A071139 -id:A071139 - cp's OEIS frontend

A089352 Numbers that are divisible by the sum of their distinct prime factors (A008472).

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 37, 41, 43, 47, 49, 53, 59, 60, 61, 64, 67, 70, 71, 73, 79, 81, 83, 84, 89, 90, 97, 101, 103, 105, 107, 109, 113, 120, 121, 125, 127, 128, 131, 137, 139, 140, 149, 150, 151, 157, 163, 167, 168, 169, 173

Offset: 1

Ramin Naimi (rnaimi(AT)oxy.edu), Dec 26 2003

The Koninck & Luca bound of x / exp(c(1 + o(1))sqrt(log x log log x)) on A158804 applies equally to this sequence. - Charles R Greathouse IV, Sep 08 2012

			84=2*2*3*7 is divisible by 2+3+7.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Jean-Marie de Koninck, Florian Luca, Integers divisible by the sum of their prime factors, Mathematika 52:1-2 (2005), pp. 69-77.

Cf. A008472 (sopf).

Different from A071139.

primeDivisors[n_] := Select[Divisors[n], PrimeQ]; primeSumDivQ[n_] := 0 == Mod[n, Apply[Plus, primeDivisors[n]]]; Select[Range[2, 300], primeSumDivQ]
Select[Range[2, 175], Divisible[#, Plus @@ First /@ FactorInteger[#]] &] (* Jayanta Basu, Aug 13 2013 *)

is(n)=my(f=factor(n)[,1]);n%sum(i=1,#f,f[i])==0 \\ Charles R Greathouse IV, Feb 01 2013

Name edited by Michel Marcus, Jul 15 2020

A326847 Heinz numbers of integer partitions of m >= 0 using divisors of m whose length also divides m.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197

Offset: 1

Gus Wiseman, Jul 26 2019

nonn

First differs from A071139, A089352 and A086486 in lacking 60. First differs from A326837 in lacking 268.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The enumeration of these partitions by sum is given by A326842.

			The sequence of terms together with their prime indices begins:
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  30: {1,2,3}
  31: {11}
  32: {1,1,1,1,1}
  37: {12}

R. J. Mathar, Table of n, a(n) for n = 1..489

Intersection of A326841 and A316413.

Cf. A001222, A018818, A056239, A067538, A112798, A316413, A326836, A326842.

isA326847 := proc(n)
    psigsu := A056239(n) ;
    for ifs in ifactors(n)[2] do
        p := op(1,ifs) ;
        psig := numtheory[pi](p) ;
        if modp(psigsu,psig) <> 0 then
            return false;
        end if;
    end do:
    psigle := numtheory[bigomega](n) ;
    if modp(psigsu,psigle) = 0 then
        true;
    else
        false;
    end if;
end proc:
n := 1:
for i from 2 to 3000 do
    if isA326847(i) then
        printf("%d %d\n",n,i);
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 09 2019

Select[Range[2,100],With[{y=Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},Divisible[Total[y],Length[y]]&&And@@IntegerQ/@(Total[y]/y)]&]

Intersection of A326841 and A316413.

A144100 Numbers k such that k is strictly greater than f(k), where f(k) = 1 if k is prime, 2 * rad(k) if 4 divides k and rad(k) otherwise.

Original entry on oeis.org

2, 3, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 24, 25, 27, 29, 31, 32, 36, 37, 40, 41, 43, 45, 47, 48, 49, 50, 53, 54, 56, 59, 61, 63, 64, 67, 71, 72, 73, 75, 79, 80, 81, 83, 88, 89, 90, 96, 97, 98, 99, 100, 101, 103, 104, 107, 108, 109, 112, 113, 117, 120, 121, 125, 126

Offset: 1

Reikku Kulon, Sep 10 2008

This is the set of all integers k such that there exists a full period linear congruential pseudorandom number generator x -> bx + c (mod k), where b is not a multiple of k, b - 1 is a multiple of f(k) and c is a positive integer relatively prime to k.
4 is the only prime power not a member of the set: f(4) = 2 * rad(4) = 4.
This sequence consists of the primes and 2*A013929. - Charlie Neder, Jan 28 2019

			2 is a member: f(2) = 1 and the sequence (0, 1, 0, ...) given by x -> x + 1 (mod 2) has period 2.
8 is a member: f(8) = 4 and the sequence (0, 1, 6, 7, 4, 5, 2, 3, 0, ...) given by x -> 5x + 1 (mod 8) has period 8.
18 is a member: f(18) = 6 and the sequence (0, 1, 14, 3, 4, 17, 6, 7, 2, 9, 10, 5, 12, 13, 8, 15, 16, 11, 0, ...) given by x -> 13x + 1 (mod 18) has period 18.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000040, A007947, A071139, A082377, A086486, A089352, A133810, A133811.

a144100 n = a144100_list !! (n-1)
a144100_list = filter (\x -> a144907 x < x) [1..]
-- Reinhard Zumkeller, Mar 12 2014

rad(n) = local(p); p=factor(n)[, 1]; prod(i=1, length(p), p[i]) ;
f(n) = if (isprime(n), 1, if ((n % 4)==0 , 2*rad(n), rad(n))); isok(n) = n > f(n); \\ Michel Marcus, Aug 09 2013

A144907(a(n)) < a(n). - Reinhard Zumkeller, Mar 12 2014

A071140 Numbers n such that sum of distinct primes dividing n is divisible by largest prime dividing n; n is neither a prime, nor a true power of prime.

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

Offset: 1

Labos Elemer, May 13 2002

nonn

a(n) are the numbers such that the difference between the largest and the smallest prime divisor equals the sum of the other distinct prime divisors. - Michel Lagneau, Nov 13 2011
The statement above is only true for 966 of the first 1000 terms. The first counterexample is a(140) = 15015. - Donovan Johnson, Apr 10 2013
Lagneau's definition can be simplified to the largest prime divisor equals the sum of the other distinct prime divisors. - Christian N. K. Anderson, Apr 15 2013

			n = 70 = 2*5*7 has a form of 2pq, where p and q are twin primes; n = 3135 = 3*5*11*19, sum = 3+5+11+19 = 38 = 2*19, divisible by 19.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.

Subsequence of A024619.

a071140 n = a071140_list !! (n-1)
a071140_list = filter (\x -> a008472 x `mod` a006530 x == 0) a024619_list
-- Reinhard Zumkeller, Apr 18 2013

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Greater[s, 1], Print[{n, ba[n]}]], {n, 2, 1000000}]
(* Second program: *)
Select[Range@ 2250, And[Length@ # > 1, Divisible[Total@ #, Last@ #]] &[FactorInteger[#][[All, 1]] ] &] (* Michael De Vlieger, Jul 18 2017 *)

A008472(n)/A006530(n) is an integer and n has at least 3 distinct prime factors.

A008472(a(n)) mod A006530(a(n)) = 0 and A010055(a(n)) = 0. - Reinhard Zumkeller, Apr 18 2013

A071147 Smallest squarefree number k with exactly n prime factors such that the sum of the prime factors is divisible by the largest prime dividing k, or 0 if no such k exists.

Original entry on oeis.org

1, 2, 0, 30, 3135, 3570, 72930, 1231230, 14804790, 497668710, 14908423530, 278196808890, 12192694624110, 550939666387110, 21275256232500270, 1458502323630662310, 87988283090327810190, 3254611619240885033130, 261462818462495728868790, 9965666894849284108299810, 557940830126698960967415390, 90544636506979071680577724410

Offset: 0

Labos Elemer, May 13 2002

nonn

No solution exists for n=2, so a(2)=0.

			a(0) =       1 = 1;
a(1) =       2 = 2;
a(3) =      30 = 2 *  3 *  5;
a(4) =    3135 = 3 *  5 * 11 * 19;
a(5) =    3570 = 2 *  3 *  5 *  7 * 17;
a(6) =   72930 = 2 *  3 *  5 * 11 * 13 * 17;
a(7) = 1231230 = 2 *  3 *  5 *  7 * 11 * 13 * 41.

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071146.

A008472(k)/A006530(k) is an integer; k is squarefree and has exactly n prime factors.

Corrected and extended by Donovan Johnson, Apr 22 2008

Name corrected by Jon E. Schoenfield, Jul 08 2018

A071142 Numbers of the form 2pq where (p,q) is a twin prime pair.

Original entry on oeis.org

30, 70, 286, 646, 1798, 3526, 7198, 10366, 20806, 23326, 38086, 44998, 64798, 73726, 78406, 103966, 115198, 145798, 159046, 194686, 242206, 352798, 373246, 426886, 544966, 649798, 719998, 763846, 824326, 871198, 1312198, 1351366, 1371166, 1472326, 1555846

Offset: 1

Labos Elemer, May 13 2002

nonn

For each term k, A008472(k)/A006530(k) = (2+p+q)/q = (q+q)/q = 2.

			a(1) = 2 * (product of 1st twin prime pair) = 2*3*5 = 30.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] amo[x_] := Abs[MoebiusMu[x]] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Equal[lf[n], 3]&& !Equal[amo[n], 0], Print[{n, ba[n]}]], {n, 2, 1000000}]

a(n) = 2*A037074(n).

Edited by Jon E. Schoenfield, Sep 30 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

A071141 Numbers n such that sum of distinct primes dividing n is divisible by the largest prime dividing n. Also n is neither a prime, nor a true power of prime and n is squarefree. Squarefree solutions of A071140.

Original entry on oeis.org

30, 70, 286, 646, 1798, 3135, 3526, 3570, 6279, 7198, 8855, 8970, 10366, 10626, 10695, 11571, 15015, 16095, 16530, 17255, 17391, 20615, 20706, 20735, 20806, 23326, 24738, 24882, 26691, 28083, 31031, 36519, 36890, 38086, 38130, 41151, 41615, 44330, 44998

Offset: 1

Labos Elemer, May 13 2002

nonn

			n = 286 = 2*11*13 has a form of 2pq, where p and q are twin primes;
n = 5414430 = 2*3*5*7*19*23*59, sum = 2+3+5+7+19+23+59 = 118 = 2*59.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] amo[x_] := Abs[MoebiusMu[x]] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Greater[lf[n], 1]&& !Equal[amo[n], 1], Print[{n, ba[n]}]], {n, 2, 1000000}]
(* Second program: *)
Select[Range@ 45000, Function[n, And[Length@ # > 1, SquareFreeQ@ n, Divisible[Total@ #, Last@ #]] &[FactorInteger[n][[All, 1]] ]]] (* Michael De Vlieger, Jul 18 2017 *)

A008472(n)/A006530(n) is an integer, n has at least 3 distinct prime factors and n is squarefree.

A071146 Numbers n such that sum of distinct primes dividing n is divisible by the largest prime dividing n. Also n has exactly 7 distinct prime factors and n is squarefree.

Original entry on oeis.org

1231230, 2062830, 2181270, 3327870, 3594990, 4224990, 4320030, 4671030, 5162430, 5411406, 5414430, 6767670, 7052430, 7432230, 7870830, 7947030, 8150142, 8273265, 8287230, 8569470, 8804334, 9378390, 10630830, 10705695, 10757838, 10776990, 10900230

Offset: 1

Labos Elemer, May 13 2002

nonn

			n = pqrstu, p<q<r<s<t<u, primes, p+q+r+s+t+u = ku; n = 9378390 = 2*3*5*7*17*37*71; sum = 2+3+5+7+17+37+71 = 142 = 2*71

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] amo[x_] := Abs[MoebiusMu[x]] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Equal[lf[n], 7]&& !Equal[amo[n], 0], Print[{n, ba[n]}]], {n, 2, 1000000}]

A008472(n)/A006530(n) is an integer; A001221(n) = 7, n is squarefree.

A336445 Integers m such that m/sopf(m) is a prime number where sopf(m) is A008472(m), the sum of the distinct primes dividing m.

Original entry on oeis.org

4, 9, 25, 30, 49, 70, 84, 105, 121, 169, 231, 234, 260, 286, 289, 361, 456, 529, 532, 627, 646, 805, 841, 897, 961, 1116, 1364, 1369, 1581, 1665, 1681, 1798, 1849, 1924, 2064, 2150, 2209, 2632, 2809, 2967, 3055, 3339, 3481, 3526, 3721, 4489, 4543, 4824, 5025, 5041

Offset: 1

Michel Marcus, Jul 22 2020

nonn

All squares of primes (A001248) are terms.

			4 is a term since sopf(4)=2 and 4/2 = 2 is a prime.
30 is a term since sopf(30)=10 and 30/10 = 3 is a prime.

Michel Marcus, Table of n, a(n) for n = 1..1049 (terms up to 10^7)

Cf. A008472 (sopf).
Subsequence of A071139.
A001248 is a subsequence.
Cf. A336296, A336297.

sopf(n)=vecsum(factor(n)[, 1]); \\ A008472
isokp(k) = my(q=k/sopf(k)); (denominator(q)==1) && isprime(q);

cp's OEIS Frontend

A089352 Numbers that are divisible by the sum of their distinct prime factors (A008472).

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A326847 Heinz numbers of integer partitions of m >= 0 using divisors of m whose length also divides m.

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A144100 Numbers k such that k is strictly greater than f(k), where f(k) = 1 if k is prime, 2 * rad(k) if 4 divides k and rad(k) otherwise.

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A071140 Numbers n such that sum of distinct primes dividing n is divisible by largest prime dividing n; n is neither a prime, nor a true power of prime.

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A071147 Smallest squarefree number k with exactly n prime factors such that the sum of the prime factors is divisible by the largest prime dividing k, or 0 if no such k exists.

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A071142 Numbers of the form 2*p*q where (p,q) is a twin prime pair.

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

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A071142 Numbers of the form 2pq where (p,q) is a twin prime pair.