A379320 -id:A379320 - cp's OEIS frontend

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

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[#]]]&]

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

A380219 Number of integer partitions of n whose product is a proper multiple of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 3, 3, 3, 0, 18, 0, 9, 21, 75, 0, 109, 0, 146, 83, 43, 0, 730, 224, 82, 806, 722, 0, 1782, 0, 4254, 733, 258, 1923, 9558, 0, 435, 1875, 16395, 0, 14625, 0, 9857, 33053, 1150, 0, 102070, 19391, 57326, 10157, 30702, 0, 207699, 47925, 200645

Offset: 1

Gus Wiseman, Jan 21 2025

nonn

			The partition y = (4,3,3,2) has product 72, which is a multiple of 12, so y is counted under a(12).
The a(8) = 3 through a(14) = 9 partitions:
  (44)    (63)    (532)   .  (66)       .  (743)
  (422)   (333)   (541)      (543)         (752)
  (2222)  (3321)  (5221)     (642)         (761)
                             (831)         (7322)
                             (4332)        (7421)
                             (4431)        (72221)
                             (5322)        (73211)
                             (6222)        (74111)
                             (6321)        (722111)
                             (6411)
                             (33222)
                             (43221)
                             (43311)
                             (62211)
                             (322221)
                             (332211)
                             (432111)
                             (3222111)

The non-proper version is A057568, case of equality A001055.
The case of strict partitions is A379733 - 1.
The case of partitions without 1's is A379734 - 1.
These partitions are ranked by A380216.
A000041 counts integer partitions, strict A000009.
A379666 counts partitions by sum and product.
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, A028422, A318950, A319000, A319916, A326152, A326156, A379320, A379671, A379735, A380218, A380221.

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

a(n) = my(nb=0); forpart(p=n, my(vp=vecprod(Vec(p))); if (!(vp%n) && (vp>n), nb++)); nb; \\ Michel Marcus, Jan 22 2025

a(n) = A057568(n) - A001055(n).

A380221 Number of strict integer partitions of n containing 1 whose product of parts is a multiple of n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 3, 3, 0, 4, 0, 9, 6, 4, 0, 22, 5, 6, 15, 28, 0, 54, 0, 49, 30, 14, 57, 134, 0, 22, 58, 219, 0, 242, 0, 180, 349, 44, 0, 722, 113, 369, 196, 404, 0, 994, 556, 1363, 338, 111, 0, 3016, 0, 150, 2569, 3150, 1485, 2815, 0

Offset: 1

Gus Wiseman, Jan 22 2025

nonn

Also the number of strict integer partitions of n - 1 not containing 1 whose product of parts is a multiple of n. These are strict integer factorizations of multiples of n summing to n - 1.

			The a(6) = 1 through a(16) = 3 partitions:
  (3,2,1) . . . (5,4,1) . (8,3,1)   . (7,6,1)   (9,5,1)     (8,4,3,1)
                          (6,3,2,1)   (7,4,2,1) (6,5,3,1)   (8,5,2,1)
                                                (5,4,3,2,1) (6,4,3,2,1)

Positions of 0 after 9 appear to be the prime numbers A000040.
The non-strict version is A379320 shifted right, ranks A380217 = A379319/2.
Not requiring 1 gives A379733.
For n instead of n+1 we have A379735 shifted left, non-strict A379734.
Partitions of this type are ranked by A379845.
The case of equality for non-strict partitions is A380218 shifted left.
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, A069016, A319000, A319916, A325042, A326150, A326152, A326156, A379671, A379678.

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

A379318 Odd numbers whose product of prime indices is a multiple of their sum of prime indices.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 165, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Gus Wiseman, Jan 16 2025

nonn

Contains all odd primes.

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 terms together with their prime indices begin:
     2: {1}         53: {16}           109: {29}
     3: {2}         59: {17}           113: {30}
     5: {3}         61: {18}           125: {3,3,3}
     7: {4}         63: {2,2,4}        127: {31}
     9: {2,2}       65: {3,6}          131: {32}
    11: {5}         67: {19}           137: {33}
    13: {6}         71: {20}           139: {34}
    17: {7}         73: {21}           149: {35}
    19: {8}         79: {22}           150: {1,2,3,3}
    23: {9}         81: {2,2,2,2}      151: {36}
    29: {10}        83: {23}           154: {1,4,5}
    30: {1,2,3}     84: {1,1,2,4}      157: {37}
    31: {11}        89: {24}           163: {38}
    37: {12}        97: {25}           165: {2,3,5}
    41: {13}       101: {26}           167: {39}
    43: {14}       103: {27}           169: {6,6}
    47: {15}       107: {28}           173: {40}
    49: {4,4}      108: {1,1,2,2,2}    179: {41}

Including evens gives A326149, counted by A057568.
For nonprime instead of odd we get A326150.
For even instead of odd we get A379319, counted by A379320.
Partitions of this type are counted by A379734, strict A379735, see A379733.
For squarefree instead of odd we get A379844, even case A379845.
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, A319916, A324851, A325041, A326152, A379671, A379678.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],OddQ[#]&&Divisible[Times@@prix[#],Total[prix[#]]]&]

A380411 Number of integer partitions of n such that the product of parts is greater than the sum of primes indexed by the parts.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 4, 8, 14, 23, 39, 58, 85, 121, 168, 228, 308, 404, 533, 691, 892, 1136, 1449, 1820, 2291, 2857, 3553, 4387, 5418, 6646, 8144, 9931, 12086, 14649, 17733, 21379, 25747, 30905, 37049, 44282, 52863, 62936, 74841, 88792, 105202, 124387

Offset: 0

Gus Wiseman, Jan 26 2025

nonn

			The partition y = (4,3,2) has product of parts 4*3*2 = 24 and sum of corresponding primes 7+5+3 = 15, so y is counted under a(9).
The a(0) = 1 through a(10) = 14 partitions:
  ()  .  .  .  .  .  .  (322)  (44)    (54)     (55)
                               (332)   (333)    (64)
                               (422)   (432)    (433)
                               (2222)  (522)    (442)
                                       (3222)   (532)
                                       (3321)   (622)
                                       (4221)   (3322)
                                       (22221)  (3331)
                                                (4222)
                                                (4321)
                                                (5221)
                                                (22222)
                                                (32221)
                                                (33211)

For parts instead of primes on the RHS we have A114324.
The version for divisibility instead of inequality is A330954.
The version for equality is A331383, ranks A331384.
These partitions are ranked by A380410.
A000040 lists the primes, differences A001223.
A000041 counts integer partitions, strict A000009.
A001414 gives sum of prime factors.
A003963 gives product of prime indices
A379666 counts partitions by sum and product.
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. A175508, A318950, A319000, A319916, A326152, A379320, A379734, A380218, A380219, A380344.

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

A380343 Number of strict integer partitions of n whose product of parts is a multiple of n + 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 3, 0, 3, 5, 5, 0, 8, 0, 15, 11, 8, 0, 42, 8, 12, 26, 49, 0, 100, 0, 90, 56, 27, 105, 246, 0, 41, 108, 414, 0, 450, 0, 332, 651, 81, 0, 1341, 210, 693, 366, 754, 0, 1869, 1044, 2579, 634, 206, 0, 5695, 0, 278, 4850, 5927, 2802

Offset: 0

Gus Wiseman, Jan 22 2025

nonn

			The a(5) = 1 through a(17) = 8 partitions (A=10, C=12):
  32  .  421  .  54  .  83   .  76    95    843   .  98
                        632     742   653   852      863
                        641     7321  A31   861      962
                                      5432  6432     C32
                                      6521  8421     7631
                                                     9431
                                                     9521
                                                     65321

The non-strict version is A379320, ranked by A380217 = A379319/2.
For n instead of n+1 we have A379733, non-strict A057568.
The case of equality for non-strict partitions is A380218.
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, A069016, A319000, A319916, A325042, A326152, A326156, A379671, A379734.

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

cp's OEIS Frontend

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

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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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A380219 Number of integer partitions of n whose product is a proper multiple of n.

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A380221 Number of strict integer partitions of n containing 1 whose product of parts is a multiple of n.

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A379318 Odd numbers whose product of prime indices is a multiple of their sum of prime indices.

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A380411 Number of integer partitions of n such that the product of parts is greater than the sum of primes indexed by the parts.

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A380343 Number of strict integer partitions of n whose product of parts is a multiple of n + 1.

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