A046347 -id:A046347 - 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

A046348 Composite palindromes divisible by the sum of their prime factors (counted with multiplicity).

Original entry on oeis.org

4, 646, 2772, 5445, 8778, 30303, 48384, 50505, 54145, 63336, 77077, 117711, 219912, 234432, 239932, 255552, 272272, 294492, 535535, 585585, 636636, 717717, 825528, 888888, 951159, 999999, 1103011, 1112111, 1345431, 2248422, 2267622

Offset: 1

Patrick De Geest, Jun 15 1998

			a(2)=646 :  2*17*19 -> 2 + 17 + 19 = 38 and 646 / 38 = 17.

David A. Corneth, Table of n, a(n) for n = 1..4180 (Terms <= 10^15. First 1000 terms from R. J. Mathar and Giovanni Resta)
David A. Corneth, PARI prog

Cf. A032350, A001414, A046346, A046347.

t={}; Do[If[!PrimeQ[n]&&Reverse[x=IntegerDigits[n]]==x&&IntegerQ[n/Total[Times@@@FactorInteger[n]]],AppendTo[t,n]],{n,4,2.5*10^6}]; t (* Jayanta Basu, Jun 04 2013 *)
cpdQ[n_]:=CompositeQ[n]&&PalindromeQ[n]&&Divisible[n,Total[ Flatten[ Table[ #[[1]],#[[2]]]&/@FactorInteger[n]]]]; Select[Range[23*10^5],cpdQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 01 2018 *)

PARI
```
See PARI link.
```

A308643 Odd squarefree composite numbers k, divisible by the sum of their prime factors, sopfr (A001414).

Original entry on oeis.org

105, 231, 627, 805, 897, 1581, 2967, 3055, 4543, 5487, 6461, 6745, 7881, 9717, 10707, 14231, 15015, 16377, 21091, 26331, 29607, 33495, 33901, 33915, 35905, 37411, 38843, 40587, 42211, 45885, 49335, 50505, 51051, 53295, 55581, 60297

Offset: 1

David James Sycamore, Jun 13 2019

nonn

Every term has an odd number of prime divisors (A001221(k) is odd), since if not, sopfr(k) would be even, and so not divide k, which is odd.
Some Carmichael numbers appear in this sequence, the first of which is 3240392401.
From Robert Israel, Jul 05 2019: (Start)
Includes p*q*r if p and q are distinct odd primes and r=(p-1)*(q-1)-1 is prime. Dickson's conjecture implies that there are infinitely many such terms for each odd prime p.  Thus for p=3, q is in A063908 (except 3), for p=5, q is in A156300 (except 2), and for p=7, q is in A153135 (except 2). (End)

			105=3*5*7; sum of prime factors = 15 and 105 = 7*15, so 105 is a term.

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

Intersection of A005117 and A046347.

Cf. A001414, A046346, A002997, A001221, A063908, A156300, A153135.

[k:k in [2*d+1: d in [1..35000]]|IsSquarefree(k) and not IsPrime(k) and k mod &+PrimeDivisors(k) eq 0]; // Marius A. Burtea, Jun 19 2019

with(NumberTheory);
N := 500;
for n from 2 to N do
S := PrimeFactors(n);
X := add(S);
if IsSquareFree(n) and not mod(n, 2) = 0 and not isprime(n) and mod(n, X) = 0 then print(n);
end if:
end do:

aQ[n_] := Module[{f = FactorInteger[n]}, p=f[[;;,1]]; e=f[[;;,2]]; Length[e] > 1 && Max[e]==1 && Divisible[n, Plus@@(p^e)]]; Select[Range[1, 61000, 2], aQ] (* Amiram Eldar, Jul 04 2019 *)

A309003 Carmichael numbers divisible by the sum of their prime factors, sopfr (A001414).

Original entry on oeis.org

3240392401, 13577445505, 14446721521, 84127131361, 203340265921, 241420757761, 334797586201, 381334973041, 461912170321, 1838314142785, 3636869821201, 10285271821441, 17624045440981, 18773053896961, 20137015596061, 24811804945201, 26863480687681, 35598629998801

Offset: 1

David James Sycamore, Jul 05 2019

nonn

Intersection of A002997 and A308643.

Intersection of A002997 and A036844.

			3240392401 = 29*37*41*73*1009, A001414(3240392401)=1189 = 29*41.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A001414, A036844, A046347, A308643.

sopfr(f) = f[, 1]~*f[, 2];
isCarmichael(n, f)= bittest(n, 0) && !for(i=1, #f~, (f[i, 2]==1 && n%(f[i, 1]-1)==1)||return) && (#f~>1);
isok(n) = my(f=factor(n)); isCarmichael(n, f) && !(n % sopfr(f)); \\ Michel Marcus, Jul 07 2019

cp's OEIS Frontend

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

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A046348 Composite palindromes divisible by the sum of their prime factors (counted with multiplicity).

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Mathematica

PARI

A308643 Odd squarefree composite numbers k, divisible by the sum of their prime factors, sopfr (A001414).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

A309003 Carmichael numbers divisible by the sum of their prime factors, sopfr (A001414).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI