A033942 -id:A033942 - cp's OEIS frontend

A101040 If n has one or two prime-factors then 1 else 0.

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

Offset: 1

Reinhard Zumkeller, Nov 28 2004

nonn

a(A033942(n))=0; for n>1: a(A037143(n))=1;
a(A000040(n))=1; a(A001358(n))=1;
A101041(n) = Sum(a(k): 1<=k<=n) + 1.
Primes counted with multiplicity. - Harvey P. Dale, Feb 16 2024

Characteristic function of A037143 (without its initial term 1).

a[n_] := If[n == 1, 0, Boole[PrimeOmega[n] <= 2]];
Array[a, 105] (* Jean-François Alcover, Dec 02 2021 *)

vector(105,k,bigomega(k)<=2&&k>1) \\ Hugo Pfoertner, Dec 02 2021

(define (A101040 n) (if (= 1 n) 0 (A063524 (A032742 (A032742 n))))) ;; Antti Karttunen, Nov 23 2017

a(n) = A010051(n)+A064911(n) = 0^floor(A001222(n)/3)-0^(n-1).

a(1) = 0; for n > 1, a(n) = A063524(A032742(A032742(n))). - Antti Karttunen, Nov 23 2017

A331488 Number of unlabeled lone-child-avoiding rooted trees with n vertices and more than two branches (of the root).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 6, 10, 20, 36, 70, 134, 263, 513, 1022, 2030, 4076, 8203, 16614, 33738, 68833, 140796, 288989, 594621, 1226781, 2536532, 5256303, 10913196, 22700682, 47299699, 98714362, 206323140, 431847121, 905074333, 1899247187, 3990145833, 8392281473

Offset: 1

Gus Wiseman, Jan 20 2020

nonn

Also the number of lone-child-avoiding rooted trees with n vertices and more than two branches.

			The a(4) = 1 through a(9) = 10 trees:
  (ooo)  (oooo)  (ooooo)   (oooooo)   (ooooooo)    (oooooooo)
                 (oo(oo))  (oo(ooo))  (oo(oooo))   (oo(ooooo))
                           (ooo(oo))  (ooo(ooo))   (ooo(oooo))
                                      (oooo(oo))   (oooo(ooo))
                                      (o(oo)(oo))  (ooooo(oo))
                                      (oo(o(oo)))  (o(oo)(ooo))
                                                   (oo(o(ooo)))
                                                   (oo(oo)(oo))
                                                   (oo(oo(oo)))
                                                   (ooo(o(oo)))

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014)
Eric Weisstein's World of Mathematics, Series-reduced tree.
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

The not necessarily lone-child-avoiding version is A331233.
The Matula-Goebel numbers of these trees are listed by A331490.
A000081 counts unlabeled rooted trees.
A001678 counts lone-child-avoiding rooted trees.
A001679 counts topologically series-reduced rooted trees.
A291636 lists Matula-Goebel numbers of lone-child-avoiding rooted trees.
A331489 lists Matula-Goebel numbers of series-reduced rooted trees.
Cf. A000014, A000669, A004250, A007097, A007821, A033942, A060313, A060356, A061775, A109082, A109129, A196050, A276625, A330943.

urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
Table[Length[Select[urt[n],Length[#]>2&&FreeQ[#,{_}]&]],{n,10}]

For n > 1, a(n) = A001679(n) - A001678(n).

a(37)-a(38) from Jinyuan Wang, Jun 26 2020

Terminology corrected (lone-child-avoiding, not series-reduced) by Gus Wiseman, May 10 2021

A346701 Heinz number of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 6, 13, 7, 5, 4, 17, 6, 19, 10, 7, 11, 23, 6, 5, 13, 9, 14, 29, 10, 31, 8, 11, 17, 7, 6, 37, 19, 13, 10, 41, 14, 43, 22, 15, 23, 47, 12, 7, 10, 17, 26, 53, 9, 11, 14, 19, 29, 59, 10, 61, 31, 21, 8, 13, 22, 67, 34, 23, 14, 71

Offset: 1

Gus Wiseman, Aug 03 2021

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.

			The partition (2,2,2,1,1) has Heinz number 108 and odd bisection (2,2,1) with Heinz number 18, so a(108) = 18.
The partitions (3,2,2,1,1), (3,2,2,2,1), (3,3,2,1,1) have Heinz numbers 180, 270, 300 and all have odd bisection (3,2,1) with Heinz number 30, so a(180) = a(270) = a(300) = 30.

Positions of last appearances are A000290 without the first term 0.
Positions of primes are A037143 (complement: A033942).
The even version is A329888.
Positions of first appearances are A342768.
The sum of prime indices of a(n) is A346699(n), non-reverse: A346697.
The non-reverse version is A346703.
The even non-reverse version is A346704.
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum, reverse A344612.
A209281 (shifted) adds up the odd bisection of standard compositions.
A316524 gives the alternating sum of prime indices, reverse A344616.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices.
A344617 gives the sign of the alternating sum of prime indices.
A346700 gives the sum of the even bisection of reversed prime indices.
Cf. A025047, A027187, A027193, A053738, A106356, A277103, A341446, A344653, A345957, A345958, A345959, A346698, A346702.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@First/@Partition[Append[Reverse[primeMS[n]],0],2],{n,100}]

a(n) * A329888(n) = n.

A056239(a(n)) = A346699(n).

A039955 Squarefree numbers congruent to 1 (mod 4).

Original entry on oeis.org

1, 5, 13, 17, 21, 29, 33, 37, 41, 53, 57, 61, 65, 69, 73, 77, 85, 89, 93, 97, 101, 105, 109, 113, 129, 133, 137, 141, 145, 149, 157, 161, 165, 173, 177, 181, 185, 193, 197, 201, 205, 209, 213, 217, 221, 229, 233, 237, 241, 249, 253, 257, 265, 269

Offset: 1

R. K. Guy

The subsequence of primes is A002144.
The subsequence of semiprimes (intersection with A001358) begins: 21, 33, 57, 65, 69, 77, 85, 93, 129, 133, 141, 145, 161, 177, 185, 201, 205, 209, 213, 217, 221, 237, 249, 253, 265.
The subsequence with more than two prime factors (intersection with A033942) begins: 105 = 3 * 5 * 7, 165 = 3 * 5 * 11, 273, 285, 345, 357, 385, 429, 465. - Jonathan Vos Post, Feb 19 2011
Except for a(1) = 1 these are the squarefree members of A079896 (i.e., squarefree determinants D of indefinite binary quadratic forms). - Wolfdieter Lang, Jun 01 2013
The asymptotic density of this sequence is 2/Pi^2 = 0.202642... (A185197). - Amiram Eldar, Feb 10 2021

Richard A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. M. Legendre, Diviseurs de la formule t^2+a*u^2, a étant de la forme 4 n + 1, Essai sur la Théorie des Nombres An VI, Table IV. See first column. [_Paul Curtz_, Aug 14 2019]

Cf. A002144, A039956, A039957, A005117, A185197.

a039955 n = a039955_list !! (n-1)
a039955_list = filter ((== 1) . (`mod` 4)) a005117_list
-- Reinhard Zumkeller, Aug 15 2011

[4*n+1: n in [0..67] | IsSquarefree(4*n+1)];  // Bruno Berselli, Mar 03 2011

fQ[n_] := Max[Last /@ FactorInteger@ n] == 1 && Mod[n, 4] == 1; Select[ Range@ 272, fQ] (* Robert G. Wilson v *)
Select[Range[1,300,4],SquareFreeQ[#]&] (* Harvey P. Dale, Mar 27 2020 *)

list(lim)=my(v=List([1])); forfactored(n=5,lim\1, if(vecmax(n[2][,2])==1 && n[1]%4==1, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 05 2017

is(n)=n%4==1 && issquarefree(n) \\ Charles R Greathouse IV, Nov 05 2017

A331490 Matula-Goebel numbers of series-reduced rooted trees with more than two branches (of the root).

Original entry on oeis.org

8, 16, 28, 32, 56, 64, 76, 98, 112, 128, 152, 172, 196, 212, 224, 256, 266, 304, 343, 344, 392, 424, 428, 448, 512, 524, 532, 602, 608, 652, 686, 688, 722, 742, 784, 848, 856, 896, 908, 931, 1024, 1048, 1052, 1064, 1204, 1216, 1244, 1304, 1372, 1376, 1444

Offset: 1

Gus Wiseman, Jan 20 2020

nonn

We say that a rooted tree is (topologically) series-reduced if no vertex has degree 2.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches. This gives a bijective correspondence between positive integers and unlabeled rooted trees.
Also Matula-Goebel numbers of lone-child-avoiding rooted trees with more than two branches.

			The sequence of all series-reduced rooted trees with more than two branches together with their Matula-Goebel numbers begins:
    8: (ooo)
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   98: (o(oo)(oo))
  112: (oooo(oo))
  128: (ooooooo)
  152: (ooo(ooo))
  172: (oo(o(oo)))
  196: (oo(oo)(oo))
  212: (oo(oooo))
  224: (ooooo(oo))
  256: (oooooooo)
  266: (o(oo)(ooo))
  304: (oooo(ooo))
  343: ((oo)(oo)(oo))
  344: (ooo(o(oo)))

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014)
Eric Weisstein's World of Mathematics, Series-reduced tree.
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

These trees are counted by A331488.
Unlabeled rooted trees are counted by A000081.
Lone-child-avoiding rooted trees are counted by A001678.
Topologically series-reduced rooted trees are counted by A001679.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Matula-Goebel numbers of series-reduced rooted trees are A331489.
Cf. A000014, A000669, A004250, A007097, A007821, A033942, A060313, A060356, A061775, A109082, A109129, A196050, A276625, A330943.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
srQ[n_]:=Or[n==1,With[{m=primeMS[n]},And[Length[m]>1,And@@srQ/@m]]];
Select[Range[1000],PrimeOmega[#]>2&&srQ[#]&]

A060278 Sum of composite divisors of n less than n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 10, 0, 0, 0, 12, 0, 15, 0, 14, 0, 0, 0, 30, 0, 0, 9, 18, 0, 31, 0, 28, 0, 0, 0, 49, 0, 0, 0, 42, 0, 41, 0, 26, 24, 0, 0, 70, 0, 35, 0, 30, 0, 60, 0, 54, 0, 0, 0, 97, 0, 0, 30, 60, 0, 61, 0, 38, 0, 59, 0, 117, 0, 0, 40, 42, 0, 71, 0, 98, 36, 0, 0, 127, 0, 0, 0

Offset: 1

Jack Brennen, Mar 28 2001

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A010051, A027751, A033942, A035322, A035321, A037143, A023891.

a060278 1 = 0
a060278 n = sum $ filter ((== 0) . a010051) $ tail $ a027751_row n
-- Reinhard Zumkeller, Apr 05 2013

for n from 1 to 300 do s := 0: for j from 2 to n-1 do if isprime(j) then else if n mod j = 0 then s := s+j fi; fi: od: printf(`%d,`,s) od:

Join[{0},Table[Total[Select[Most[Rest[Divisors[n]]],!PrimeQ[#]&]],{n,2,90}]] (* Harvey P. Dale, Oct 25 2011 *)
a[n_] := DivisorSigma[1, n] - Plus @@ FactorInteger[n][[;; , 1]] - If[PrimeQ[n], 0, n] - 1; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 20 2022 *)

a(n) = sumdiv(n, d, if ((d1) && !isprime(d), d)); \\ Michel Marcus, Jan 13 2020

From Reinhard Zumkeller, Apr 05 2013: (Start)
a(n) = Sum_{k=2..A000005(n)-1} A010051(A027751(n,k));
a(A037143(n)) = 0;
a(A033942(n)) > 0. (End)

More terms from James Sellers and Matthew Conroy, Mar 29 2001

A342768 a(n) = A342767(n, n).

Original entry on oeis.org

1, 2, 3, 8, 5, 12, 7, 32, 27, 20, 11, 48, 13, 28, 45, 128, 17, 108, 19, 80, 63, 44, 23, 192, 125, 52, 243, 112, 29, 180, 31, 512, 99, 68, 175, 432, 37, 76, 117, 320, 41, 252, 43, 176, 405, 92, 47, 768, 343, 500, 153, 208, 53, 972, 275, 448, 171, 116, 59, 720

Offset: 1

Rémy Sigrist, Apr 02 2021

nonn

This sequence has similarities with A087019.

These are the positions of first appearances of each positive integer in A346701, and also in A346703. - Gus Wiseman, Aug 09 2021

			For n = 42:
- 42 = 2 * 3 * 7, so:
          2 3 7
        x 2 3 7
        -------
          2 3 7
        2 3 3
    + 2 2 2
    -----------
      2 2 3 3 7
- hence a(42) = 2 * 2 * 3 * 3 * 7 = 252.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A342768
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A007947, A087019, A342767.
The sum of prime indices of a(n) is 2*A056239(n) - A061395(n).
The version for even indices is A129597(n) = 2*a(n) for n > 1.
The sorted version is A346635.
These are the positions of first appearances in A346701 and in A346703.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A027193 counts partitions of odd length, ranked by A026424.
A209281 adds up the odd bisection of standard compositions (even: A346633).
A346697 adds up the odd bisection of prime indices (reverse: A346699).
Cf. A000290, A006530, A033942, A037143, A344653, A345957, A346698, A346700, A346704.

Table[n^2/FactorInteger[n][[-1,1]],{n,100}] (* Gus Wiseman, Aug 09 2021 *)

PARI
```
See Links section.
```

a(n) = n iff n = 1 or n is a prime number.
a(p^k) = p^(2*k-1) for any k > 0 and any prime number p.
A007947(a(n)) = A007947(n).
A001222(a(n)) = 2*A001222(n) - 1 for any n > 1.
From Gus Wiseman, Aug 09 2021: (Start)
A001221(a(n)) = A001221(n).
If g = A006530(n) is the greatest prime factor of n, then a(n) = n^2/g.
a(n) = A129597(n)/2.
(End)

A346635 Numbers whose division (or multiplication) by their greatest prime factor yields a perfect square. Numbers k such that k*A006530(k) is a perfect square.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 12, 13, 17, 19, 20, 23, 27, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 48, 52, 53, 59, 61, 63, 67, 68, 71, 73, 76, 79, 80, 83, 89, 92, 97, 99, 101, 103, 107, 108, 109, 112, 113, 116, 117, 124, 125, 127, 128, 131, 137, 139, 148, 149, 151, 153

Offset: 1

Gus Wiseman, Aug 10 2021

nonn

This is the sorted version of A342768(n) = position of first appearance of n in A346701 (but A346703 works also).

			The terms together with their prime indices begin:
     1: {}          31: {11}            71: {20}
     2: {1}         32: {1,1,1,1,1}     73: {21}
     3: {2}         37: {12}            76: {1,1,8}
     5: {3}         41: {13}            79: {22}
     7: {4}         43: {14}            80: {1,1,1,1,3}
     8: {1,1,1}     44: {1,1,5}         83: {23}
    11: {5}         45: {2,2,3}         89: {24}
    12: {1,1,2}     47: {15}            92: {1,1,9}
    13: {6}         48: {1,1,1,1,2}     97: {25}
    17: {7}         52: {1,1,6}         99: {2,2,5}
    19: {8}         53: {16}           101: {26}
    20: {1,1,3}     59: {17}           103: {27}
    23: {9}         61: {18}           107: {28}
    27: {2,2,2}     63: {2,2,4}        108: {1,1,2,2,2}
    28: {1,1,4}     67: {19}           109: {29}
    29: {10}        68: {1,1,7}        112: {1,1,1,1,4}

Removing 1 gives a subset of A026424.
The unsorted even version is A129597.
The unsorted version is A342768(n) = A342767(n,n).
Except the first term, the even version is 2*a(n).
A000290 lists squares.
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A006530 gives the greatest prime factor.
A061395 gives the greatest prime index.
A027193 counts partitions of odd length.
A056239 adds up prime indices, row sums of A112798.
A209281 = odd bisection sum of standard compositions (even: A346633).
A316524 = alternating sum of prime indices (sign: A344617, rev.: A344616).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices.
A346697 = odd bisection sum of prime indices (weights of A346703).
A346699 = odd bisection sum of reversed prime indices (weights of A346701).
Cf. A028260, A033942, A035363, A037143, A341446, A344653, A345957, A345958, A345959, A346698, A346700, A346704.

filter:= proc(n) issqr(n/max(numtheory:-factorset(n))) end proc:
filter(1):= true:
select(filter, [$1..200]); # Robert Israel, Nov 26 2022

sqrQ[n_]:=IntegerQ[Sqrt[n]];
Select[Range[100],sqrQ[#*FactorInteger[#][[-1,1]]]&]

isok(m) = (m==1) || issquare(m/vecmax(factor(m)[,1])); \\ Michel Marcus, Aug 12 2021

a(n) = A129597(n)/2 for n > 1.

A109810 Number of permutations of the positive divisors of n, where every element is coprime to its adjacent elements.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 0, 2, 4, 2, 0, 2, 4, 4, 0, 2, 0, 2, 0, 4, 4, 2, 0, 2, 4, 0, 0, 2, 0, 2, 0, 4, 4, 4, 0, 2, 4, 4, 0, 2, 0, 2, 0, 0, 4, 2, 0, 2, 0, 4, 0, 2, 0, 4, 0, 4, 4, 2, 0, 2, 4, 0, 0, 4, 0, 2, 0, 4, 0, 2, 0, 2, 4, 0, 0, 4, 0, 2, 0, 0, 4, 2, 0, 4, 4, 4, 0, 2, 0, 4, 0, 4, 4, 4, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0

Offset: 1

Leroy Quet, Aug 16 2005

nonn

Depends only on prime signature. - Reinhard Zumkeller, May 24 2010

			The divisors of 6 are 1, 2, 3 and 6. Of the permutations of these integers, only (6,1,2,3), (6,1,3,2), (2,3,1,6) and (3,2,1,6) are such that every pair of adjacent elements is coprime.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A178254. - Reinhard Zumkeller, May 24 2010

a(1)=1, a(p) = 2, a(p^2) = 2, a(p*q) = 4 (where p and q are distinct primes), all other terms are 0.

a(A033942(n))=0; a(A037143(n))>0; a(A000430(n))=2; a(A006881(n))=4. - Reinhard Zumkeller, May 24 2010

Terms 17 to 59 from Diana L. Mecum, Jul 18 2008

More terms from David Wasserman, Oct 01 2008

A341610 Nonprimitive terms of A246282: numbers k that have more than one divisor d|k such that A003961(d) > 2*d.

Original entry on oeis.org

8, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 126, 128, 130, 132, 135, 136, 138, 140, 144, 147, 148, 150, 152, 153, 154, 156, 160, 162, 164, 165

Offset: 1

Antti Karttunen, Feb 22 2021

nonn

Numbers k for which A337345(k) > 1, or equally, for which A337346(k) > 0.
Obviously A337346(n) = 0 for any noncomposite and for any semiprime, thus this is a subsequence of A033942. The first term of A033942 not present here is 125, as A337345(125) = 1.
Empirically checked: in range 1 .. 2^31, all abundant numbers are found in this sequence. For proving this, we should concentrate only on checking A091191, as the set A005101 \ A091191 (non-primitive abundant numbers) is certainly included, as for any divisor d for which sigma(d) > 2*d (or even sigma(d) >= 2*d), we also have A003961(d) > 2*d.

Cf. A337345.
Positions of nonzero terms in A337346.
Setwise difference A246282 \ A337372.
Conjectured subsequence: A005101. (Clearly abundant numbers are all in A246282).
Cf. A341611, A341614.
Differs from its subsequence A033942 for the first time at n=52, with a(52) = 126, while A033942(52) = 125.

Block[{nn = 165, s}, s = {1}~Join~Array[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] &, nn - 1, 2]; Select[Range[nn], 1 < DivisorSum[#, 1 &, s[[#]] > 2 # &] &]] (* Michael De Vlieger, Feb 22 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA341610(n) = (1A003961(d)>(d+d)));

cp's OEIS Frontend

A101040 If n has one or two prime-factors then 1 else 0.

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A331488 Number of unlabeled lone-child-avoiding rooted trees with n vertices and more than two branches (of the root).

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A346701 Heinz number of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.

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A039955 Squarefree numbers congruent to 1 (mod 4).

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A331490 Matula-Goebel numbers of series-reduced rooted trees with more than two branches (of the root).

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A060278 Sum of composite divisors of n less than n.

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A342768 a(n) = A342767(n, n).

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