A319000 -id:A319000 - cp's OEIS frontend

A379681 Sum plus product of the multiset of prime indices of n.

1, 2, 4, 3, 6, 5, 8, 4, 8, 7, 10, 6, 12, 9, 11, 5, 14, 9, 16, 8, 14, 11, 18, 7, 15, 13, 14, 10, 20, 12, 22, 6, 17, 15, 19, 10, 24, 17, 20, 9, 26, 15, 28, 12, 19, 19, 30, 8, 24, 16, 23, 14, 32, 15, 23, 11, 26, 21, 34, 13, 36, 23, 24, 7, 27, 18, 38, 16, 29, 20

Offset: 1

Gus Wiseman, Jan 05 2025

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.

Includes all positive integers.

For prime factors instead of indices we have A075254, subtracted A075255.
Positions of first appearances are A379682.
For minus instead of plus we have A325036, which takes the following values:
- zero: A301987, counted by A001055
- negative: A325037, counted by A114324
- positive: A325038, counted by A096276 shifted right
- negative one: A325041, counted by A028422
- one: A325042, counted by A001055 shifted right
- nonnegative: A325044, counted by A096276
- nonpositive: A379721, counted by A319005
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Cf. A000720, A003963, A178503, A175508, A319000, A325032, A325033, A325034, A325035, A325040.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Plus@@prix[n]+Times@@prix[n],{n,30}]

a(n) = A056239(n) + A003963(n).

The last position of k is 2^(k-1).

A379734 Number of integer partitions of n into parts > 1 whose product is a multiple of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 4, 3, 2, 1, 8, 1, 4, 8, 27, 1, 32, 1, 40, 24, 13, 1, 175, 56, 22, 188, 166, 1, 387, 1, 874, 166, 61, 410, 1833, 1, 98, 391, 3028, 1, 2704, 1, 1828, 5893, 239, 1, 16756, 3446, 9742, 1865, 5276, 1, 32927, 8179, 31643, 3840, 814, 1, 82958, 1

Offset: 1

Gus Wiseman, Jan 07 2025

nonn

Allowing 1's gives A057568.

			The a(n) partitions for n = 6, 10, 12, 15, 22:
  (6)  (10)     (12)         (15)         (22)
       (5,3,2)  (6,6)        (6,5,4)      (11,6,5)
                (5,4,3)      (7,5,3)      (11,7,4)
                (6,4,2)      (10,3,2)     (11,8,3)
                (4,3,3,2)    (5,4,3,3)    (11,9,2)
                (5,3,2,2)    (5,5,3,2)    (11,4,4,3)
                (6,2,2,2)    (6,5,2,2)    (11,5,4,2)
                (3,3,2,2,2)  (5,3,3,2,2)  (11,6,3,2)
                                          (11,7,2,2)
                                          (11,3,3,3,2)
                                          (11,4,3,2,2)
                                          (11,5,2,2,2)
                                          (11,3,2,2,2,2)

These partitions are ranked by the odd terms of A326149.
The strict version is A379735, allowing 1's A379733.
A000041 counts integer partitions, strict A000009.
A002865 counts partitions into parts > 1.
A379666 counts partitions by sum and product, without 1's A379668.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A069016, A318950, A319000, A319916, A324851, A325041, A326152, A379671, A379678.

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=1, 1, 0), `if`(i<2, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, min(i, n-i), t/igcd(i, t)))))
    end:
a:= n-> `if`(isprime(n), 1, b(n$3)):
seq(a(n), n=1..70);  # Alois P. Heinz, Jan 07 2025

Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&Divisible[Times@@#,n]&]],{n,30}]

A379319 Even numbers whose product of prime indices is a multiple of their sum of prime indices.

Original entry on oeis.org

2, 30, 84, 108, 150, 154, 190, 198, 200, 264, 364, 390, 442, 468, 490, 506, 580, 624, 630, 658, 700, 714, 810, 840, 846, 874, 900, 918, 952, 988, 1020, 1080, 1110, 1118, 1120, 1224, 1254, 1330, 1430, 1440, 1480, 1596, 1632, 1666, 1708, 1710, 1716, 1786, 1794

Offset: 1

Gus Wiseman, Jan 17 2025

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 and product of prime indices are A056239 and A003963 respectively.

			The prime indices of 150 are {1,2,3,3}, with sum 9 and product 18, so 150 is in the sequence.
The terms together with their prime indices begin:
     2: {1}
    30: {1,2,3}
    84: {1,1,2,4}
   108: {1,1,2,2,2}
   150: {1,2,3,3}
   154: {1,4,5}
   190: {1,3,8}
   198: {1,2,2,5}
   200: {1,1,1,3,3}
   264: {1,1,1,2,5}
   364: {1,1,4,6}
   390: {1,2,3,6}
   442: {1,6,7}
   468: {1,1,2,2,6}
   490: {1,3,4,4}

Even terms of A326149, which is counted by A057568 (strict A379733).
For nonprime instead of even we have A326150.
For odd instead of even we have A379318, counted by A379734 (strict A379735).
Partitions of this type are counted by A379320.
For squarefree instead of even we have A379844.
The squarefree case is A379845.
Divide all terms by 2 to get A380217.
A000040 lists the prime numbers, differences A001223.
A003963 multiplies together prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A069016, A301988, A318950, A319000, A324851, A325041, A379671, A379678.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],MemberQ[prix[#],1]&&Divisible[Times@@prix[#],Total[prix[#]]]&]

A379735 Number of strict integer partitions of n into parts > 1 whose product is a multiple of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 3, 4, 4, 1, 8, 1, 11, 9, 7, 1, 26, 7, 10, 18, 33, 1, 67, 1, 56, 37, 20, 69, 158, 1, 27, 70, 252, 1, 280, 1, 207, 402, 52, 1, 834, 133, 423, 226, 465, 1, 1132, 635, 1541, 388, 129, 1, 3377, 1, 171, 2891, 3561, 1674, 3154

Offset: 1

Gus Wiseman, Jan 07 2025

nonn

These partitions are ranked by the odd squarefree terms of A326149.

			The a(n) partitions for n = 2, 9, 12, 15, 18, 20, 21:
  (2)  (9)    (12)     (15)      (18)       (20)         (21)
       (6,3)  (5,4,3)  (6,5,4)   (12,6)     (8,7,5)      (8,7,6)
              (6,4,2)  (7,5,3)   (9,5,4)    (10,6,4)     (9,7,5)
                       (10,3,2)  (9,6,3)    (10,8,2)     (11,7,3)
                                 (9,7,2)    (11,5,4)     (12,7,2)
                                 (6,5,4,3)  (12,5,3)     (14,4,3)
                                 (7,6,3,2)  (7,6,5,2)    (7,6,5,3)
                                 (9,4,3,2)  (8,5,4,3)    (9,7,3,2)
                                            (9,5,4,2)    (7,5,4,3,2)
                                            (10,5,3,2)
                                            (6,5,4,3,2)

Allowing 1's gives A379733.
The non-strict version is A379734, allowing 1's A057568.
A000041 counts integer partitions, strict A000009.
A002865 counts partitions into parts > 1.
A379666 counts partitions by sum and product, without 1's A379668.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A069016, A319000, A319057, A319916, A324851, A326152, A379671, A379678.

Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&UnsameQ@@#&&Divisible[Times@@#,n]&]],{n,30}]

A379682 Least number whose prime indices have sum + product = n.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 10, 7, 14, 11, 15, 13, 26, 17, 25, 19, 33, 23, 35, 29, 58, 31, 51, 37, 74, 41, 65, 43, 69, 47, 85, 53, 105, 59, 93, 61, 122, 67, 115, 71, 123, 73, 145, 79, 158, 83, 141, 89, 161, 97, 185, 101, 177, 103, 205, 107, 214, 109, 201, 113, 226, 127

Offset: 1

Gus Wiseman, Jan 05 2025

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 positions of 11 in A379681 are: 15, 22, 56, 72, 160, 384, 1024, so a(11) = 15.

Position of first appearance of n in A379681.
The subtraction A325036 takes the following values:
- zero: A301987, counted by A001055
- negative: A325037, counted by A114324
- positive: A325038, counted by A096276 shifted right
- negative one: A325041, counted by A028422
- one: A325042, counted by A001055 shifted right
- nonnegative: A325044, counted by A096276
- nonpositive: A379721, counted by A319005
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Cf. A000720, A003963, A075254, A178503, A175508, A319000, A325032, A325033, A325034, A325035, A325040.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sp=Table[Plus@@prix[n]+Times@@prix[n],{n,1000}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[sp,n][[1,1]],{n,mnrm[sp]}]

A379320 Number of integer partitions of n whose product is a multiple of n + 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 2, 3, 0, 14, 0, 7, 15, 53, 0, 81, 0, 110, 61, 32, 0, 562, 170, 62, 621, 560, 0, 1400, 0, 3387, 569, 199, 1515, 7734, 0, 339, 1486, 13374, 0, 11926, 0, 8033, 27164, 913, 0, 85326, 15947, 47588, 8294, 25430, 0, 174779, 39748, 169009

Offset: 0

Gus Wiseman, Jan 18 2025

nonn

Also the number of integer partitions of n containing 1 whose product is a multiple of n. Without requiring a 1 we get A057568.

			The a(5) = 1 through a(11) = 14 partitions:
  (3,2)  .  (4,2,1)    (3,3,2)    (5,4)      .  (8,3)
            (2,2,2,1)  (3,3,1,1)  (5,2,2)       (4,4,3)
                                  (5,2,1,1)     (6,3,2)
                                                (6,4,1)
                                                (4,3,2,2)
                                                (4,3,3,1)
                                                (6,2,2,1)
                                                (3,2,2,2,2)
                                                (3,3,2,2,1)
                                                (4,3,2,1,1)
                                                (6,2,1,1,1)
                                                (3,2,2,2,1,1)
                                                (4,3,1,1,1,1)
                                                (3,2,2,1,1,1,1)

For n instead of n+1 we have A057568 (strict A379733), ranks A326149.
These partitions are ranked by A380217 = A379319/2 = (even case of A326149)/2.
The case of equality is A380218, see also A028422 = A001055 - 1 (ranks A325041).
A000041 counts integer partitions, strict A000009.
A379666 counts partitions by sum and product.
A380219 counts partitions of n whose product is a proper multiple of n, ranks A380216.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568, ranks A326149
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A003963, A036844, A069016, A318950, A319000, A319916, A324851, A325042, A326150, A326152, A326156, A379671, A379678, A379734.

Table[Length[Select[IntegerPartitions[n],Divisible[Times@@#,n+1]&]],{n,0,30}]

a(n) = my(nb=0); forpart(p=n, if (!(vecprod(Vec(p)) % (n+1)), nb++)); nb; \\ Michel Marcus, Jan 21 2025

A380217 Numbers whose product of prime indices is a multiple of their sum of prime indices plus one.

Original entry on oeis.org

1, 15, 42, 54, 75, 77, 95, 99, 100, 132, 182, 195, 221, 234, 245, 253, 290, 312, 315, 329, 350, 357, 405, 420, 423, 437, 450, 459, 476, 494, 510, 540, 555, 559, 560, 612, 627, 665, 715, 720, 740, 798, 816, 833, 854, 855, 858, 893, 897, 899, 979, 1026, 1064

Offset: 1

Gus Wiseman, Jan 18 2025

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 and product of prime indices are A056239 and A003963 respectively.

			The prime indices of 75 are {2,3,3}, with product 18 and sum 8, and since 18 is a multiple of 8+1, 75 is in the sequence.
The terms together with their prime indices begin:
     1: {}
    15: {2,3}
    42: {1,2,4}
    54: {1,2,2,2}
    75: {2,3,3}
    77: {4,5}
    95: {3,8}
    99: {2,2,5}
   100: {1,1,3,3}
   132: {1,1,2,5}
   182: {1,4,6}
   195: {2,3,6}
   221: {6,7}
   234: {1,2,2,6}
   245: {3,4,4}

The case of equality is A325041, counted by A380218 = A028422 except n=3.
Without "plus one" we get A326149, counted by A057568, see A379733, A379734, A379735.
Double all terms to get A379319.
Partitions of this type are counted by A379320.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568, ranks A326149
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A069016, A301988, A318950, A319000, A324851, A379318, A379671, A379844, A379845.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Divisible[Times@@prix[#],1+Total[prix[#]]]&]

vpind(n)=my(v=List(), f=factor(n)); for(i=1, #f~, for(j=1, f[i, 2], listput(v, primepi(f[i, 1])))); Vec(v); \\ A112798
isok(k) = my(vind = vpind(k)); (vecprod(vind) % (vecsum(vind)+1)) == 0; \\ Michel Marcus, Jan 21 2025

a(n) = A379319(n)/2.

A380218 Number of integer partitions of n with product n+1.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 4, 0, 3, 0, 3, 1, 1, 0, 6, 1, 1, 2, 3, 0, 4, 0, 6, 1, 1, 1, 8, 0, 1, 1, 6, 0, 4, 0, 3, 3, 1, 0, 11, 1, 3, 1, 3, 0, 6, 1, 6, 1, 1, 0, 10, 0, 1, 3, 10, 1, 4, 0, 3, 1, 4, 0, 15, 0, 1, 3, 3, 1, 4, 0, 11, 4, 1, 0, 10, 1, 1, 1, 6, 0, 10, 1, 3, 1, 1, 1, 18, 0, 3, 3, 8, 0, 4, 0, 6, 4, 1

Offset: 0

Gus Wiseman, Jan 21 2025

nonn

For n instead of n+1 we have 0 followed by A001055.

Also the number of integer factorizations of n with sum < n. [When the sequence is interpreted as having offset 1 instead of 0. Clarified by Antti Karttunen, Jan 28 2025]

			The a(5) = 1 through a(15) = 4 partitions with product n+1:
  32  .  421   3311  5211  .  62111    .  721111  53111111  8211111
         2221                 431111                        441111111
                              3221111                       4221111111
                                                            22221111111
The a(1) = 1 through a(12) = 3 factorizations with sum < n:
  ()  .  .  .  .  (2*3)  .  (2*4)    (3*3)  (2*5)  .  (2*6)
                            (2*2*2)                   (3*4)
                                                      (2*2*3)

Antti Karttunen, Table of n, a(n) for n = 0..20000

Same as A028422 = A001055-1 except initial terms.
These partitions are ranked by A325041.
The version for divisibility instead of equality is A379320.
A000041 counts integer partitions, strict A000009.
A379666 counts partitions by sum and product.
A380219 counts partitions of n whose product is a proper multiple of n, ranks A380216.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568 (strict A379733), ranks A326149, see A379319, A380217.
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A003963, A069016, A318950, A319000, A319916, A325042, A326152, A326156, A379734.

Table[Length[Select[IntegerPartitions[n],SameQ[Times@@#,n+1]&]],{n,0,30}]

A380218off1(n, m=n, e=n) = if(1==n, (e>0), sumdiv(n, d, if((d>1)&&(d<=m),  A380218off1(n/d, d, e-d))));
A380218off0(n) = A380218off1(1+n); \\ Antti Karttunen, Jan 28 2025

More terms from Antti Karttunen, Jan 28 2025

A353506 Number of integer partitions of n whose parts have the same product as their multiplicities.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 1, 2, 1, 2, 0, 3, 3, 2, 3, 2, 0, 2, 3, 2, 1, 3, 1, 6, 3, 2, 3, 3, 2, 3, 4, 1, 2, 3, 6, 3, 2, 2, 3, 3, 1, 2, 6, 6, 4, 7, 2, 3, 6, 4, 3, 3, 0, 4, 5, 3, 5, 5, 6, 5, 3, 3, 3, 6, 5, 5, 6, 6, 3, 3, 3, 4, 4, 4, 6, 7, 2, 5, 7, 6, 2, 3, 4, 6, 11, 9, 4, 4, 1, 5, 6, 4, 7, 9, 6, 4

Offset: 0

Gus Wiseman, May 17 2022

nonn

			The a(0) = 1 through a(18) = 2 partitions:
  n= 0: ()
  n= 1: (1)
  n= 2:
  n= 3:
  n= 4: (211)
  n= 5:
  n= 6: (3111) (2211)
  n= 7:
  n= 8: (41111)
  n= 9:
  n=10: (511111)
  n=11: (32111111)
  n=12: (6111111) (22221111)
  n=13: (322111111)
  n=14: (71111111) (4211111111)
  n=15:
  n=16: (811111111) (4411111111) (42211111111)
  n=17: (521111111111) (332111111111) (322211111111)
  n=18: (9111111111) (333111111111)
For example, the partition y = (322111111) has multiplicities (1,2,6) with product 12, and the product of parts is also 3*2*2*1*1*1*1*1*1 = 12, so y is counted under a(13).

LHS (product of parts) is ranked by A003963, counted by A339095.
RHS (product of multiplicities) is ranked by A005361, counted by A266477.
For shadows instead of prime exponents we have A008619, ranked by A003586.
Taking sum instead of product of parts gives A266499.
For shadows instead of prime indices we have A353398, ranked by A353399.
These partitions are ranked by A353503.
Taking sum instead of product of multiplicities gives A353698.
A008284 counts partitions by length.
A098859 counts partitions with distinct multiplicities, ranked by A130091.
A353507 gives product of multiplicities (of exponents) in prime signature.
Cf. A085629, A114640, A116608, A118914, A124010, A319000, A325702, A353394, A353500.
Cf. A000792, A266480.

Table[Length[Select[IntegerPartitions[n], Times@@#==Times@@Length/@Split[#]&]],{n,0,30}]

a(n) = {my(nb=0); forpart(p=n, my(s=Set(p), v=Vec(p)); if (vecprod(vector(#s, i, #select(x->(x==s[i]), v))) == vecprod(v), nb++);); nb;} \\ Michel Marcus, May 20 2022

a(71)-a(100) from Alois P. Heinz, May 20 2022

A379840 Numbers that are the sum + product of a unique multiset of positive integers > 1.

Original entry on oeis.org

1, 4, 6, 10, 11, 12, 15, 16, 17, 18, 22, 27, 28, 30, 31, 43, 52, 58, 61, 67, 70, 73, 91, 97, 100, 102, 108, 115, 130, 145, 147, 148, 162, 165, 171, 217, 262, 277, 283, 291, 361, 430, 481, 508, 577, 633, 652, 682, 763, 1093, 1137, 1201, 1513, 1705, 2257, 2401, 2653, 3133, 4123, 5113, 5905

Offset: 1

Gus Wiseman, Jan 08 2025

nonn

			The only multiset with no 1's and sum + product = 165 is {2,3,5,5}, so 165 is in the sequence.

Positions of 1 in A379669 (zeros A379670).
For sets instead of multisets we have A379842, see A379841.
Arrays counting multisets by sum and product:
- partitions: A379666, antidiagonal sums A379667
- partitions without ones: A379668, antidiagonal sums A379669
- strict partitions: A379671, antidiagonal sums A379672
- strict partitions without ones: A379678, antidiagonal sums A379679
Counting and ranking multisets by comparing sum and product:
- same: A001055 (strict A045778), ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A326156, A326172, A379733
- greater: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721
- different: A379736, ranks A379722, see A111133
A002865 counts partitions into parts > 1, strict A025147.
A318950 counts factorizations by sum.
Cf. A003963, A069016, A319000, A319057, A319916, A325036, A326152, A379680.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Join@@Position[Table[Length[Select[Join@@Array[facs,n], Total[#]+Times@@#==n&]],{n,100}],1]

More terms from Jinyuan Wang, Jan 12 2025

cp's OEIS Frontend

A379681 Sum plus product of the multiset of prime indices of n.

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A379734 Number of integer partitions of n into parts > 1 whose product is a multiple of n.

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A379319 Even numbers whose product of prime indices is a multiple of their sum of prime indices.

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A379735 Number of strict integer partitions of n into parts > 1 whose product is a multiple of n.

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A379682 Least number whose prime indices have sum + product = n.

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A379320 Number of integer partitions of n whose product is a multiple of n + 1.

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A380217 Numbers whose product of prime indices is a multiple of their sum of prime indices plus one.

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A380218 Number of integer partitions of n with product n+1.

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