A331378 -id:A331378 - cp's OEIS frontend

A036844 Numbers k such that k / sopfr(k) is an integer, where sopfr = sum-of-prime-factors, A001414.

2, 3, 4, 5, 7, 11, 13, 16, 17, 19, 23, 27, 29, 30, 31, 37, 41, 43, 47, 53, 59, 60, 61, 67, 70, 71, 72, 73, 79, 83, 84, 89, 97, 101, 103, 105, 107, 109, 113, 127, 131, 137, 139, 149, 150, 151, 157, 163, 167, 173, 179, 180, 181, 191, 193, 197, 199, 211, 220, 223

Offset: 1

Robert A. Stump (bee_ess107(AT)yahoo.com), Jan 09 2002

nonn

Union of A046346 and the primes. - T. D. Noe, Feb 20 2007
These are the Heinz numbers of the partitions counted by A330953. - Gus Wiseman, Jan 17 2020
Alladi and Erdős (1977) noted that sopfr(k) = k if k is a prime or k = 4. They called the terms for which k/sopfr(k) > 1 "special numbers", and proved that there are infinitely many such terms that are squarefree. - Amiram Eldar, Nov 02 2020

			a(12) = 27 because sopfr(27) = 3 + 3 + 3 = 9 and 27 is divisible by 9.

Amarnath Murthy, Generalization of Partition function and introducing Smarandache Factor Partition, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring-2000.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 89.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Krishnaswami Alladi and Paul Erdős, On an additive arithmetic function, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294, alternative link.
Mohan Lal, Iterates of a number-theoretic function, Math. Comp., Vol. 23, No. 105 (1969), pp. 181-183.

sopfr(n) is defined in A001414.
The version for prime indices instead of prime factors is A324851.
Partitions whose Heinz number is divisible by their sum: A330950.
Partitions whose Heinz number is divisible by their sum of primes: A330953.
Partitions whose product is divisible by their sum of primes: A330954.
Partitions whose product divides their sum of primes: A331381.
Product of prime indices is divisible by sum of prime factors: A331378.
Sum of prime factors is divisible by sum of prime indices: A331380.
Product of prime indices equals sum of prime factors: A331384.
Cf. A056239, A112798, A120383, A238525, A331379, A331382, A331383.

a036844 n = a036844_list !! (n-1)
a036844_list = filter ((== 0). a238525) [2..]
-- Reinhard Zumkeller, Jul 21 2014

Select[Range[2, 224], Divisible[#, Plus @@ Times @@@ FactorInteger[#]] &] (* Jayanta Basu, Aug 13 2013 *)

is_A036844(n)=n>1 && !(n%A001414(n)) \\ M. F. Hasler, Mar 01 2014

A238525(a(n)) = 0. - Reinhard Zumkeller, Jul 21 2014

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

A331416 Irregular triangle read by rows where T(n,k) is the number of integer partitions y of n such that Sum_i prime(y_i) = k.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 3, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 5, 3, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 6, 3, 4, 2

Offset: 0

Gus Wiseman, Jan 17 2020

			Triangle begins:
  1
  0 0 1
  0 0 0 1 1
  0 0 0 0 0 2 1
  0 0 0 0 0 0 1 3 1
  0 0 0 0 0 0 0 0 2 3 1 1
  0 0 0 0 0 0 0 0 0 1 4 3 1 2
  0 0 0 0 0 0 0 0 0 0 0 2 5 3 2 2 0 1
  0 0 0 0 0 0 0 0 0 0 0 0 1 4 6 3 4 2 0 2
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 6 6 4 6 2 1 2 0 1
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 8 6 6 7 2 4 2 0 1 0 0 0 1
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 6 9 7 9 7 3 7 2 1 1 0 0 0 2
Row n = 8 counts the following partitions (empty column not shown):
  (2222)  (332)    (44)      (41111)    (53)        (611)   (8)
          (422)    (431)     (311111)   (62)        (5111)  (71)
          (3221)   (3311)    (2111111)  (521)
          (22211)  (4211)               (11111111)
                   (32111)
                   (221111)
Column k = 19 counts the following partitions:
  (8)   (6111)   (532)        (443)       (33222)
  (71)  (51111)  (622)        (4331)      (42222)
                 (5221)       (4421)      (322221)
                 (4111111)    (33311)     (2222211)
                 (31111111)   (43211)
                 (211111111)  (332111)
                              (422111)
                              (3221111)
                              (22211111)

Row lengths are A331417.
Row sums are A000041.
Column sums are A000607.
Shifting row n to the left n times gives A331385.
Partitions whose Heinz number is divisible by their sum of primes: A330953.
Partitions of whose sum of primes is divisible by their sum are A331379.
Partitions whose product divides their sum of primes are A331381.
Partitions whose product equals their sum of primes are A331383.
Cf. A000040, A001414, A014689, A056239, A330950, A330954, A331378, A331387, A331415, A331418.

maxm[n_]:=Max@@Table[Total[Prime/@y],{y,IntegerPartitions[n]}];
Table[Length[Select[IntegerPartitions[n],Total[Prime/@#]==k&]],{n,0,10},{k,0,maxm[n]}]

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}]

A331384 Numbers whose sum of prime factors is equal to their product of prime indices.

Original entry on oeis.org

35, 65, 95, 98, 154, 324, 364, 476, 623, 763, 791, 812, 826, 938, 994, 1036, 1064, 1106, 1144, 1148, 1162, 1288, 1484, 1708, 1736, 2044, 2408, 2632, 4320, 5408, 6688, 6974, 8000, 10208, 12623, 12701, 12779, 14144, 19624, 23144, 25784, 26048, 44176, 47696

Offset: 1

Gus Wiseman, Jan 16 2020

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.

Numbers k such that A001414(k) = A003963(k). - Jason Yuen, Dec 19 2024

			The sequence of terms together with their prime indices begins:
     35: {3,4}
     65: {3,6}
     95: {3,8}
     98: {1,4,4}
    154: {1,4,5}
    324: {1,1,2,2,2,2}
    364: {1,1,4,6}
    476: {1,1,4,7}
    623: {4,24}
    763: {4,29}
    791: {4,30}
    812: {1,1,4,10}
    826: {1,4,17}
    938: {1,4,19}
    994: {1,4,20}
   1036: {1,1,4,12}
   1064: {1,1,1,4,8}
   1106: {1,4,22}
   1144: {1,1,1,5,6}
   1148: {1,1,4,13}
For example, 476 has prime factors {2,2,7,17} and prime indices {1,1,4,7}, and 2+2+7+17 = 28 = 1*1*4*7, so 476 is in the sequence.

These are the Heinz numbers of the partitions counted by A331383.
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, A003963, A324850, A330954, A331378, A331379, A331382, A331415, A331416.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Times@@primeMS[#]==Plus@@Prime/@primeMS[#]&]

A331415 Sum of prime factors minus sum of prime indices of n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 2, 3, 6, 3, 7, 4, 3, 4, 10, 3, 11, 4, 4, 7, 14, 4, 4, 8, 3, 5, 19, 4, 20, 5, 7, 11, 5, 4, 25, 12, 8, 5, 28, 5, 29, 8, 4, 15, 32, 5, 6, 5, 11, 9, 37, 4, 8, 6, 12, 20, 42, 5, 43, 21, 5, 6, 9, 8, 48, 12, 15, 6, 51, 5, 52, 26, 5, 13, 9, 9

Offset: 1

Gus Wiseman, Jan 17 2020

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 prime factors of 12 are {2,2,3}, while the prime indices are {1,1,2}, so a(12) = 7 - 4 = 3.

The number of k's is A331387(k) = sum of k-th column of A331385.
The sum of prime factors of n is A001414(n).
The sum of prime indices of n is A056239(n).
Numbers divisible by the sum of their prime factors are A036844.
Sum of prime factors is divisible by sum of prime indices: A331380
Product of prime indices equals sum of prime factors: A331384.
Cf. A000040, A000720, A001222, A014689, A056239, A112798, A301987, A318995, A325036, A330953, A331378, A331379, A331416, A331418.

Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>k*(p-PrimePi[p])]],{n,30}]

Totally additive with a(prime(k)) = prime(k) - k = A014689(k).

a(n) = A001414(n) - A056239(n).

A331380 Numbers whose sum of prime factors is divisible by their sum of prime indices.

Original entry on oeis.org

2, 4, 8, 16, 32, 33, 39, 55, 64, 65, 66, 74, 77, 78, 86, 91, 110, 128, 130, 132, 154, 156, 164, 182, 188, 220, 256, 260, 264, 308, 312, 364, 371, 411, 440, 459, 512, 513, 520, 528, 530, 616, 624, 636, 689, 728, 746, 755, 765, 766, 855, 880, 906, 915, 918, 1007

Offset: 1

Gus Wiseman, Jan 16 2020

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 sequence of terms together with their prime indices begins:
     2: {1}
     4: {1,1}
     8: {1,1,1}
    16: {1,1,1,1}
    32: {1,1,1,1,1}
    33: {2,5}
    39: {2,6}
    55: {3,5}
    64: {1,1,1,1,1,1}
    65: {3,6}
    66: {1,2,5}
    74: {1,12}
    77: {4,5}
    78: {1,2,6}
    86: {1,14}
    91: {4,6}
   110: {1,3,5}
   128: {1,1,1,1,1,1,1}
   130: {1,3,6}
   132: {1,1,2,5}
For example, 132 has prime factors {2,2,3,11} and prime indices {1,1,2,5}, and 18 is divisible by 9, so 132 is in the sequence.

These are the Heinz numbers of the partitions counted 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.
Heinz number is divisible by 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, A330954, A331378, A331379, A331382, A331415, A331416.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],Divisible[Plus@@Prime/@primeMS[#],Plus@@primeMS[#]]&]

cp's OEIS Frontend

A036844 Numbers k such that k / sopfr(k) is an integer, where sopfr = sum-of-prime-factors, A001414.

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

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

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A331416 Irregular triangle read by rows where T(n,k) is the number of integer partitions y of n such that Sum_i prime(y_i) = k.

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

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

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

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A331384 Numbers whose sum of prime factors is equal to their product of prime indices.

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A331415 Sum of prime factors minus sum of prime indices of n.

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