A014311 -id:A014311 - cp's OEIS frontend

A338557 Products of three distinct prime numbers of even index.

273, 399, 609, 741, 777, 903, 1113, 1131, 1281, 1443, 1491, 1653, 1659, 1677, 1729, 1869, 2067, 2109, 2121, 2247, 2373, 2379, 2451, 2639, 2751, 2769, 2919, 3021, 3081, 3171, 3219, 3367, 3423, 3471, 3477, 3633, 3741, 3801, 3857, 3913, 3939, 4047, 4053, 4173

Offset: 1

Gus Wiseman, Nov 08 2020

nonn

All terms are odd.
Also sphenic numbers (A007304) with all even prime indices (A031215).
Also Heinz numbers of strict integer partitions with 3 parts, all of which are even. These partitions are counted by A001399.

			The sequence of terms together with their prime indices begins:
     273: {2,4,6}     1869: {2,4,24}    3219: {2,10,12}
     399: {2,4,8}     2067: {2,6,16}    3367: {4,6,12}
     609: {2,4,10}    2109: {2,8,12}    3423: {2,4,38}
     741: {2,6,8}     2121: {2,4,26}    3471: {2,6,24}
     777: {2,4,12}    2247: {2,4,28}    3477: {2,8,18}
     903: {2,4,14}    2373: {2,4,30}    3633: {2,4,40}
    1113: {2,4,16}    2379: {2,6,18}    3741: {2,10,14}
    1131: {2,6,10}    2451: {2,8,14}    3801: {2,4,42}
    1281: {2,4,18}    2639: {4,6,10}    3857: {4,8,10}
    1443: {2,6,12}    2751: {2,4,32}    3913: {4,6,14}
    1491: {2,4,20}    2769: {2,6,20}    3939: {2,6,26}
    1653: {2,8,10}    2919: {2,4,34}    4047: {2,8,20}
    1659: {2,4,22}    3021: {2,8,16}    4053: {2,4,44}
    1677: {2,6,14}    3081: {2,6,22}    4173: {2,6,28}
    1729: {4,6,8}     3171: {2,4,36}    4179: {2,4,46}

For the following, NNS means "not necessarily strict".
A007304 allows all prime indices (not just even) (NNS: A014612).
A046389 allows all odd primes (NNS: A046316).
A258117 allows products of any length (NNS: A066207).
A307534 is the version for odds instead of evens (NNS: A338471).
A337453 is a different ranking of ordered triples (NNS: A014311).
A338556 is the NNS version.
A001399(n-6) counts strict 3-part partitions (NNS: A001399(n-3)).
A005117 lists squarefree numbers, with even case A039956.
A078374 counts 3-part relatively prime strict partitions (NNS: A023023).
A075819 lists even Heinz numbers of strict triples (NNS: A075818).
A220377 counts 3-part pairwise coprime strict partitions (NNS: A307719).
A258116 lists squarefree numbers with all odd prime indices (NNS: A066208).
A285508 lists Heinz numbers of non-strict triples.
Cf. A000217, A001221, A001222, A037144, A056239, A112798, A337605.

Select[Range[1000],SquareFreeQ[#]&&PrimeOmega[#]==3&&OddQ[Times@@(1+PrimePi/@First/@FactorInteger[#])]&]

isok(m) = my(f=factor(m)); (bigomega(f)==3) && (omega(f)==3) && (#select(x->(x%2), apply(primepi, f[,1]~)) == 0); \\ Michel Marcus, Nov 10 2020

from itertools import filterfalse
from math import isqrt
from sympy import primepi, primerange, nextprime, integer_nthroot
def A338557(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum((primepi(x//(k*m))>>1)-(b>>1) for a,k in filterfalse(lambda x:x[0]&1,enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2)) for b,m in filterfalse(lambda x:x[0]&1,enumerate(primerange(nextprime(k)+1,isqrt(x//k)+1),a+2))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

A038461 Sums of 10 distinct powers of 2.

Original entry on oeis.org

1023, 1535, 1791, 1919, 1983, 2015, 2031, 2039, 2043, 2045, 2046, 2559, 2815, 2943, 3007, 3039, 3055, 3063, 3067, 3069, 3070, 3327, 3455, 3519, 3551, 3567, 3575, 3579, 3581, 3582, 3711, 3775, 3807, 3823, 3831, 3835, 3837, 3838, 3903

Offset: 1

Olivier Gérard

Ivan Neretin, Table of n, a(n) for n = 1..10000
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. See p. 18 for Mathematica code irwinSums.m.

Base 2 interpretation of A038452.

Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691 (Hamming weight = 1, 2, ..., 9).

Select[Range[4000], DigitCount[#, 2, 1] == 10 &] (* Amiram Eldar, Feb 14 2022 *)

isok(n) = hammingweight(n) == 10; \\ Michel Marcus, Feb 29 2016

from itertools import islice
def A038461_gen(): # generator of terms
    yield (n:=1023)
    while True: yield (n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b)
A038461_list = list(islice(A038461_gen(),20)) # Chai Wah Wu, Mar 10 2025

Sum_{n>=1} 1/a(n) = 1.386312271262110321181505974797071257205562524228381227122302929089588534920... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

Offset changed to 1 by Ivan Neretin, Feb 28 2016

A091890 Number of partitions of n into sums of exactly three distinct powers of 2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 1, 1, 3, 2, 0, 1, 3, 3, 2, 5, 2, 1, 1, 5, 6, 3, 9, 5, 4, 5, 10, 9, 8, 13, 8, 10, 8, 16, 17, 15, 22, 18, 18, 20, 25, 28, 27, 34, 31, 32, 33, 44, 49, 44, 64, 53, 56, 61, 71, 77, 77, 100, 88, 94, 99, 123, 125, 132, 162, 147, 154

Offset: 1

Reinhard Zumkeller, Feb 10 2004

nonn

			a(14)=2: 14 = (2^3+2^2+2^1) = (2^2+2^1+2^0)+(2^2+2^1+2^0).

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A014311, A018819, A091889, A000120, A000041, A091893, A091891.

With[{max = 80}, m = Select[Range[max], DigitCount[#, 2, 1] == 3 &]; a[n_] := Length@ IntegerPartitions[n, n, m]; Array[a, max]] (* Amiram Eldar, Aug 01 2023 *)

A091891 Number of partitions of n into parts which are a sum of exactly as many distinct powers of 2 as n has 1's in its binary representation.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 2, 1, 10, 3, 2, 1, 5, 1, 2, 1, 36, 6, 12, 1, 11, 3, 2, 1, 24, 3, 3, 1, 5, 1, 2, 1, 202, 67, 55, 9, 93, 4, 5, 1, 112, 8, 13, 1, 10, 3, 2, 1, 304, 22, 18, 1, 20, 3, 3, 1, 34, 3, 3, 1, 5, 1, 2, 1, 1828, 1267, 1456, 71, 1629, 77, 100, 2, 2342, 99, 123, 9, 132, 4, 3, 1

Offset: 0

Reinhard Zumkeller, Feb 10 2004

			a(9) = 3 because there are 3 partitions of 9 into parts of size 3, 5, 6, 9 which are the numbers that have two 1's in their binary representations. The 3 partitions are: 9, 6 + 3 and 3 + 3 + 3. - _Andrew Howroyd_, Apr 20 2021

Alois P. Heinz, Table of n, a(n) for n = 0..16384 (terms n = 1..1000 from Andrew Howroyd)

Cf. A000079, A000120, A000041, A018819, A018900, A014311.

Cf. A091889, A091890, A091892, A091893.

H:= proc(n) option remember; add(i, i=Bits[Split](n)) end:
v:= proc(n, k) option remember; `if`(n<1, 0,
      `if`(H(n)=k, n, v(n-1, k)))
    end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, v(i-1, k), k)+b(n-i, v(min(n-i, i), k), k)))
    end:
a:= n-> b(n$2, H(n)):
seq(a(n), n=0..80);  # Alois P. Heinz, Dec 12 2021

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}] b[n - j], {j, 1, n}]/n]; b];
EulerT[v_List] := With[{q = etr[v[[#]]&]}, q /@ Range[Length[v]]];
a[n_] := EulerT[Table[DigitCount[k, 2, 1] == DigitCount[n, 2, 1] // Boole, {k, 1, n}]][[n]];
Array[a, 100] (* Jean-François Alcover, Dec 12 2021, after Andrew Howroyd *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
a(n) = {EulerT(vector(n,k,hammingweight(k)==hammingweight(n)))[n]} \\ Andrew Howroyd, Apr 20 2021

a(A000079(n)) = A018819(n);
a(A018900(n)) = A091889(n);
a(A014311(n)) = A091890(n);
a(A091892(n)) = 1.

a(0)=1 prepended by Alois P. Heinz, Dec 12 2021

A321773 Number of compositions of n into parts with distinct multiplicities and with exactly three parts.

Original entry on oeis.org

1, 3, 6, 4, 9, 9, 10, 12, 15, 13, 18, 18, 19, 21, 24, 22, 27, 27, 28, 30, 33, 31, 36, 36, 37, 39, 42, 40, 45, 45, 46, 48, 51, 49, 54, 54, 55, 57, 60, 58, 63, 63, 64, 66, 69, 67, 72, 72, 73, 75, 78, 76, 81, 81, 82, 84, 87, 85, 90, 90, 91, 93, 96, 94, 99, 99

Offset: 3

Alois P. Heinz, Nov 18 2018

nonn

			From _Gus Wiseman_, Nov 11 2020: (Start)
Also the number of 3-part non-strict compositions of n. For example, the a(3) = 1 through a(11) = 15 triples are:
  111   112   113   114   115   116   117   118   119
        121   122   141   133   161   144   181   155
        211   131   222   151   224   171   226   191
              212   411   223   233   225   244   227
              221         232   242   252   262   272
              311         313   323   333   334   335
                          322   332   414   343   344
                          331   422   441   424   353
                          511   611   522   433   434
                                      711   442   443
                                            622   515
                                            811   533
                                                  551
                                                  722
                                                  911
(End)

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=3 of A242887.
A235451 counts 3-part compositions with distinct run-lengths
A001399(n-6) counts 3-part compositions in the complement.
A014311 intersected with A335488 ranks these compositions.
A140106 is the unordered case, with Heinz numbers A285508.
A261982 counts non-strict compositions of any length.
A001523 counts unimodal compositions, with complement A115981.
A007318 and A097805 count compositions by length.
A032020 counts strict compositions.
A047967 counts non-strict partitions, with Heinz numbers A013929.
A242771 counts triples that are not strictly increasing.
Cf. A000212, A000217, A001840, A128422, A156040, A332834, A337461, A337484, A337603, A337604.

Table[Length[Join@@Permutations/@Select[IntegerPartitions[n,{3}],!UnsameQ@@#&]],{n,0,100}] (* Gus Wiseman, Nov 11 2020 *)

Conjectures from Colin Barker, Dec 11 2018: (Start)
G.f.: x^3*(1 + 3*x + 5*x^2) / ((1 - x)^2*(1 + x)*(1 + x + x^2)).
a(n) = a(n-2) + a(n-3) - a(n-5) for n>7. (End)
Conjecture: a(n) = (3*n-k)/2 where k value has a cycle of 6 starting from n=3 of (7,6,3,10,3,6). - Bill McEachen, Aug 12 2025

A038462 Sums of 11 distinct powers of 2.

Original entry on oeis.org

2047, 3071, 3583, 3839, 3967, 4031, 4063, 4079, 4087, 4091, 4093, 4094, 5119, 5631, 5887, 6015, 6079, 6111, 6127, 6135, 6139, 6141, 6142, 6655, 6911, 7039, 7103, 7135, 7151, 7159, 7163, 7165, 7166, 7423, 7551, 7615, 7647, 7663, 7671

Offset: 1

Olivier Gérard

Ivan Neretin, Table of n, a(n) for n = 1..10000
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. See p. 18 for Mathematica code irwinSums.m.

Base 2 interpretation of A038453.

Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691, A038461 (Hamming weight = 1, 2, ..., 10).

Select[Range[8000], DigitCount[#, 2, 1] == 11 &] (* Amiram Eldar, Feb 14 2022 *)

from itertools import islice
def A038462_gen(): # generator of terms
    yield (n:=2047)
    while True: yield (n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b)
A038462_list = list(islice(A038462_gen(),20)) # Chai Wah Wu, Mar 10 2025

Sum_{n>=1} 1/a(n) = 1.386300330514503033229968047555778179200262625510401687087371496738972082061... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

Offset changed to 1 by Ivan Neretin, Feb 28 2016

A173589 Integers whose binary representation contains exactly three 1's, no two 1's being adjacent.

Original entry on oeis.org

21, 37, 41, 42, 69, 73, 74, 81, 82, 84, 133, 137, 138, 145, 146, 148, 161, 162, 164, 168, 261, 265, 266, 273, 274, 276, 289, 290, 292, 296, 321, 322, 324, 328, 336, 517, 521, 522, 529, 530, 532, 545, 546, 548, 552, 577, 578, 580, 584, 592, 641, 642, 644, 648

Offset: 1

David Koslicki (koslicki(AT)math.psu.edu), Feb 22 2010

Subsequence of A014311. [R. J. Mathar, Feb 24 2010]

A000120(a(n))=3; A023416(a(n))>1; 1 < A087116(a(n))<=3. [Reinhard Zumkeller, Mar 11 2010]

			a(1) = 21 = 10101_2.
a(2) = 37 = 100101_2.
a(3) = 41 = 101001_2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000120, A014311, A023416, A087116.

seq(seq(seq(2^a+2^b+2^c, c=0..b-2),b=2..a-2),a=4..10); # Robert Israel, Dec 19 2016

e31sQ[n_]:=Module[{idn2=IntegerDigits[n,2]},Total[idn2]==3 && SequenceCount[ idn2,{1,1}]==0]; Select[Range[700],e31sQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 20 2018 *)

from math import isqrt, comb
from sympy import integer_nthroot
def A173589(n): return (1<<(r:=n-1-comb((m:=integer_nthroot(6*n,3)[0])+(t:=(n>comb(m+2,3)))+1,3))-comb((k:=isqrt(b:=r+1<<1))+(b>k*(k+1)),2))+(1<<(a:=isqrt(s:=n-comb(m-(t^1)+2,3)<<1))+((s<<2)>(a<<2)*(a+1)+1)+1)+(1<Chai Wah Wu, Apr 07 2025

More terms from R. J. Mathar, Feb 24 2010

A212190 Squares that are the sum of exactly three distinct powers of 2.

Original entry on oeis.org

25, 49, 81, 100, 196, 289, 324, 400, 529, 784, 1089, 1156, 1296, 1600, 2116, 3136, 4225, 4356, 4624, 5184, 6400, 8464, 12544, 16641, 16900, 17424, 18496, 20736, 25600, 33856, 50176, 66049, 66564, 67600, 69696, 73984, 82944, 102400, 135424, 200704, 263169

Offset: 1

Reinhard Zumkeller, May 03 2012

nonn

Squares with exactly three ones in their binary representation: A000120(a(n)) = 3;
squares in A014311;
a(n) = A212191(n)^2.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)

Cf. A010052, A212192.

a212190 n = a212190_list !! (n-1)
a212190_list = filter ((== 1) . a010052) a014311_list

Select[Range[1000]^2, DigitCount[#, 2, 1] == 3&] (* Jean-François Alcover, Nov 07 2016 *)

A242771 Number of integer points in a certain quadrilateral scaled by a factor of n (another version).

Original entry on oeis.org

0, 0, 1, 3, 6, 9, 14, 19, 25, 32, 40, 48, 58, 68, 79, 91, 104, 117, 132, 147, 163, 180, 198, 216, 236, 256, 277, 299, 322, 345, 370, 395, 421, 448, 476, 504, 534, 564, 595, 627, 660, 693, 728, 763, 799, 836, 874, 912, 952, 992, 1033, 1075, 1118, 1161, 1206

Offset: 1

Michael Somos, May 22 2014

nonn

The quadrilateral is given by four vertices [(1/2, 1/3), (0, 1), (0, 0), (1, 0)] as an example on page 22 of Ehrhart 1967. Here the open line segment from (1/2, 1/3) to (0, 1) is included but the rest of the boundary is not. The sequence is denoted by d'(n).
From Gus Wiseman, Oct 18 2020: (Start)
Also the number of ordered triples of positive integers summing to n that are not strictly increasing. For example, the a(3) = 1 through a(7) = 14 triples are:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)
           (1,2,1)  (1,2,2)  (1,3,2)  (1,3,3)
           (2,1,1)  (1,3,1)  (1,4,1)  (1,4,2)
                    (2,1,2)  (2,1,3)  (1,5,1)
                    (2,2,1)  (2,2,2)  (2,1,4)
                    (3,1,1)  (2,3,1)  (2,2,3)
                             (3,1,2)  (2,3,2)
                             (3,2,1)  (2,4,1)
                             (4,1,1)  (3,1,3)
                                      (3,2,2)
                                      (3,3,1)
                                      (4,1,2)
                                      (4,2,1)
                                      (5,1,1)
A001399(n-6) counts the complement (unordered strict triples).
A014311 \ A333255 ranks these compositions.
A140106 is the unordered version.
A337484 is the case not strictly decreasing either.
A337698 counts these compositions of any length, with complement A000009.
A001399(n-6) counts unordered strict triples.
A001523 counts unimodal compositions, with complement A115981.
A007318 and A097805 count compositions by length.
A069905 counts unordered triples.
A218004 counts strictly increasing or weakly decreasing compositions.
A337483 counts triples either weakly increasing or weakly decreasing.
Cf. A332834, A337461, A337481, A337482, A337604.
(End)

			G.f. = x^3 + 3*x^4 + 6*x^5 + 9*x^6 + 14*x^7 + 19*x^8 + 25*x^9 + 32*x^10 + ...

E. Ehrhart, Sur un problème de géométrie diophantienne linéaire I, (Polyèdres et réseaux), J. Reine Angew. Math. 226 1967 1-29. MR0213320 (35 #4184).
E. Ehrhart, Sur un problème de géométrie diophantienne linéaire I, (Polyèdres et réseaux), J. Reine Angew. Math. 226 1967 1-29. MR0213320 (35 #4184). [Annotated scanned copy of pages 16 and 22 only]
E. Ehrhart, Sur un problème de géométrie diophantienne linéaire II. Systemes diophantiens lineaires, J. Reine Angew. Math. 227 1967 25-49. [Annotated scanned copy of pages 47-49 only]
Wikipedia, Ehrhart polynomial
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A002789, A010761, A147874.

Cf. A000212, A000217, A001840, A046691, A128422, A156040.

[Floor((5*n-7)*(n-1)/12): n in [1..60]]; // Vincenzo Librandi, Jun 27 2015

a[ n_] := Quotient[ 7 - 12 n + 5 n^2, 12];
a[ n_] := With[ {o = Boole[ 0 < n], c = Boole[ 0 >= n], m = Abs@n}, Length @ FindInstance[ 0 < c + x && 0 < c + y && (2 x < c + m && 4 x + 3 y < o + 3 m || m < o + 2 x && 2 x + 3 y < c + 2 m), {x, y}, Integers, 10^9]];
LinearRecurrence[{1,1,0,-1,-1,1},{0,0,1,3,6,9},90] (* Harvey P. Dale, May 28 2015 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],!Less@@#&]],{n,0,15}] (* Gus Wiseman, Oct 18 2020 *)

PARI
```
{a(n) = (7 - 12*n + 5*n^2) \ 12};
```

{a(n) = if( n<0, polcoeff( x * (2 + x^2 + x^3 + x^4) / ((1 - x)^2 * (1 - x^6)) + x * O(x^-n), -n), polcoeff( x^3 * (1 + x + x^2 + 2*x^4) / ((1 - x)^2 * (1 - x^6)) + x * O(x^n), n))};

G.f.: x^3 * (1 + 2*x + 2*x^2) / (1 - x - x^2 + x^4 + x^5 - x^6) = (x^3 + x^4 + x^5 + 2*x^7) / ((1 - x)^2 * (1 - x^6)).
a(n) = floor( A147874(n) / 12).
a(-n) = A002789(n).
a(n+1) - a(n) = A010761(n).
For n >= 6, a(n) = A000217(n-2) - A001399(n-6). - Gus Wiseman, Oct 18 2020

A337459 Numbers k such that the k-th composition in standard order is a unimodal triple.

Original entry on oeis.org

7, 11, 13, 14, 19, 21, 25, 26, 28, 35, 37, 41, 42, 49, 50, 52, 56, 67, 69, 73, 74, 81, 82, 84, 97, 98, 100, 104, 112, 131, 133, 137, 138, 145, 146, 161, 162, 164, 168, 193, 194, 196, 200, 208, 224, 259, 261, 265, 266, 273, 274, 289, 290, 292, 321, 322, 324

Offset: 1

Gus Wiseman, Sep 07 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.
A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence together with the corresponding triples begins:
      7: (1,1,1)     52: (1,2,3)    133: (5,2,1)
     11: (2,1,1)     56: (1,1,4)    137: (4,3,1)
     13: (1,2,1)     67: (5,1,1)    138: (4,2,2)
     14: (1,1,2)     69: (4,2,1)    145: (3,4,1)
     19: (3,1,1)     73: (3,3,1)    146: (3,3,2)
     21: (2,2,1)     74: (3,2,2)    161: (2,5,1)
     25: (1,3,1)     81: (2,4,1)    162: (2,4,2)
     26: (1,2,2)     82: (2,3,2)    164: (2,3,3)
     28: (1,1,3)     84: (2,2,3)    168: (2,2,4)
     35: (4,1,1)     97: (1,5,1)    193: (1,6,1)
     37: (3,2,1)     98: (1,4,2)    194: (1,5,2)
     41: (2,3,1)    100: (1,3,3)    196: (1,4,3)
     42: (2,2,2)    104: (1,2,4)    200: (1,3,4)
     49: (1,4,1)    112: (1,1,5)    208: (1,2,5)
     50: (1,3,2)    131: (6,1,1)    224: (1,1,6)

Eric Weisstein's World of Mathematics, Unimodal Sequence

A337460 is the non-unimodal version.
A000217(n - 2) counts 3-part compositions.
6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1) counts strict 3-part compositions.
A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.
A001399(n - 6) = A069905(n - 3) = A211540(n - 1) counts strict 3-part partitions.
A001523 counts unimodal compositions.
A007052 counts unimodal patterns.
A011782 counts unimodal permutations.
A115981 counts non-unimodal compositions.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Triples are A014311, with strict case A337453.
- Sum is A070939.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Heinz number is A333219.
- Combinatory separations are counted by A334030.
- Non-unimodal compositions are A335373.
- Non-co-unimodal compositions are A335374.
Cf. A007304, A014612, A072706, A156040, A211540, A227038, A332743, A337452, A337461, A337604.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Length[stc[#]]==3&&!MatchQ[stc[#],{x_,y_,z_}/;x>y

Complement of A335373 in A014311.

cp's OEIS Frontend

A338557 Products of three distinct prime numbers of even index.

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A038461 Sums of 10 distinct powers of 2.

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A091890 Number of partitions of n into sums of exactly three distinct powers of 2.

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A091891 Number of partitions of n into parts which are a sum of exactly as many distinct powers of 2 as n has 1's in its binary representation.

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A321773 Number of compositions of n into parts with distinct multiplicities and with exactly three parts.

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A038462 Sums of 11 distinct powers of 2.

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A173589 Integers whose binary representation contains exactly three 1's, no two 1's being adjacent.

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