A071142 -id:A071142 - cp's OEIS frontend

A046346 Composite numbers that are divisible by the sum of their prime factors (counted with multiplicity).

4, 16, 27, 30, 60, 70, 72, 84, 105, 150, 180, 220, 231, 240, 256, 286, 288, 308, 378, 440, 450, 476, 528, 540, 560, 576, 588, 594, 624, 627, 646, 648, 650, 728, 800, 805, 840, 884, 897, 900, 945, 960, 1008, 1040, 1056, 1080, 1100, 1122, 1134, 1160, 1170, 1248

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

If m is in the sequence and d|m, then m^d is also a term. Note that this sequence contains all infinite subsequences of the form p^(p^k) for k>0, where p is a prime. - Amiram Eldar and Thomas Ordowski, Feb 06 2019

If one selects some composite k, k >= 8, and decomposes (k - sopfr(k)) into an additive partition having only prime parts, then those parts, when taken as a product with k, yield an element of this sequence. - Christopher Hohl, Jul 30 2019

			a(38) = 884 = 2 * 2 * 13 * 17 -> 2 + 2 + 13 + 17 = 34 so 884 / 34 = 26.

François Huppé, Table of n, a(n) for n = 1..50000 (terms 1..1000 from T. D. Noe)
K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294. See "special numbers" on page 287.

Cf. A036844, A046347, A046348, A001414.

Contains A071142.

m=1;for u=2:1200 if and(isprime(u)==0,mod(u,sum(factor(u)))==0); sol(m)=u; m=m+1; end; end;sol % Marius A. Burtea, Jul 31 2019

[k:k in [2..1200]| not IsPrime(k) and  k mod (&+[m[1]*m[2]: m in Factorization(k)]) eq 0]; // Marius A. Burtea, Jul 31 2019

isA046346 := proc(n)
    if isprime(n) then
        false;
    elif modp(n,A001414(n)) = 0 then
        true;
    else
        false;
    end if;
end proc:
for n from 2 to 1000 do
    if isA046346(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Jan 12 2016

Select[Range[2,1170],!PrimeQ[#]&&IntegerQ[#/Total[Times@@@FactorInteger[#]]]&] (* Jayanta Basu, Jun 02 2013 *)

sopfr(n) = {my(f=factor(n)); sum(k=1, #f~, f[k,1]*f[k,2]);}
lista(nn) = forcomposite(n=2, nn, if (! (n % sopfr(n)), print1(n, ", "));); \\ Michel Marcus, Jan 06 2016

from sympy import factorint
def ok(n):
  f = factorint(n)
  return sum(f[p] for p in f) > 1 and n % sum(p*f[p] for p in f) == 0
print(list(filter(ok, range(1250)))) # Michael S. Branicky, Apr 16 2021

Description corrected by Robert A. Stump (bee_ess107(AT)yahoo.com), Jan 09 2002

A365795 Numbers k such that omega(k) = 3 and its prime factors satisfy the equation p_1 + p_2 = p_3.

Original entry on oeis.org

30, 60, 70, 90, 120, 140, 150, 180, 240, 270, 280, 286, 300, 350, 360, 450, 480, 490, 540, 560, 572, 600, 646, 700, 720, 750, 810, 900, 960, 980, 1080, 1120, 1144, 1200, 1292, 1350, 1400, 1440, 1500, 1620, 1750, 1798, 1800, 1920, 1960, 2160, 2240, 2250, 2288, 2400, 2430, 2450

Offset: 1

Stefano Spezia, Sep 19 2023

nonn

The lower prime factor p_1 is equal to 2 and the other two are twin primes: p_3 - p_2 = 2.

			60 is a term since 60 = 2^2*3*5 and 2 + 3 = 5.
286 is a term since 286 = 2*11*13 and 2 + 11 = 13.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A001221, A001359, A006512, A071142.

Subsequence of A033992 and of A071140.

Select[Range[2500],PrimeNu[#]==3&&Part[First/@FactorInteger[#],1]+Part[First/@FactorInteger[#],2]==Part[First/@FactorInteger[#],3]&]

isok(k) = if (omega(k)==3, my(f=factor(k)[,1]); f[1] + f[2] == f[3]); \\ Michel Marcus, Sep 19 2023

A221054 Numbers whose distinct prime factors can be partitioned into two equal sums.

Original entry on oeis.org

1, 30, 60, 70, 90, 120, 140, 150, 180, 240, 270, 280, 286, 300, 350, 360, 450, 480, 490, 540, 560, 572, 600, 646, 700, 720, 750, 810, 900, 960, 980, 1080, 1120, 1144, 1200, 1292, 1350, 1400, 1440, 1500, 1620, 1750, 1798, 1800, 1920, 1960, 2145, 2160, 2240, 2250, 2288, 2310, 2400, 2430, 2450, 2584, 2700, 2730, 2800, 2880, 3000, 3135

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Apr 15 2013

nonn

This is a superset of 2*product of twin primes, A071142.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Table of n, a(n), and equal sums of factors for n=1..10000

Cf. A175592 (multiplicity of prime factors allowed).
Cf. A071139-A071147, especially A071140.
Cf. A008472, A037074, A071142.
Cf. A027748, A005843, A083207.

a221054 n = a221054_list !! (n-1)
a221054_list = filter (z 0 0 . a027748_row) $ tail a005843_list where
   z u v []     = u == v
   z u v (p:ps) = z (u + p) v ps || z u (v + p) ps
-- Reinhard Zumkeller, Apr 18 2013

q[n_] := Module[{p = FactorInteger[n][[;; , 1]], sum, x}, sum = Total[p]; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, p}], x][[1 + sum/2]] > 0]; Select[Range[3200], q] (* Amiram Eldar, May 31 2025 *)

isok(k) = my(f=factor(k), nb=#f~); for (i=0,2^nb-1, my(v=Vec(Vecrev(binary(i)), nb)); if (sum(k=1, nb, if (v[k], f[k,1])) == sum(k=1, nb, if (!v[k], f[k,1])), return(1));); \\ Michel Marcus, May 31 2025

Missing terms inserted by Michel Marcus, May 31 2025

A379033 Numbers that are the product of exactly three (not necessarily distinct) primes and these primes are sides of a nondegenerate triangle.

Original entry on oeis.org

8, 12, 18, 27, 45, 50, 75, 98, 105, 125, 147, 175, 242, 245, 338, 343, 363, 385, 429, 507, 539, 578, 605, 637, 715, 722, 845, 847, 867, 969, 1001, 1058, 1083, 1105, 1183, 1309, 1331, 1445, 1547, 1573, 1587, 1615, 1682, 1729, 1805, 1859, 1922, 2023, 2057, 2185, 2197

Offset: 1

Felix Huber, Dec 24 2024

nonn

Subsequence of A014612 and of A145784.

Numbers that are the product of exactly three (not necessarily distinct) primes and these primes are sides of a degenerate triangle are in A071142.

			12 = 2*2*3 is in the sequence because 2 + 2 > 3.
20 = 2*2*5 is not in the sequence because 2 + 2 < 5.
30 = 2*3*5 is not in the sequence because 2 + 3 = 5.

Felix Huber, Table of n, a(n) for n = 1..10000

Cf. A000040, A001222, A014612, A070088, A071142, A145784, A306678, A366398, A378819.

A379033:=proc(n)
   option remember;
   local a,i,j,P;
   if n=1 then
      8
   else
      for a from procname(n-1)+1 do
         P:=[];
         if NumberTheory:-Omega(a)=3 then
            for i in ifactors(a)[2] do
               j:=0;
               while jP[3] then
               return a
            fi
         fi
      od
   fi	
end proc;
seq(A379033(n),n=1..51);

A384498 Squarefree numbers whose distinct prime factors can be partitioned into two sets with equal sums.

Original entry on oeis.org

1, 30, 70, 286, 646, 1798, 2145, 2310, 2730, 3135, 3526, 3570, 4641, 4845, 5005, 5610, 6006, 6279, 6630, 7198, 7410, 7854, 8778, 8855, 8970, 9177, 10366, 10374, 10626, 10695, 11305, 11571, 11730, 13110, 13485, 13566, 13585, 15470, 16095, 16302, 16422, 16530

Offset: 1

Alois P. Heinz, May 31 2025

nonn

			2145 = 3*5*11*13 is a term because it is squarefree and 3+13 = 5+11.
16422 = 2*3*7*17*23 is squarefree and 2+7+17 = 3+23.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Intersection of A005117 and A221054.

Cf. A071141, A071142, A071312.

q:= n-> (l-> {l[.., 2][]} minus {1}={} and (s->
        (m-> m::even and coeff(mul(1+x^j, j=s), x, m/2)>0)
        (add(i, i=s)))({l[.., 1][]}))(ifactors(n)[2]):
select(q, [$1..20000])[];

Join[{1},Select[Range[16600],SquareFreeQ[#]&&MemberQ[Total/@Subsets[First/@FactorInteger[#]],Total[First/@FactorInteger[#]]/2]&]] (* James C. McMahon, Jun 02 2025 *)

A323640 Numbers m having at least one pair (x,y) of divisors with x

Original entry on oeis.org

6, 20, 56, 70, 110, 182, 272, 286, 308, 506, 646, 650, 812, 884, 992, 1150, 1406, 1672, 1748, 1798, 1892, 2162, 2756, 2990, 3422, 3526, 3782, 4030, 4466, 4556, 4606, 4930, 5402, 5510, 5704, 6032, 6068, 6806, 7198, 7310, 7378, 7832, 7904, 8084, 8170, 8246, 8584, 8710

Offset: 1

David A. Corneth, Aug 31 2019

nonn

Primitive terms of A094519.
From Bernard Schott, Aug 31 2019: (Start)
Some subsequences (this list is not exhaustive):
1) Oblong numbers of the form (3*k+1)*(3*k+2). These are in A001504 and the pair (x,y) = (1,3*k+1). Only 6 is oblong and not of this form. The first few terms are 20, 56, 110, 182, 272, ...
2) Numbers of the form 2*p*q  where (p, q) is a twin prime pair. These terms are precisely A071142 \ {30} and the pair (x,y) = (2,p). The first few terms are 70, 286, 646, ...
3) Numbers of the form 2^2 * p * q where p and q = p+4 are primes and p > 3. These primes p are in A023200 \ {3} and the pair (x,y) = (4,p). The first few terms are 308, 884, ...
4) More generally, numbers of the form 2^k * p * q where p and q = p+2^k are primes and the pair (x,y) = (2^k,p). For k = 3, the smallest such term is 1672 with p = 11. (End)

			56 is in the sequence as 1, 7 and 1 + 7 = 8 are divisors of 56 and no divisor of 56 is in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..11106 (terms <= 5*10^7)

Cf. A094519.

filter:= proc(n) local D,i,j,nD;
  D:= numtheory:-divisors(n);
  nD:= nops(D);
  for i from 1 to nD-1 do
    for j from i+1 to nD do
      if (n/(D[i]+D[j]))::integer then return true fi
  od od;
  false
end proc:
N:= 10000: # for terms <= N
C:= Vector(N):
R:= NULL:
for i from 1 to N do
  if C[i]=0 and filter(i) then
    R:= R, i;
    C[[seq(i*j,j=2..N/i)]]:= 1
  fi
od:
R; # Robert Israel, Sep 02 2019

upto(n) = {my(charprim = vector(n, i, 1), res = List()); for(i = 1, n, if(charprim[i] == 1, if(isA094519(i), listput(res, i); for(k = 2, n \ i, charprim[i*k] = 0 ) , charprim[i] = 0; ) ) ); res }
isA094519(n) = {my(d = divisors(n)); for(i = 1, #d - 2, for(j = i + 1, #d - 1, if(n % (d[i] + d[j]) == 0, return(1) ) ) ); 0 }

cp's OEIS Frontend

A046346 Composite numbers that are divisible by the sum of their prime factors (counted with multiplicity).

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A365795 Numbers k such that omega(k) = 3 and its prime factors satisfy the equation p_1 + p_2 = p_3.

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Mathematica

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A221054 Numbers whose distinct prime factors can be partitioned into two equal sums.

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Haskell

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Extensions

A379033 Numbers that are the product of exactly three (not necessarily distinct) primes and these primes are sides of a nondegenerate triangle.

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Maple

A384498 Squarefree numbers whose distinct prime factors can be partitioned into two sets with equal sums.

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Maple

Mathematica

A323640 Numbers m having at least one pair (x,y) of divisors with x

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Programs

Maple

PARI