A036844 -id:A036844 - cp's OEIS frontend

A104465 a(n) is the least k with n prime factors (counting multiplicity) such that the sum of these n factors divides k. First member of A036844 with n prime factors.

2, 4, 27, 16, 72, 240, 288, 256, 2688, 3840, 4608, 13824, 30720, 61440, 73728, 65536, 294912, 983040, 4325376, 5898240, 11010048, 15728640, 18874368, 141557760, 169869312, 654311424, 1107296256, 1006632960, 2818572288, 4026531840

Offset: 1

David Wasserman, Mar 09 2005

A001222(a(n)) = n; A001414(a(n)) divides a(n).

			a(5) = 72 = 2*2*2*3*3; 2+2+2+3+3 = 12 divides 72.

Cf. A001414, A001222, A036844, A086487.

A104466 a(n) is the least k with n distinct prime factors such that the sum of its prime factors (counting multiplicity) divides k, or 0 if no such k exists. First member of A036844 with n distinct prime factors.

Original entry on oeis.org

2, 0, 30, 840, 2730, 72930, 870870, 9699690, 340510170, 9592993410, 265257422430, 8624101075590, 304250263527210, 14299762385778870, 1164365758518632670, 43657174563782890110, 1987938667108592728530, 172494415819766906755890

Offset: 1

David Wasserman, Mar 09 2005

A001221(a(n)) = n; A001414(a(n)) divides a(n). a(4) = 840; A086487(4) = 1122. This is probably the only difference between these two sequences.

			a(4) = 840 = 2^3*3*5*7; 2+2+2+3+5+7 = 21 divides 840.

Cf. A001414, A001221, A086487, A104465.

A324850 Numbers divisible by the product of their prime indices.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 28, 30, 32, 36, 48, 56, 60, 64, 72, 96, 112, 120, 128, 144, 152, 156, 168, 180, 192, 216, 224, 240, 256, 288, 304, 312, 330, 336, 360, 384, 432, 448, 476, 480, 512, 576, 608, 624, 660, 672, 720, 768, 784, 828, 840, 848, 864, 888, 896

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, with product A003963(n). For example, the prime indices of 30 are {1,2,3}, with product 6, which divides 30, so 30 is in the sequence.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  12: {1,1,2}
  16: {1,1,1,1}
  24: {1,1,1,2}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  48: {1,1,1,1,2}
  56: {1,1,1,4}
  60: {1,1,2,3}
  64: {1,1,1,1,1,1}
  72: {1,1,1,2,2}
  96: {1,1,1,1,1,2}

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

Cf. A003963, A036844, A120383, A324847, A324756, A324758, A324847, A324848, A324849, A324852, A324853, A324856, A324923, A324924, A324925, A324931.

Select[Range[100],Divisible[#,Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]]&]

isok(n) = my(f=factor(n)); !(n % prod(k=1, #f~, primepi(f[k,1])^f[k,2])); \\ Michel Marcus, Mar 22 2019

n/A003963(n) = A324933(n)/A324934(n).

A324851 Numbers > 1 divisible by the sum of their prime indices.

Original entry on oeis.org

2, 4, 6, 12, 15, 16, 20, 30, 35, 36, 42, 48, 56, 88, 99, 112, 120, 126, 130, 135, 143, 144, 160, 162, 180, 192, 210, 216, 220, 221, 228, 231, 242, 250, 256, 270, 275, 280, 288, 297, 300, 308, 322, 330, 338, 360, 396, 400, 408, 429, 435, 440, 455, 468, 480, 493

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The sum of prime indices of n is A056239(n). For example, the prime indices of 99 are {2,2,5}, with sum 9, a divisor of 99, so 99 is in the sequence.

For any k>=2, let d be a divisor of k such that d > A056239(k). Then 2^(d-A056239(k))*k is in the sequence. Similarly if k is in the sequence with d = A056239(k), then 2^d*k is in the sequence. - Robert Israel, Mar 19 2019

			The sequence of terms together with their prime indices begins:
    2: {1}
    4: {1,1}
    6: {1,2}
   12: {1,1,2}
   15: {2,3}
   16: {1,1,1,1}
   20: {1,1,3}
   30: {1,2,3}
   35: {3,4}
   36: {1,1,2,2}
   42: {1,2,4}
   48: {1,1,1,1,2}
   56: {1,1,1,4}
   88: {1,1,1,5}
   99: {2,2,5}
  112: {1,1,1,1,4}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  130: {1,3,6}
  135: {2,2,2,3}

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

Cf. A003963, A036844, A056239, A112798, A120383, A290822, A306844.

Cf. A324695, A324741, A324743, A324847, A324756, A324758, A324765, A324847, A324848, A324849, A324850, A324852.

filter:= proc(n) local t; n mod add(numtheory:-pi(t[1])*t[2],t=ifactors(n)[2]) = 0 end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 19 2019

Select[Range[2,100],Divisible[#,Plus@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]*k]]&]

isok(n) = {my(f = factor(n)); (n!=1) && !(n % sum(k=1, #f~, primepi(f[k,1])*f[k,2]));} \\ Michel Marcus, Mar 19 2019

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

Original entry on oeis.org

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

A330953 Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by their sum of primes of parts.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 3, 4, 6, 3, 12, 10, 12, 14, 27, 38, 44, 52, 48, 77, 101, 106, 127, 206, 268, 377, 392, 496, 602, 671, 821, 1090, 1318, 1568, 1926, 2260, 2703, 3258, 3942, 4858, 5923, 6891, 8286, 9728, 11676, 13775, 16314, 19749, 23474, 27793, 32989, 38775

Offset: 1

Gus Wiseman, Jan 15 2020

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The a(1) = 1 through a(11) = 12 partitions: (A = 10, B = 11):
  1  2   3  4     5  6    7      8         9        A         B
     11     1111     222  3211   431       432      5311      542
                     321  22111  4211      3321     22111111  5411
                                 11111111  32211              33221
                                           321111             42221
                                           2211111            53111
                                                              322211
                                                              431111
                                                              521111
                                                              2222111
                                                              3311111
                                                              32111111
For example, the partition (3,3,2,2,1) is counted under a(11) because 5*5*3*3*2 = 450 is divisible by 5+5+3+3+2 = 18.

The Heinz numbers of these partitions are given by A036844.
Numbers divisible by the sum of their prime indices are A324851.
Partitions whose product is divisible by their sum are A057568.
Partitions whose Heinz number is divisible by all parts are A330952.
Partitions whose Heinz number is divisible by their product are A324925.
Partitions whose Heinz number is divisible by their sum are A330950.
Partitions whose product is divisible by their sum of primes are A330954.
Cf. A001414, A003963, A056239, A112798, A120383, A326149, A326155, A331378, A331379, A331381, A331383, A331415, A331416, A331417.

Table[Length[Select[IntegerPartitions[n],Divisible[Times@@Prime/@#,Plus@@Prime/@#]&]],{n,30}]

A331383 Number of integer partitions of n whose sum of primes of parts is equal to their product of parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 1, 1, 1, 2, 2, 2, 1, 4, 2, 2, 2, 4, 2, 3, 4, 1, 3, 4, 5, 0, 3, 3, 1, 6, 2, 1, 5, 4, 2, 3, 4, 2, 2, 3, 1, 5, 2, 3, 4, 6, 5, 2, 7, 1, 3, 5, 3, 4, 2, 5, 5, 4, 7, 3, 6, 4, 4, 2, 4, 4, 3, 9, 4, 3, 5, 3, 5, 4, 4, 4, 3, 7, 4, 2, 8, 2, 3

Offset: 1

Gus Wiseman, Jan 16 2020

nonn

			The a(n) partitions for n = 7, 9, 18, 24:
  (4,3)  (6,3)    (12,4,1,1)                 (19,4,1)
         (4,4,1)  (11,4,1,1,1)               (18,4,1,1)
                  (8,5,1,1,1,1,1)            (9,6,1,1,1,1,1,1,1,1,1)
                  (4,2,2,2,1,1,1,1,1,1,1,1)
For example, (4,4,1) has sum of primes of parts 7+7+2 = 16 and product of parts 4*4*1 = 16, so is counted under a(9).

The Heinz numbers of these partitions are given by A331384.
Numbers divisible by the sum of their prime factors are A036844.
Partitions whose product is divisible by their sum are A057568.
Numbers divisible by the sum of their prime indices are A324851.
Product of prime indices is divisible by sum of prime indices: A326149.
Partitions whose Heinz number is divisible by their sum are A330950.
Partitions whose Heinz number is divisible by their sum of primes: A330953.
Sum of prime factors is divisible by sum of prime indices: A331380
Partitions whose product divides their sum of primes are A331381.
Cf. A000040, A001414, A324850, A330954, A331378, A331379, A331382, A331415, A331416.

Table[Length[Select[IntegerPartitions[n],Times@@#==Plus@@Prime/@#&]],{n,30}]

a(n) = my(c=0); forpart(v=n, if(vecprod(Vec(v))==sum(i=1, #v, prime(v[i])), c++)); c; \\ Jinyuan Wang, Feb 14 2025

a(71)-a(87) from Robert Price, Apr 10 2020

A330954 Number of integer partitions of n whose product is divisible by the sum of primes of their parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 2, 3, 4, 2, 3, 9, 8, 18, 15, 25, 35, 44, 50, 70, 71, 93, 141, 158, 226, 286, 337, 439, 532, 648, 789, 1013, 1261, 1454, 1776, 2176, 2701, 3258, 3823, 4606, 5521, 6613, 7810, 9202, 11074, 13145, 15498, 18413, 21818, 25774, 30481, 35718

Offset: 1

Gus Wiseman, Jan 15 2020

nonn

			The a(7) = 1 through a(15) = 8 partitions (empty column not shown):
  43  63   541     83     552   6322   4433       5532
      441  4222    3332   6411  7411   7322       6522
           222211  5222         62221  44321      84111
                   33221               63311      333222
                                       65111      432222
                                       72221      3322221
                                       433211     32222211
                                       4322111    333111111
                                       322211111
For example, the partition (3,3,2,2,1) has product 3 * 3 * 2 * 2 * 1 = 36 and sum of primes 5 + 5 + 3 + 3 + 2 = 18, and 36 is divisible by 18, so (3,3,2,2,1) is counted under a(11).

The Heinz numbers of these partitions are given by A331378.
Partitions whose product is divisible by their sum are A057568.
Numbers divisible by the sum of their prime indices are A324851.
Partitions whose sum of primes divides their product of primes are A330953.
Partitions whose sum of primes divides of their product are A331381.
Partitions whose product equals their sum of primes are A331383.
Cf. A000040, A001414, A036844, A056239, A324850, A326149, A330950, A331379, A331382, A331384, A331415, A331416.

Table[Length[Select[IntegerPartitions[n],Divisible[Times@@#,Plus@@Prime/@#]&]],{n,30}]

A331379 Number of integer partitions of n whose sum of primes of parts is divisible by n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 4, 6, 7, 7, 7, 9, 11, 18, 24, 33, 39, 44, 51, 55, 66, 83, 106, 121, 145, 167, 193, 232, 253, 300, 342, 427, 469, 557, 628, 729, 846, 936, 1088, 1195, 1450, 1601, 1895, 2097, 2482, 2782, 3220, 3592, 4073, 4641, 5202, 5911, 6494, 7443, 8294

Offset: 1

Gus Wiseman, Jan 17 2020

nonn

			The a(6) = 1 through a(11) = 7 partitions:
  111111  52       53        54         64          641
          1111111  62        63         541         5411
                   521       531        631         6311
                   11111111  621        5311        53111
                             5211       6211        62111
                             111111111  52111       521111
                                        1111111111  11111111111
For example, the partition (5,4,1,1) has sum of primes 11+7+2+2 = 22, which is divisible by 5+4+1+1 = 11, so (5,4,1,1) is counted under a(11).

The Heinz numbers of these partitions are given by A331380.
Numbers divisible by the sum of their prime factors are A036844.
Partitions whose product is divisible by their sum are A057568.
Numbers divisible by the sum of their prime indices are A324851.
Product of prime indices is divisible by sum of prime indices: A326149.
Partitions whose Heinz number is divisible by their sum are A330950.
Partitions whose Heinz number is divisible by their sum of primes: A330953.
Partitions whose product divides their sum of primes are A331381.
Partitions whose product is equal to their sum of primes are A331383.
Product of prime indices equals sum of prime factors: A331384.
Cf. A000040, A001414, A056239, A324850, A330954, A331378, A331382.

Table[Length[Select[IntegerPartitions[n],Divisible[Plus@@Prime/@#,n]&]],{n,30}]

A331381 Number of integer partitions of n whose sum of primes of parts is divisible by their product of parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 5, 2, 6, 6, 5, 5, 7, 4, 7, 7, 7, 10, 8, 9, 6, 10, 9, 9, 15, 7, 12, 10, 14, 10, 10, 8, 8, 15, 10, 7, 16, 13, 9, 10, 14, 12, 10, 8, 14, 11, 13, 11, 16, 15, 14, 15, 15, 10, 14, 18, 11, 12, 13, 13, 18, 21, 15, 16, 19, 16, 15, 8, 17, 17

Offset: 0

Gus Wiseman, Jan 16 2020

nonn

			The a(n) partitions for n = 1, 5, 7, 8, 9, 13, 14:
  1  221    43       311111    63         7411           65111
     311    511      11111111  441        721111         322211111
     11111  3211               711        43111111       311111111111
            22111              42111      421111111      11111111111111
            1111111            2211111    3211111111
                               111111111  22111111111
                                          1111111111111

The Heinz numbers of these partitions are given by A331382.
Numbers divisible by the sum of their prime factors are A036844.
Partitions whose product is divisible by their sum are A057568.
Numbers divisible by the sum of their prime indices are A324851.
Product of prime indices is divisible by sum of prime indices: A326149.
Partitions whose Heinz number is divisible by their sum are A330950.
Sum of prime factors is divisible by sum of prime indices: A331380
Partitions whose product is equal to their sum of primes are A331383.
Product of prime indices equals sum of prime factors: A331384.
Cf. A000040, A001414, A324850, A330953, A330954, A331378, A331379, A331415, A331416.

Table[Length[Select[IntegerPartitions[n],Divisible[Plus@@Prime/@#,Times@@#]&]],{n,0,30}]

cp's OEIS Frontend

A104465 a(n) is the least k with n prime factors (counting multiplicity) such that the sum of these n factors divides k. First member of A036844 with n prime factors.

Views

Author

Keywords

Comments

Examples

Crossrefs

A104466 a(n) is the least k with n distinct prime factors such that the sum of its prime factors (counting multiplicity) divides k, or 0 if no such k exists. First member of A036844 with n distinct prime factors.

Views

Author

Keywords

Comments

Examples

Crossrefs

A324850 Numbers divisible by the product of their prime indices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A324851 Numbers > 1 divisible by the sum of their prime indices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

MATLAB

Magma

Maple

Mathematica

PARI

Python

Extensions

A330953 Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by their sum of primes of parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A331383 Number of integer partitions of n whose sum of primes of parts is equal to their product of parts.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A330954 Number of integer partitions of n whose product is divisible by the sum of primes of their parts.

Views

Author

Keywords

Examples

Crossrefs