A046346 -id:A046346 - cp's OEIS frontend

A078177 Composite numbers with an integer arithmetic mean of all prime factors.

4, 8, 9, 15, 16, 20, 21, 25, 27, 32, 33, 35, 39, 42, 44, 49, 50, 51, 55, 57, 60, 64, 65, 68, 69, 77, 78, 81, 85, 87, 91, 92, 93, 95, 105, 110, 111, 112, 114, 115, 116, 119, 121, 123, 125, 128, 129, 133, 140, 141, 143, 145, 155, 156, 159, 161, 164, 169, 170, 177, 180

Offset: 1

Reinhard Zumkeller, Nov 20 2002

nonn

That is, composite numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is an integer.

			60 = 2*2*3*5: (2+2+3+5)/4 = 3, therefore 60 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A078175, A001222, A100118, A046363, A133620, A133621.

Cf. A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134333, A134334, A134344, A134376.

Select[Range[200], CompositeQ[#] && IntegerQ[Mean[Flatten[Table[#[[1]], #[[2]]]& /@ FactorInteger[#]]]]&] (* Jean-François Alcover, Aug 03 2018 *)

lista(nn) = {forcomposite(n=1, nn, my(f = factor(n)); if (! (sum(k=1, #f~, f[k,1]*f[k,2]) % vecsum(f[,2])), print1(n, ", ")););} \\ Michel Marcus, Feb 22 2016

A001414(a(n)) == 0 modulo A001222(a(n)).

Edited by N. J. A. Sloane, May 30 2008 at the suggestion of R. J. Mathar

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.
```

A352989 Composites k such that the k-th triangular number is divisible by the integer log of k.

Original entry on oeis.org

8, 15, 16, 27, 44, 54, 72, 84, 90, 95, 105, 125, 143, 150, 180, 195, 231, 256, 264, 287, 288, 308, 315, 319, 328, 351, 390, 423, 440, 483, 495, 512, 528, 540, 558, 559, 560, 576, 588, 608, 624, 627, 645, 648, 650, 728, 800, 805, 819, 840, 855, 870, 884, 896, 897, 924, 935, 945, 960, 975, 987

Offset: 1

J. M. Bergot and Robert Israel, Apr 13 2022

nonn

Composites k such that A000217(k) is divisible by A001414(k).

Contains all odd members of A046346, and in particular p^(p^k) if p is an odd prime and k >= 1.

			a(5) = 44 = 2*2*11 is a term because it is composite and A000217(44) = 44*45/2 = 990 is divisible by 2+2+11 = 15.

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

Cf. A000217, A001414, A046346.

filter:= proc(n) local t; not isprime(n) and (n*(n+1)/2/add(t[1]*t[2],t=ifactors(n)[2]))::integer end proc:
select(filter, [$4..1000]);

Select[Range[1000], CompositeQ[#] && Divisible[#*(# + 1)/2, Plus @@ Times @@@ FactorInteger[#]] &] (* Amiram Eldar, Apr 13 2022 *)

A036345 Divisible by its 'even' sum of prime factors (counted with multiplicity).

Original entry on oeis.org

2, 4, 16, 30, 60, 70, 72, 84, 220, 240, 256, 286, 288, 308, 378, 440, 450, 476, 528, 540, 560, 576, 594, 624, 646, 648, 728, 800, 884, 900, 960, 1040, 1056, 1080, 1160, 1170, 1248, 1276, 1404, 1456, 1496, 1530, 1748, 1776, 1798, 1824, 1976, 2322, 2408, 2464

Offset: 1

Patrick De Geest, Dec 15 1998

nonn

			646 = 2*17*19 so the sum of prime factors (with multiplicity) is 2+17+19 = 38 which is even and a divisor of 646 so 646 is in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000 (terms a(n) for n = 2..1002 from Harvey P. Dale).

Cf. A036346, A046346.

dspfQ[n_]:=Module[{spf=Total[Times@@@FactorInteger[n]]},EvenQ[spf] && Divisible[n,spf]]; Select[Range[4,2500,2],dspfQ] (* Harvey P. Dale, Oct 06 2011 *)

is(n) = my(f = factor(n), s = sum(i = 1, #f~, f[i, 1] * f[i, 2])); s > 0 && s % 2 == 0 && n % s == 0 \\ David A. Corneth, Feb 07 2019

Offset corrected and a(1) = 2 added by Thomas Ordowski, Feb 07 2019

A036346 Composites n such that A001414(n) is odd and divides n.

Original entry on oeis.org

27, 105, 150, 180, 231, 588, 627, 650, 805, 840, 897, 945, 1008, 1100, 1122, 1134, 1581, 1755, 2079, 2106, 2625, 2660, 2800, 2958, 2964, 2967, 2970, 3055, 3125, 3150, 3432, 3564, 3750, 3861, 4000, 4070, 4185, 4500, 4543, 4760, 4800, 5292, 5304, 5355

Offset: 1

Patrick De Geest, Dec 15 1998

nonn

			5445 = 3*3*5*11*11 -> sum = 3+3+5+11+11 = 33 (=odd) so 33 divides 5445 exactly.

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

Cf. A001414, A046346, A036345.

filter:= proc(n) local F,s;
  if isprime(n) then return false fi;
  F:=  ifactors(n)[2];
  s:= add(t[1]*t[2],t=F);
  s::odd and ( n mod s = 0)
end proc:
select(filter, [$1..10000]); # Robert Israel, Jul 15 2020

With[{s = Map[{#, Total[Times @@@ FactorInteger[#]]} &, Select[Range[4, 6000], CompositeQ]]}, Select[s, Mod[#1, #2] == 0 && OddQ[#2] & @@ # &][[All, 1]] ] (* Michael De Vlieger, Jul 15 2020 *)

Name and offset corrected by Robert Israel, Jul 15 2020

A134335 Numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is an integer, but not a prime.

Original entry on oeis.org

15, 35, 39, 42, 50, 51, 55, 65, 77, 78, 87, 91, 92, 95, 110, 111, 114, 115, 119, 123, 140, 141, 143, 155, 159, 161, 164, 170, 183, 185, 186, 187, 189, 201, 203, 204, 209, 215, 219, 221, 222, 225, 230, 235, 236, 242, 247, 258, 259, 264, 267, 284, 285, 287, 290

Offset: 1

Hieronymus Fischer, Oct 23 2007

nonn

			a(1) = 15, since 15 = 3*5 and (3+5)/2 = 4 is not prime.
a(5) = 50, since 50 = 2*5*5 and (2+5+5)/3 = 4 is not prime.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A000040, A001222, A100118, A046363, A133620, A133621.

Cf. A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134333, A134334.

fp[{a_,b_}]:=a*b;s={};Do[If[q=Total[fp/@FactorInteger[n]]/Total[Last/@FactorInteger[n]];IntegerQ[q]&&!PrimeQ[q],AppendTo[s,n]],{n,2,290}];s (* James C. McMahon, Apr 05 2025 *)

Definition clarified by the author, May 06 2013

A175729 Numbers n such that the sum of the prime factors with multiplicity of n divides n-1.

Original entry on oeis.org

6, 21, 45, 52, 225, 301, 344, 441, 697, 1225, 1333, 1540, 1625, 1680, 1695, 1909, 2025, 2041, 2145, 2295, 2466, 2601, 2926, 3051, 3104, 3146, 3400, 3510, 3738, 3888, 3901, 4030, 4186, 4251, 4375, 4641, 4675, 4693, 4930, 5005, 5085, 5244, 5425, 6025, 6105

Offset: 1

K. T. Lee (7x3(AT)21cn.com), Aug 23 2010

			For example, 21=7x3, 7+3=10 which divides 21-1=20.

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

Disjoint from A130871 and A046346.
Cf. A001414.
Contains A164643.

[k:k in [2..6200]| IsIntegral((k-1)/( &+[m[1]*m[2]: m in Factorization(k)]))]; // Marius A. Burtea, Sep 16 2019

A001414 := proc(n) ifactors(n)[2] ; add( op(1,p)*op(2,p),p=%) ; end proc:
isA175729 := proc(n) if (n-1) mod A001414(n) = 0 then true; else false; end if; end proc:
for n from 2 to 10000 do if isA175729(n) then printf("%d,",n) ; end if; end do:
# R. J. Mathar, Aug 24 2010

fQ[n_] := Mod[n - 1, Plus @@ Flatten[ Table[ #1, {#2}] & @@@ FactorInteger@ n]] == 0; Select[ Range@ 6174, fQ] (* Robert G. Wilson v, Aug 25 2010 *)

from sympy import factorint
def ok(n): return n>1 and (n-1)%sum(p*e for p, e in factorint(n).items())==0
print([k for k in range(10**4) if ok(k)]) # Michael S. Branicky, Sep 30 2022

{n : A001414(n) | (n-1)}. [R. J. Mathar, Aug 24 2010]

Extended by R. J. Mathar and Robert G. Wilson v, Aug 24 2010

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 *)

A370354 Composite numbers k that share a factor with sopfr(k), the sum of the primes dividing k, with repetition.

Original entry on oeis.org

4, 8, 9, 16, 18, 24, 25, 27, 30, 32, 36, 42, 49, 50, 60, 64, 66, 70, 72, 78, 81, 84, 98, 100, 102, 105, 110, 114, 120, 121, 125, 126, 128, 130, 132, 138, 140, 144, 150, 154, 156, 160, 162, 168, 169, 170, 174, 180, 182, 186, 190, 192, 195, 196, 200, 204, 216, 220, 222, 228, 230, 231, 234, 238, 240

Offset: 1

Scott R. Shannon, May 06 2024

nonn

All prime powers, see A246547, are terms.

			18 is a term as 18 = 2 * 3 * 3, and soprf(18) = 2 + 3 + 3 = 8, which shares a factor with 18.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A001414, A002808, A027746, A246547, A046346, A372524.

fn[{a_,b_}]:=a*b;Select[Range[240],CompositeQ[#]&&ContainsAny[First/@FactorInteger[#],First/@FactorInteger[ Total[fn/@FactorInteger[#]]]]&] (* James C. McMahon, May 06 2024 *)

A091340 Amicable numbers with property that each member m of the corresponding amicable pair is divisible by sopfr(m) (the sum of prime factors with repetition, A001414).

Original entry on oeis.org

821921625, 988676775, 4024942087978, 4179223134422, 100733767393275, 110452715806725

Offset: 1

Sven Simon, Dec 31 2003

Both members of the amicable pair appear in the sequence.
Conjecture: sequence is finite (even though there are many known amicable numbers - about 8*10^7 currently).
Sergei Chernykh and the teams @Boinc found a new pair with their still ongoing search in the 21-digit range, 109297847965212832096 with sopfr(m)=118618 and 109392896505354817184 with sopfr(m)=39152. - Sven Simon, Sep 19 2020

			a(1): 821921625 = 3^2*5^3*7*29*59*61, sopfr(n) = 177 = 3*59.
a(2): 988676775 = 3^2*5^2*71*199*311, sopfr(n) = 597 = 3*199.

Sergei Chernykh, Amicable pairs list
J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]

Cf. A001414, A046346, A063990, A259180 (amicable pairs).

Cf. A267076 (amicable pairs that have the same sopfr).

cp's OEIS Frontend

A078177 Composite numbers with an integer arithmetic mean of all prime factors.

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

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Mathematica

PARI

A352989 Composites k such that the k-th triangular number is divisible by the integer log of k.

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Maple

Mathematica

A036345 Divisible by its 'even' sum of prime factors (counted with multiplicity).

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Mathematica

PARI

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A036346 Composites n such that A001414(n) is odd and divides n.

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A134335 Numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is an integer, but not a prime.

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A175729 Numbers n such that the sum of the prime factors with multiplicity of n divides n-1.

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Magma

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