A033942 -id:A033942 - cp's OEIS frontend

A350352 Products of three or more distinct prime numbers.

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 210, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 330, 345, 354, 357, 366, 370, 374, 385, 390, 399, 402, 406, 410, 418, 426, 429, 430

Offset: 1

Gus Wiseman, Jan 11 2022

nonn

First differs from A336568 in lacking 420.

			The terms and their prime indices begin:
     30: {1,2,3}     182: {1,4,6}      285: {2,3,8}
     42: {1,2,4}     186: {1,2,11}     286: {1,5,6}
     66: {1,2,5}     190: {1,3,8}      290: {1,3,10}
     70: {1,3,4}     195: {2,3,6}      310: {1,3,11}
     78: {1,2,6}     210: {1,2,3,4}    318: {1,2,16}
    102: {1,2,7}     222: {1,2,12}     322: {1,4,9}
    105: {2,3,4}     230: {1,3,9}      330: {1,2,3,5}
    110: {1,3,5}     231: {2,4,5}      345: {2,3,9}
    114: {1,2,8}     238: {1,4,7}      354: {1,2,17}
    130: {1,3,6}     246: {1,2,13}     357: {2,4,7}
    138: {1,2,9}     255: {2,3,7}      366: {1,2,18}
    154: {1,4,5}     258: {1,2,14}     370: {1,3,12}
    165: {2,3,5}     266: {1,4,8}      374: {1,5,7}
    170: {1,3,7}     273: {2,4,6}      385: {3,4,5}
    174: {1,2,10}    282: {1,2,15}     390: {1,2,3,6}

Michael De Vlieger, Table of n, a(n) for n = 1..10000

This is the squarefree case of A033942.
Including squarefree semiprimes gives A120944.
The squarefree complement consists of 1 and A167171.
These are the Heinz numbers of the partitions counted by A347548.
A000040 lists prime numbers (exactly 1 prime factor).
A005117 lists squarefree numbers.
A006881 lists squarefree numbers with exactly 2 prime factors.
A007304 lists squarefree numbers with exactly 3 prime factors.
A046386 lists squarefree numbers with exactly 4 prime factors.
Cf. A000009, A000111, A001250, A002808, A027383, A349796.

Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]>=3&]

is(n,f=factor(n))=my(e=f[,2]); #e>2 && vecmax(e)==1 \\ Charles R Greathouse IV, Jul 08 2022

list(lim)=my(v=List()); forsquarefree(n=30,lim\1, if(#n[2][,2]>2, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Jul 08 2022

from sympy import factorint
def ok(n): f = factorint(n, multiple=True); return len(f) == len(set(f)) > 2
print([k for k in range(431) if ok(k)]) # Michael S. Branicky, Jan 14 2022

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A350352(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(3,x.bit_length())))
    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
    return bisection(f,n,n) # Chai Wah Wu, Sep 11 2024

A063928 Largest nonprime proper divisor of n (with a(1)=1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 8, 1, 9, 1, 10, 1, 1, 1, 12, 1, 1, 9, 14, 1, 15, 1, 16, 1, 1, 1, 18, 1, 1, 1, 20, 1, 21, 1, 22, 15, 1, 1, 24, 1, 25, 1, 26, 1, 27, 1, 28, 1, 1, 1, 30, 1, 1, 21, 32, 1, 33, 1, 34, 1, 35, 1, 36, 1, 1, 25, 38, 1, 39, 1, 40, 27, 1, 1, 42, 1, 1, 1, 44, 1

Offset: 1

Henry Bottomley, Aug 15 2001

nonn

a(m)*a(n) <= a(m*n); a(m)*a(n) = a(m*n) iff m and n are prime or = 1. - Reinhard Zumkeller, Apr 11 2008

R. Zumkeller, Table of n, a(n) for n = 1..1000

a(n)=1 if n is 1, prime (A000040), or the product of two primes (A001358), i.e., if n is in A037143, otherwise, with n in A033942, a(n)=A032742(n). Cf. A006530.

{ for (n=1, 1000, if (n==1, a=1, d=divisors(n); m=length(d); until (!isprime(a), m--; a=d[m])); write("b063928.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 02 2009

A104016 Devaraj numbers: squarefree r-prime-factor (r>1) integers N=p1...pr such that phi(N)=(p1-1)...(pr-1) divides gcd(p1-1,...,pr-1)^2*(N-1)^(r-2).

Original entry on oeis.org

561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 11305, 15841, 29341, 39865, 41041, 46657, 52633, 62745, 63973, 75361, 96985, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 278545, 294409, 314821, 334153, 340561, 399001, 401401, 410041

Offset: 1

Max Alekseyev, Feb 25 2005

nonn

A.K. Devaraj conjectured that these numbers are exactly Carmichael numbers. It was proved (see Alekseyev link) that every Carmichael number is indeed a Devaraj number, but the converse is not true. Devaraj numbers that are not Carmichael are given by A104017.

These numbers can't be even, since phi(N) is always even (N>2) but p1=2 implies that gcd{pi-1}=1 and N-1 is odd. - M. F. Hasler, Apr 03 2009

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
Max Alekseyev, Pomerance's proof, June 2005.

Subsequence of A350352 and hence of A033942.

Cf. A104017, A002997.

Devaraj() = for(n=2,10^8, f=factorint(n); if(vecmax(f[,2])>1,next); f=f[,1]; r=length(f); if(r==1,next); d=f[1]-1; p=f[1]-1; for(i=2,r,d=gcd(d,f[i]-1); p*=f[i]-1); if( ((n-1)^(r-2)*d^2)%p==0, print1(" ",n)) )

isA104016(n)= local(f=factor(n)); vecmax(f[,2])==1 && #(f*=[1,-1]~)>1 && gcd(f)^2*(n-1)^(#f-2)%prod(i=1,#f,f[i])==0
/* To print the list: */ forstep( n=3, 10^6, 2, vecmax((f=factor(n))[,2])>1 && next; #(f*=[1,-1]~)>1 || next; gcd(f)^2*(n-1)^(#f-2)%prod(i=1,#f,f[i]) || print1(n","))
/* The following version could be efficient for large omega(n) */
isA104016(n) = issquarefree(n) && !isprime(n) && Mod(n-1,prod(i=1,#n=factor(n)*[1,-1]~,n[i]))^(#n-2)*gcd(n)^2==0 \\ M. F. Hasler, Apr 03 2009

A230594 Number of ways to write n as n = x*y, where x, y = noncomposite numbers (A008578), 1 <= x <= n, 1 <= y <= n.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Oct 27 2013

nonn

Dirichlet convolution of A080339(n) with itself, where A080339 = characteristic function of noncomposite numbers (A008578).
Dirichlet convolution of functions b(n) and c(n) is function a(n) = Sum_{d|n} b(d) * c(n/d).
a(n) = 0, 1 or 2. a(n) = 0 for numbers n from A033942 (numbers with least 3 prime factors (counted with multiplicity)); a(n) = 1 for n = p^2, p = prime; a(n) = 2 for numbers n from A167171 (A006881 union A000040).

			For n = 6: a(6) = Sum_(d|6) A080339(d) * A080339(6/d) = 1*0 + 1*1 + 1*1 + 0*1 = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A080339, A033942, A167171, A006881, A000040, A230595.

a[n_] := Switch[FactorInteger[n][[;;, 2]], {2}, 1, {1}, 2, {1, 1}, 2, , 0]; a[1] = 1; Array[a, 100] (* _Amiram Eldar, Aug 29 2023 *)

A080339(n) = ((1==n)||isprime(n));
A230594(n) = sumdiv(n,d,(A080339(d)*A080339(n/d))); \\ Antti Karttunen, Jul 27 2017

from sympy import factorint
def A230594(n): return 0 if sum(f:=factorint(n).values())>2 else (1 if n==1 or max(f)==2 else 2) # Chai Wah Wu, Jul 23 2024

a(n) = Sum_{d|n} A080339(d) * A080339(n/d).

A331233 Number of unlabeled rooted trees with n vertices and more than two branches of the root.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 12, 30, 75, 194, 501, 1317, 3485, 9302, 24976, 67500, 183290, 500094, 1369939, 3766831, 10391722, 28756022, 79794407, 221987348, 619019808, 1729924110, 4844242273, 13590663071, 38195831829, 107523305566, 303148601795, 855922155734, 2419923253795

Offset: 1

Gus Wiseman, Jan 21 2020

nonn

			The a(4) = 1 through a(7) = 12 rooted trees:
  (ooo)  (oooo)   (ooooo)    (oooooo)
         (oo(o))  (oo(oo))   (oo(ooo))
                  (ooo(o))   (ooo(oo))
                  (o(o)(o))  (oooo(o))
                  (oo((o)))  (o(o)(oo))
                             (oo((oo)))
                             (oo(o)(o))
                             (oo(o(o)))
                             (ooo((o)))
                             ((o)(o)(o))
                             (o(o)((o)))
                             (oo(((o))))

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 500 terms from Andrew Howroyd)

The Matula-Goebel numbers of these trees are given by A033942.
The series-reduced case is A331488.
The lone-child-avoiding case is (also) A331488.
The labeled version is A331577.
Unlabeled rooted trees are counted by A000081.
Cf. A001678, A001679, A004111, A033185, A060313, A206429, A331490, A331578.

g:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
      `if`(i<1, 0, add(binomial(g(i-1$2, 0)+j-1, j)*
         g(n-i*j, i-1, max(0, t-j)), j=0..n/i)))
    end:
a:= n-> g(n-1$2, 3):
seq(a(n), n=1..40);  # Alois P. Heinz, Jan 22 2020

urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
Table[Length[Select[urt[n],Length[#]>2&]],{n,10}]
(* Second program: *)
g[n_, i_, t_] := g[n, i, t] = If[n == 0, If[t == 0, 1, 0],
     If[i < 1, 0, Sum[Binomial[g[i - 1, i - 1, 0] + j - 1, j]*
     g[n - i*j, i - 1, Max[0, t - j]], {j, 0, n/i}]]];
a[n_] := g[n-1, n-1, 3];
Array[a, 40] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

\\ TreeGf gives gf of A000081.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={my(g=TreeGf(n)); Vec(g - x*(1 + g + (g^2 + subst(g, x, x^2))/2), -n)} \\ Andrew Howroyd, Jan 22 2020

For n > 1, a(n) = Sum_{k > 2} A033185(n - 1, k).

G.f.: f(x) - x*(1 + f(x) + (f(x)^2 + f(x^2))/2) where f(x) is the g.f. of A000081. - Andrew Howroyd, Jan 22 2020

A111087 Neither primes nor semiprimes.

Original entry on oeis.org

1, 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, 125, 126, 128, 130, 132, 135, 136, 138, 140, 144

Offset: 1

Noelle Clou (keynews.tv(AT)skynet.be), Oct 12 2005

1 together with A033942.

f[n_] := Plus @@ Last /@ FactorInteger[n]; Select[ Range[146], f[ # ] != 1 && f[ # ] != 2 &] (* Robert G. Wilson v *)
Join[{1},Select[Range[200],PrimeOmega[#]>2&]] (* Harvey P. Dale, Mar 12 2018 *)

is(n)=bigomega(n)>2 || n==1 \\ Charles R Greathouse IV, Nov 13 2016

a(n) = n + O(n log log n/log n). - Charles R Greathouse IV, Nov 13 2016

More terms from Robert G. Wilson v, Oct 12 2005

A321516 Number of composite divisors of n that are < n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 1, 2, 0, 3, 0, 3, 0, 0, 0, 5, 0, 0, 0, 4, 0, 3, 0, 2, 2, 0, 0, 6, 0, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 7, 0, 0, 2, 4, 0, 3, 0, 2, 0, 3, 0, 8, 0, 0, 2, 2, 0, 3, 0, 6, 2, 0, 0, 7, 0, 0, 0

Offset: 1

Felix Fröhlich, Nov 12 2018

nonn

Different from A294902 (see A321517).

a(n) > 0 iff n is a term of A033942.

			For n = 24: The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Four of those divisors, namely 4, 6, 8 and 12 are composite and < 24, so a(24) = 4.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A033942, A294902, A321517.

a[n_] := Length[Select[Most[Divisors[n]], CompositeQ]]; Array[a, 87] (* Amiram Eldar, Nov 12 2018 *)

a(n) = my(d=divisors(n), i=0); for(k=2, #d-1, if(!ispseudoprime(d[k]), i++)); i

a(n) = sumdiv(n, d, (d1) && !isprime(d)); \\ Michel Marcus, Nov 12 2018

A330936 Number of nontrivial factorizations of n into factors > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 5, 0, 0, 1, 2, 0, 3, 0, 5, 0, 0, 0, 7, 0, 0, 0, 5, 0, 3, 0, 2, 2, 0, 0, 10, 0, 2, 0, 2, 0, 5, 0, 5, 0, 0, 0, 9, 0, 0, 2, 9, 0, 3, 0, 2, 0, 3, 0, 14, 0, 0, 2, 2, 0, 3, 0, 10, 3, 0, 0, 9, 0, 0

Offset: 1

Gus Wiseman, Jan 04 2020

nonn

The trivial factorizations of a number are (1) the case with only one factor, and (2) the factorization into prime numbers.

			The a(n) nontrivial factorizations of n = 8, 12, 16, 24, 36, 48, 60, 72:
  (2*4)  (2*6)  (2*8)    (3*8)    (4*9)    (6*8)      (2*30)    (8*9)
         (3*4)  (4*4)    (4*6)    (6*6)    (2*24)     (3*20)    (2*36)
                (2*2*4)  (2*12)   (2*18)   (3*16)     (4*15)    (3*24)
                         (2*2*6)  (3*12)   (4*12)     (5*12)    (4*18)
                         (2*3*4)  (2*2*9)  (2*3*8)    (6*10)    (6*12)
                                  (2*3*6)  (2*4*6)    (2*5*6)   (2*4*9)
                                  (3*3*4)  (3*4*4)    (3*4*5)   (2*6*6)
                                           (2*2*12)   (2*2*15)  (3*3*8)
                                           (2*2*2*6)  (2*3*10)  (3*4*6)
                                           (2*2*3*4)            (2*2*18)
                                                                (2*3*12)
                                                                (2*2*2*9)
                                                                (2*2*3*6)
                                                                (2*3*3*4)

Positions of nonzero terms are A033942.
Positions of 1's are A030078.
Positions of 2's are A054753.
Nontrivial integer partitions are A007042.
Nontrivial set partitions are A008827.
Nontrivial divisors are A070824.
Cf. A001055, A003238, A005121, A317145, A317176, A318812, A330665, A330935.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[DeleteCases[Rest[facs[n]],{_}]],{n,100}]

For prime n, a(n) = 0; for nonprime n, a(n) = A001055(n) - 2.

A347709 Number of distinct rational numbers of the form x * z / y for some factorization x * y * z = n, 1 < x <= y <= z.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 1, 0, 1, 1, 0, 0, 3, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 4, 0, 0, 1, 2, 0, 1, 0, 1, 0, 1, 0, 4, 0, 0, 1, 1, 0, 1, 0, 3, 1, 0, 0, 4, 0, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0, 4, 0, 1, 1, 2, 0, 1, 0, 2, 1, 0, 0, 4, 0, 1, 0, 3, 0, 1, 0, 1, 1, 0, 0, 5

Offset: 1

Gus Wiseman, Oct 14 2021

nonn

This is also the number of distinct possible alternating products of length-3 factorizations of n, where we define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)), and where a factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

			Representative factorizations for each of the a(360) = 9 alternating products:
   (2,2,90) -> 90
   (2,3,60) -> 40
   (2,4,45) -> 45/2
   (2,5,36) -> 72/5
   (2,6,30) -> 10
   (2,9,20) -> 40/9
  (2,10,18) -> 18/5
  (2,12,15) -> 5/2
   (3,8,15) -> 45/8

Antti Karttunen, Table of n, a(n) for n = 1..65537

Allowing factorizations of any length <= 3 gives A033273.
Positions of positive terms are A033942.
Positions of 0's are A037143.
The length-2 version is A072670.
Allowing any length gives A347460, reverse A038548.
Allowing any odd length gives A347708.
A001055 counts factorizations (strict A045778, ordered A074206).
A122179 counts length-3 factorizations.
A292886 counts knapsack factorizations, by sum A293627.
A301957 counts distinct subset-products of prime indices.
A304792 counts distinct subset-sums of partitions, positive A276024.
Cf. A000040, A001358, A002033, A046951, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456, A347461.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@Select[facs[n],Length[#]==3&]]],{n,100}]

A347709(n) = { my(rats=List([])); fordiv(n,z,my(yx=n/z); fordiv(yx, y, my(x = yx/y); if((y <= z) && (x <= y) && (x > 1), listput(rats,x*z/y)))); #Set(rats); }; \\ Antti Karttunen, Jan 29 2025

More terms from Antti Karttunen, Jan 29 2025

A367175 a(n) = Sum_{d|n} (-1)^[d is prime] * d, where [] denotes the Iverson bracket.

Original entry on oeis.org

1, -1, -2, 3, -4, 2, -6, 11, 7, 4, -10, 18, -12, 6, 8, 27, -16, 29, -18, 28, 12, 10, -22, 50, 21, 12, 34, 38, -28, 52, -30, 59, 20, 16, 24, 81, -36, 18, 24, 76, -40, 72, -42, 58, 62, 22, -46, 114, 43, 79, 32, 68, -52, 110, 40, 102, 36, 28, -58, 148, -60, 30

Offset: 1

Peter Luschny, Nov 08 2023

sign

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000203, A000040, A033942, A037143, A008472, A367176.

Isprime := n -> if isprime(n) then 1 else 0 fi:
a := n -> local d; add((-1)^Isprime(d) * d, d in NumberTheory:-Divisors(n)):
seq(a(n), n = 1..62);

Array[DivisorSum[#, (-1)^Boole[PrimeQ[#]]*# &] &, 62] (* Michael De Vlieger, Nov 10 2023 *)

a(n) = sumdiv(n, d, (-1)^isprime(d)*d); \\ Michel Marcus, Nov 10 2023

from sympy import divisor_sigma, primefactors
def A367175(n): return divisor_sigma(n)-(sum(primefactors(n))<<1) # Chai Wah Wu, Nov 10 2023

def A367175(n): return sum((-1)^is_prime(d)*d for d in divisors(n))
print([A367175(n) for n in range(1, 63)])

{k: a(k) < 0} = {A000040}.
{k: a(k) > k} = {A033942}.
{k: a(k) < k} = {A037143} \ {1}.
sigma(n) - a(n) = 2 * A008472(n).
Conjecture: {k: a(k) divides sigma(k)} = {1, 2, 3, 6, 14, 15, 35}.

cp's OEIS Frontend

A350352 Products of three or more distinct prime numbers.

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A063928 Largest nonprime proper divisor of n (with a(1)=1).

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A104016 Devaraj numbers: squarefree r-prime-factor (r>1) integers N=p1*...*pr such that phi(N)=(p1-1)*...*(pr-1) divides gcd(p1-1,...,pr-1)^2*(N-1)^(r-2).

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A230594 Number of ways to write n as n = x*y, where x, y = noncomposite numbers (A008578), 1 <= x <= n, 1 <= y <= n.

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A331233 Number of unlabeled rooted trees with n vertices and more than two branches of the root.

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Maple

Mathematica

PARI

Formula

A111087 Neither primes nor semiprimes.

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Mathematica

PARI

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A321516 Number of composite divisors of n that are < n.

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Mathematica

PARI

A104016 Devaraj numbers: squarefree r-prime-factor (r>1) integers N=p1...pr such that phi(N)=(p1-1)...(pr-1) divides gcd(p1-1,...,pr-1)^2*(N-1)^(r-2).