQ[Total/@Select[msubs[primeMS[#]],SameQ@@#&]]&]

A325792 Positive integers with as many proper divisors as the sum of their prime indices.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 18, 20, 32, 42, 54, 56, 64, 100, 128, 162, 176, 204, 234, 256, 260, 294, 308, 315, 350, 392, 416, 486, 500, 512, 690, 696, 798, 920, 1024, 1026, 1064, 1088, 1116, 1122, 1190, 1365, 1430, 1458, 1496, 1755, 1936, 1968, 2025, 2048, 2058, 2079

Offset: 1

Gus Wiseman, May 23 2019

nonn

First differs from A325780 in having 204.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, with sum A056239(n).

			The term 42 is in the sequence because it has 7 proper divisors (1, 2, 3, 6, 7, 14, 21) and its sum of prime indices is also 1 + 2 + 4 = 7.
The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    32: {1,1,1,1,1}
    42: {1,2,4}
    54: {1,2,2,2}
    56: {1,1,1,4}
    64: {1,1,1,1,1,1}
   100: {1,1,3,3}
   128: {1,1,1,1,1,1,1}
   162: {1,2,2,2,2}
   176: {1,1,1,1,5}
   204: {1,1,2,7}
   234: {1,2,2,6}
   256: {1,1,1,1,1,1,1,1}

Positions of 1's in A325794.
Heinz numbers of the partitions counted by A325828.
Cf. A000005, A002033, A056239, A112798, A299702, A304793.
Cf. A325694, A325780, A325781, A325793, A325795, A325796, A325797, A325798.

Select[Range[100],DivisorSigma[0,#]-1==Total[Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]]&]

A347461 Number of distinct possible alternating products of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 10, 12, 16, 19, 23, 27, 34, 41, 49, 57, 67, 78, 91, 106, 125, 147, 166, 187, 215, 245, 277, 317, 357, 405, 460, 524, 592, 666, 740, 829, 928, 1032, 1147, 1273, 1399, 1555, 1713, 1892, 2087, 2298, 2523, 2783, 3070, 3383, 3724, 4104, 4504

Offset: 0

Gus Wiseman, Oct 06 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

			Partitions representing each of the a(7) = 10 alternating products are:
     (7) -> 7
    (61) -> 6
    (52) -> 5/2
   (511) -> 5
    (43) -> 4/3
   (421) -> 2
  (4111) -> 4
   (331) -> 1
   (322) -> 3
  (3211) -> 3/2

The version for alternating sum is A004526.
Counting only integers gives A028310, reverse A347707.
The version for factorizations is A347460, reverse A038548.
The reverse version is A347462.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A108917 counts knapsack partitions, ranked by A299702.
A122768 counts distinct submultisets of partitions.
A126796 counts complete partitions.
A293627 counts knapsack factorizations by sum.
A301957 counts distinct subset-products of prime indices.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
A304793 counts distinct positive subset-sums of prime indices.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
Cf. A000070, A001055, A002033, A002219, A028983, A119620, A325768, A345926, A347443, A347444, A347445, A347446.

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@IntegerPartitions[n]]],{n,0,30}]

A353865 Number of complete rucksack partitions of n. Partitions whose weak run-sums are distinct and cover an initial interval of nonnegative integers.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 04 2022

nonn

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). A weak run-sum is the sum of any consecutive constant subsequence.

Do all positive integers appear only finitely many times in this sequence?

			The a(n) compositions for n = 1, 3, 9, 15, 18:
  (1)  (21)   (4311)       (54321)            (543321)
       (111)  (51111)      (532221)           (654111)
              (111111111)  (651111)           (7611111)
                           (81111111)         (111111111111111111)
                           (111111111111111)
For example, the weak runs of y = {7,5,4,4,3,3,3,1,1} are {}, {1}, {1,1}, {3}, {4}, {5}, {3,3}, {7}, {4,4}, {3,3,3}, with sums 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, which are all distinct and cover an initial interval, so y is counted under a(31).

Perfect partitions are counted by A002033, ranked by A325780.
Knapsack partitions are counted by A108917, ranked by A299702.
This is the complete case of A353864, ranked by A353866.
These partitions are ranked by A353867.
A000041 counts partitions, strict A000009.
A275870 counts collapsible partitions, ranked by A300273.
A304442 counts partitions with all equal run-sums, ranked by A353833.
A353832 represents the operation of taking run-sums of a partition.
A353836 counts partitions by number of distinct run-sums.
A353837 counts partitions with distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353850 counts compositions with all distinct run-sums, ranked by A353852.
A353863 counts partitions whose weak run-sums cover an initial interval.
Cf. A008284, A018818, A225485, A325239, A325862, A353834, A353839.

norqQ[m_]:=Sort[m]==Range[0,Max[m]];
msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
Table[Length[Select[IntegerPartitions[n],norqQ[Total/@Select[msubs[#],SameQ@@#&]]&]],{n,0,15}]

a(n) = my(c=0, s, v); if(n, forpart(p=n, if(p[1]==1, v=List([s=1]); for(i=2, #p, if(p[i]==p[i-1], listput(v, s+=p[i]), listput(v, s=p[i]))); s=#v; listsort(v, 1); if(s==#v&&s==v[s], c++))); c, 1); \\ Jinyuan Wang, Feb 21 2025

More terms from Jinyuan Wang, Feb 21 2025

A051709 a(n) = sigma(n) + phi(n) - 2n.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 1, 2, 0, 8, 0, 2, 2, 7, 0, 9, 0, 10, 2, 2, 0, 20, 1, 2, 4, 12, 0, 20, 0, 15, 2, 2, 2, 31, 0, 2, 2, 26, 0, 24, 0, 16, 12, 2, 0, 44, 1, 13, 2, 18, 0, 30, 2, 32, 2, 2, 0, 64, 0, 2, 14, 31, 2, 32, 0, 22, 2, 28, 0, 75, 0, 2, 14, 24, 2, 36, 0

Offset: 1

Jud McCranie and Carlos Rivera

nonn

Sigma is the sum of divisors (A000203), and phi is the Euler totient function (A000010). - Michael B. Porter, Jul 05 2013
Because sigma and phi are multiplicative functions, it is easy to show that (1) if a(n)=0, then n is prime or 1 and (2) if a(n)=2, then n is the product of two distinct prime numbers. Note that a(n) is the n-th term of the Dirichlet series whose generating function is given below. Using the generating function, it is theoretically possible to compute a(n). Hence a(n)=0 could be used as a primality test and a(n)=2 could be used as a test for membership in P2 (A006881). - T. D. Noe, Aug 01 2002
It appears that a(n) - A002033(n) = zeta(s-1) * (zeta(s) - 2 + 1/zeta(s)) + 1/(zeta(s)-2). - Eric Desbiaux, Jul 04 2013
a(n) = 1 if and only if n = prime(k)^2 (n is in A001248). It seems that a(n) = k has only finitely many solutions for k >= 3. - Jianing Song, Jun 27 2021

			a(5) = sigma(5) + phi(5) - 2*5 = 6 + 4 - 10 = 0.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (First 1000 terms from T. D. Noe.)
Carlos Rivera, Puzzle 76. z(n)=sigma(n) + phi(n) - 2n, The Prime Puzzles and Problems Connection.

Cf. A000010, A000203, A001065, A001248, A005843, A006881, A051612, A051953, A065387, A072780, A228498 (= a(n^2)), A297159, A324048, A344994, A344995, A344996, A345048, A345054.
Cf. A278373 (range of this sequence), A056996 (numbers not present).
Cf. also A344753, A345001 (analogous sequences).

Table[DivisorSigma[1,n]+EulerPhi[n]-2n,{n,80}] (* Harvey P. Dale, Apr 08 2015 *)

a(n)=sigma(n)+eulerphi(n)-2*n \\ Charles R Greathouse IV, Jul 05 2013

A051709(n) = -sumdiv(n,d,(dAntti Karttunen, Mar 02 2018

Dirichlet g.f.: zeta(s-1) * (zeta(s) - 2 + 1/zeta(s)). - T. D. Noe, Aug 01 2002
From Antti Karttunen, Mar 02 2018: (Start)
a(n) = A001065(n) - A051953(n).  [Difference between the sum of proper divisors of n and their Moebius-transform.]
a(n) = -Sum_{d|n, dA008683(n/d)*A001065(d). (End)
Sum_{k=1..n} a(k) = (3/(Pi^2) + Pi^2/12 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 03 2023

A325799 Sum of the prime indices of n minus the number of distinct positive subset-sums of the prime indices of n.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 2, 1, 4, 0, 5, 2, 2, 0, 6, 0, 7, 0, 3, 3, 8, 0, 4, 4, 3, 1, 9, 0, 10, 0, 4, 5, 4, 0, 11, 6, 5, 0, 12, 0, 13, 2, 2, 7, 14, 0, 6, 2, 6, 3, 15, 0, 5, 0, 7, 8, 16, 0, 17, 9, 4, 0, 6, 1, 18, 4, 8, 2, 19, 0, 20, 10, 3, 5, 6, 2, 21, 0, 4, 11

Offset: 1

Gus Wiseman, May 23 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, with sum A056239(n). A positive subset-sum of an integer partition is any sum of a nonempty submultiset of it.

			The prime indices of 21 are {2,4}, with positive subset-sums {2,4,6}, so a(21) = 6 - 3 = 3.

Positions of 1's are A325800.
Positions of nonzero terms are A325798.
Cf. A000005, A002033, A056239, A108917, A112798, A276024, A299702, A304793.
Cf. A325694, A325780, A325794, A325801, A325802.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p] k]];
Table[hwt[n]-Length[Union[hwt/@Rest[Divisors[n]]]],{n,30}]

a(n) = A056239(n) - A304793(n).

A347462 Number of distinct possible reverse-alternating products of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 11, 13, 17, 22, 28, 33, 42, 51, 59, 69, 84, 100, 117, 137, 163, 191, 222, 256, 290, 332, 378, 429, 489, 564, 643, 729, 819, 929, 1040, 1167, 1313, 1473, 1647, 1845, 2045, 2272, 2521, 2785, 3076, 3398, 3744, 4115, 4548, 5010, 5524, 6086

Offset: 0

Gus Wiseman, Oct 06 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.

			Partitions representing each of the a(7) = 11 reverse-alternating products:
     (7) -> 7
    (61) -> 1/6
    (52) -> 2/5
   (511) -> 5
    (43) -> 3/4
   (421) -> 2
  (4111) -> 1/4
   (331) -> 1
   (322) -> 3
  (3211) -> 2/3
  (2221) -> 1/2

The version for non-reverse alternating sum instead of product is A004526.
Counting only integers gives A028310, non-reverse A347707.
The version for factorizations is A038548, non-reverse A347460.
The non-reverse version is A347461.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A108917 counts knapsack partitions, ranked by A299702.
A122768 counts distinct submultisets of partitions.
A126796 counts complete partitions.
A293627 counts knapsack factorizations by sum.
A301957 counts distinct subset-products of prime indices.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
A304793 counts distinct positive subset-sums of prime indices.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
Cf. A000070, A001055, A002033, A002219, A028983, A119620, A325768, A345926, A347443, A347444, A347445, A347446.

revaltprod[q_]:=Product[Reverse[q][[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[revaltprod/@IntegerPartitions[n]]],{n,0,30}]

A325878 Number of maximal subsets of {1..n} such that every orderless pair of distinct elements has a different sum.

Original entry on oeis.org

1, 1, 1, 1, 4, 5, 8, 22, 40, 56, 78, 124, 222, 390, 616, 892, 1220, 1620, 2182, 3042, 4392, 6364, 9054, 12608, 16980, 22244, 28482, 36208, 45864, 58692, 75804, 98440, 128694, 168250, 218558, 281210, 357594, 449402, 560034, 693332, 853546, 1050118, 1293458, 1596144, 1975394

Offset: 0

Gus Wiseman, Jun 02 2019

nonn

			The a(1) = 1 through a(6) = 8 subsets:
  {1}  {1,2}  {1,2,3}  {1,2,3}  {1,2,4}    {1,2,3,5}
                       {1,2,4}  {2,3,4}    {1,2,3,6}
                       {1,3,4}  {2,4,5}    {1,2,4,6}
                       {2,3,4}  {1,2,3,5}  {1,3,4,5}
                                {1,3,4,5}  {1,3,5,6}
                                           {1,4,5,6}
                                           {2,3,4,6}
                                           {2,4,5,6}

Andrew Howroyd, Table of n, a(n) for n = 0..60

The subset case is A196723.
The integer partition case is A325857.
The strict integer partition case is A325877.
Heinz numbers of the counterexamples are given by A325991.
Cf. A002033, A108917, A143823, A143824, A276024.
Cf. A325858, A325859, A325864, A325865, A325867, A325879, A325880, A382398.

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],UnsameQ@@Plus@@@Subsets[Union[#],{2}]&]]],{n,0,10}]

a(n)={
   my(ismaxl(b,w)=for(k=1, n, if(!bittest(b,k) && !bitand(w,b< n, ismaxl(b,w),
         my(s=self()(k+1, r, b, w));
         if(!bitand(w,b<Andrew Howroyd, Mar 23 2025

a(21) onwards from Andrew Howroyd, Mar 23 2025

A325879 Number of maximal subsets of {1..n} such that every ordered pair of distinct elements has a different difference.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 14, 20, 24, 36, 64, 110, 176, 238, 294, 370, 504, 736, 1086, 1592, 2240, 2982, 3788, 4700, 5814, 7322, 9396, 12336, 16552, 22192, 29310, 38046, 48368, 60078, 73722, 89416, 108208, 131310, 160624, 198002, 247408, 310410, 390924, 490818, 613344, 758518

Offset: 0

Gus Wiseman, Jun 02 2019

nonn

Also the number of maximal subsets of {1..n} such that every orderless pair of (not necessarily distinct) elements has a different sum.

			The a(0) = 1 through a(7) = 20 subsets:
  {}  {1}  {1,2}  {1,2}  {2,3}    {1,2,4}  {1,2,4}  {1,2,4}
                  {1,3}  {1,2,4}  {1,2,5}  {1,2,5}  {1,2,6}
                  {2,3}  {1,3,4}  {1,3,4}  {1,2,6}  {1,3,4}
                                  {1,4,5}  {1,3,4}  {1,4,5}
                                  {2,3,5}  {1,3,6}  {1,4,6}
                                  {2,4,5}  {1,4,5}  {1,5,6}
                                           {1,4,6}  {2,3,5}
                                           {1,5,6}  {2,3,6}
                                           {2,3,5}  {2,3,7}
                                           {2,3,6}  {2,4,5}
                                           {2,4,5}  {2,4,7}
                                           {2,5,6}  {2,5,6}
                                           {3,4,6}  {2,6,7}
                                           {3,5,6}  {3,4,6}
                                                    {3,4,7}
                                                    {3,5,6}
                                                    {4,5,7}
                                                    {4,6,7}
                                                    {1,2,5,7}
                                                    {1,3,6,7}

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..100
Wikipedia, Sidon sequence.
Index entries for sequences related to Golomb rulers.

The subset case is A143823.
The integer partition case is A325858.
The strict integer partition case is A325876.
Heinz numbers of the counterexamples are given by A325992.
Cf. A002033, A108917, A143824, A196723, A382397.
Cf. A325859, A325861, A325865, A325867, A325869, A325878, A325992.

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],UnsameQ@@Subtract@@@Subsets[Union[#],{2}]&]]],{n,0,10}]

a(n)={
  my(ismaxl(b,w)=for(k=1, n, if(!bittest(b,k) && !bitand(w,bitor(b,1< n, ismaxl(b,w),
         my(s=self()(k+1, b,w));
         b+=1<Andrew Howroyd, Mar 27 2025

a(21)-a(45) from Fausto A. C. Cariboni, Feb 08 2022

cp's OEIS Frontend

A143104 Infinite Redheffer matrix read by upwards antidiagonals.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Excel

Maple

Mathematica

PARI

Formula

Extensions

A353867 Heinz numbers of integer partitions where every partial run (consecutive constant subsequence) has a different sum, and these sums include every integer from 0 to the greatest part.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A325792 Positive integers with as many proper divisors as the sum of their prime indices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A347461 Number of distinct possible alternating products of integer partitions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A353865 Number of complete rucksack partitions of n. Partitions whose weak run-sums are distinct and cover an initial interval of nonnegative integers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A051709 a(n) = sigma(n) + phi(n) - 2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A325799 Sum of the prime indices of n minus the number of distinct positive subset-sums of the prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A347462 Number of distinct possible reverse-alternating products of integer partitions of n.

Views