A089352 -id:A089352 - cp's OEIS frontend

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

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

A066031 Composite numbers n the sum of whose prime factors divides n, but which are not themselves powers of primes.

Original entry on oeis.org

30, 60, 70, 84, 90, 105, 120, 140, 150, 168, 180, 231, 234, 240, 252, 260, 270, 280, 286, 300, 315, 336, 350, 360, 450, 456, 468, 480, 490, 504, 520, 525, 528, 532, 540, 560, 572, 588, 600, 627, 646, 672, 693, 700, 702, 720, 735, 750, 756, 805, 810, 897

Offset: 1

Joseph L. Pe, Dec 12 2001

nonn

Primes and powers of primes have been excluded from the sequence because they trivially satisfy the condition "the sum of the prime factors of n divides n". Call a term of the sequence "primitive" if it is not a multiple of some previous term; for example, 70 is primitive while 60 is not. Are there infinitely many primitive terms? See A064623.

Intersection of A089352 and A024619. - Michel Marcus, Feb 03 2016

			The sum of the prime factors of 70 is 2 + 5 + 7 = 14, which divides 70.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Jean-Marie de Koninck, Florian Luca, Integers divisible by sums of powers of their prime factors, Journal of Number Theory, Volume 128, Issue 3, March 2008, Pages 557-563.
J. Pe, The Prime-perfect Numbers

Cf. A024619, A089352.

Select[ Range[2, 900], IntegerQ[ # / Apply[ Plus, First[ Transpose[ FactorInteger[ # ]]]]] && Mod[ #, # - EulerPhi[ # ]] != 0 & ]

isok(n) = if (omega(n)<2, return(0)); my(f = factor(n)) ; (n % vecsum(f[,1])) == 0; \\ Michel Marcus, Feb 03 2016

More terms from Robert G. Wilson v, Dec 12 2001

A086486 Numbers k such that the sum of the distinct prime divisors divides rad(k)=A007947(k).

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, 60, 61, 64, 67, 70, 71, 73, 79, 81, 83, 89, 90, 97, 101, 103, 105, 107, 109, 113, 120, 121, 125, 127, 128, 131, 137, 139, 140, 149, 150, 151, 157, 163, 167

Offset: 1

Amarnath Murthy, Jul 28 2003

Every prime power is a member.
Numbers with exactly two distinct prime divisors are not members of the sequence. - Victoria A Sapko (vsapko(AT)canes.gsw.edu), Sep 23 2003
Numbers k such that A008472(k) divides A007947(k).

			30 is a member. The prime divisors of 30 are 2, 3 and 5 and 2+3+5 = 10, divides 30.
84, however, is not a member because the sum of its distinct prime divisors (2+3+7=12) does not divide the product of its distinct prime divisors (2*3*7=42), even though 12 does divide 84. - _Harvey P. Dale_, Nov 26 2011, based on a comment from _Ray Chandler_

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A086487, A066031. A proper subset of A089352.

sdpQ[n_]:=Module[{dpds=Transpose[FactorInteger[n]][[1]]}, Divisible[ Times@@dpds,Total[dpds]]]; Select[Range[2,200],sdpQ] (* Harvey P. Dale, Nov 26 2011 *)

More terms from Victoria A Sapko (vsapko(AT)canes.gsw.edu), Sep 23 2003

Edited by Franz Vrabec, Sep 03 2005

A336296 The least prime p such that equation x = p*sopf(x) (where sopf(x) is the sum of distinct prime factors of x) has exactly n solutions in positive integers.

Original entry on oeis.org

2, 3, 7, 19, 71, 431, 1259, 4679, 9719, 23399, 7559, 42839, 134399, 181439, 477359, 241919, 262079, 453599

Offset: 1

Vladimir Letsko, Jul 16 2020

It seems that a(n) is the least number for which equation x = p*sopf(x) has exactly n solutions in positive integers not only for prime numbers.

			a(3) = 7 because there are 3 solutions of the equation x = 7*sopf(x), which are {49, 84, 105}, and this is the smallest prime that gives 3 solutions.

Vladimir Letsko, Mathematical Marathon, Problem 227 (in Russian)

Cf. A008472, A089352, A336098, A336099, A336297, A157190 (note overlap of values).

A114887 Multiperfect numbers sigma(n) = k*n, which are divisible by the sum of their prime factors without repetition.

Original entry on oeis.org

120, 672, 32760, 2178540, 1379454720, 14182439040, 518666803200, 30823866178560, 71065075104190073088, 154345556085770649600, 9186050031556349952000, 680489641226538823680000

Offset: 1

Sven Simon, Feb 19 2006

nonn

From a list of about 5000 multiperfect numbers, 38 numbers were found with the property, all having k <= 9, the largest was the only one having k=9. A091443 uses sopfr with repetition.

Conjecture: the sequence is finite.

			a(0) = 120 = 2^3*3*5, sopf(120) = 2+3+5 = 10.

Sven Simon, Table of n, a(n) for n = 1..38 [Conjectured to be complete]
Achim Flammenkamp, The Multiply Perfect Numbers Page (See here for the latest information about the search)
Eric Weisstein's World of Mathematics, Multiperfect numbers

Cf. A091443.

Intersection of A007691 and A089352. - Michel Marcus, Oct 08 2017

A143321 Positive integers k whose sum of distinct prime divisors divides k+1.

Original entry on oeis.org

15, 20, 24, 35, 54, 95, 98, 104, 119, 135, 143, 144, 160, 189, 207, 209, 224, 287, 319, 323, 324, 351, 363, 375, 377, 384, 390, 459, 464, 527, 539, 559, 608, 779, 845, 864, 875, 899, 923, 989, 999, 1000, 1007, 1029, 1189, 1199, 1215, 1280, 1343, 1349, 1375

Offset: 1

Leroy Quet, Aug 07 2008

nonn

			The distinct primes dividing 24 are 2 and 3, since 24 is factored as 2^3 *3^1. 2 + 3 = 5 is a divisor of 24 + 1 = 25. So 24 is a term of this sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A008472, A089352, A143322.

with(numtheory): a:= proc(n) local f: f:=factorset(n); if `mod`(n+1, add(i, i=f))=0 then n end if end proc: seq(a(n), n=2..1200); # Emeric Deutsch, Aug 14 2008

Select[Range[2,1500],Divisible[#+1,Total[FactorInteger[#][[All,1]]]]&] (* Harvey P. Dale, Aug 27 2022 *)

is(n) = n > 1 && (n + 1) % vecsum(factor(n)[, 1]) == 0 \\ David A. Corneth, Mar 10 2019

More terms from Emeric Deutsch, Aug 14 2008

More terms from Max Alekseyev, Mar 10 2009

A143322 Positive integers k whose sum of distinct prime divisors divides k-1.

Original entry on oeis.org

6, 21, 28, 36, 50, 96, 99, 216, 225, 301, 325, 352, 400, 441, 486, 495, 496, 576, 630, 676, 697, 784, 847, 925, 1225, 1296, 1333, 1521, 1536, 1587, 1695, 1701, 1792, 1909, 2025, 2041, 2133, 2145, 2500, 2601, 2624, 2916, 2926, 3025, 3200, 3220, 3276, 3456

Offset: 1

Leroy Quet, Aug 07 2008

nonn

			The distinct primes dividing 28 are 2 and 7, since 28 is factored as 2^2 * 7^1. 2 + 7 = 9 is a divisor of 28 - 1 = 27. So 28 is included in this sequence.

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

Cf. A008472, A089352, A143321.

with(numtheory): a:=proc(n) local f: f:= factorset(n): if `mod`(n-1, add(f[i], i=1..nops(f)))=0 then n else end if end proc: seq(a(n),n=2..4000); # Emeric Deutsch, Aug 16 2008

Select[Range[2,5000],Divisible[#-1,Total[Transpose[FactorInteger[#]][[1]]]]&] (* Harvey P. Dale, Aug 03 2014 *)

isok(k) = (k!=1) && (((k-1) % vecsum(factor(k)[,1])) == 0); \\ Michel Marcus, Dec 04 2020

Extended by Emeric Deutsch, Aug 16 2008

A161656 The largest multiple of {the sum of the distinct prime divisors of n} that is <=n.

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 7, 8, 9, 7, 11, 10, 13, 9, 8, 16, 17, 15, 19, 14, 20, 13, 23, 20, 25, 15, 27, 27, 29, 30, 31, 32, 28, 19, 24, 35, 37, 21, 32, 35, 41, 36, 43, 39, 40, 25, 47, 45, 49, 49, 40, 45, 53, 50, 48, 54, 44, 31, 59, 60, 61, 33, 60, 64, 54, 64, 67, 57, 52, 70, 71, 70, 73

Offset: 1

Leroy Quet, Jun 15 2009

nonn

a(n)=n iff n belongs to A089352. - Ivan Neretin, May 25 2016

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A008472, A161657.

A161656 := proc(n)
    local sd;
    if n <= 1 then
        0;
    else
        sd := A008472(n) ;
        sd*floor(n/sd) ;
    end if;
end proc: # R. J. Mathar, Mar 14 2014

Join[{0}, Table[Floor[#1/#2]*#2 &[n, Plus @@ FactorInteger[n][[All, 1]]], {n, 2, 73}] ](* Ivan Neretin, May 25 2016 *)

More terms from Sean A. Irvine, Sep 29 2009

A161657 a(n) = the smallest multiple of {the sum of the distinct prime divisors of n} that is >= n.

Original entry on oeis.org

2, 3, 4, 5, 10, 7, 8, 9, 14, 11, 15, 13, 18, 16, 16, 17, 20, 19, 21, 30, 26, 23, 25, 25, 30, 27, 36, 29, 30, 31, 32, 42, 38, 36, 40, 37, 42, 48, 42, 41, 48, 43, 52, 48, 50, 47, 50, 49, 56, 60, 60, 53, 55, 64, 63, 66, 62, 59, 60, 61, 66, 70, 64, 72, 80, 67, 76, 78, 70, 71, 75, 73

Offset: 2

Leroy Quet, Jun 15 2009

nonn

a(n) = n iff n belongs to A089352. - Ivan Neretin, May 25 2016

Ivan Neretin, Table of n, a(n) for n = 2..10000

Cf. A089352, A161656.

Table[Ceiling[#1/#2]*#2 &[n, Plus @@ FactorInteger[n][[All, 1]]], {n, 2, 73}] (* Ivan Neretin, May 25 2016 *)

More terms from Sean A. Irvine, Sep 29 2009

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

PARI

Formula

A066031 Composite numbers n the sum of whose prime factors divides n, but which are not themselves powers of primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A086486 Numbers k such that the sum of the distinct prime divisors divides rad(k)=A007947(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A336296 The least prime p such that equation x = p*sopf(x) (where sopf(x) is the sum of distinct prime factors of x) has exactly n solutions in positive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A114887 Multiperfect numbers sigma(n) = k*n, which are divisible by the sum of their prime factors without repetition.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A143321 Positive integers k whose sum of distinct prime divisors divides k+1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A143322 Positive integers k whose sum of distinct prime divisors divides k-1.

Views

Author

Keywords