A326150 -id:A326150 - cp's OEIS frontend

A326149 Numbers whose product of prime indices is divisible by their sum of prime indices.

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 49, 53, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 108, 109, 113, 125, 127, 131, 137, 139, 149, 150, 151, 154, 157, 163, 165, 167, 169, 173, 179, 181, 190, 191, 193, 197

Offset: 1

Gus Wiseman, Jun 09 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 Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose product of parts is divisible by their sum of parts. The enumeration of these partitions by sum is given by A057568.

			The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   23: {9}
   29: {10}
   30: {1,2,3}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   47: {15}
   49: {4,4}
   53: {16}
   59: {17}

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

Satisfies A056239(a(n))|A003963(a(n)).
The nonprime case is A326150, with squarefree case A326158.
Cf. A000720, A001222, A057567, A057568, A112798, A301987.
Cf. A325037, A325042, A325044, A326151, A326153/A326154, A326155, A326156.

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

A326155 Positive integers whose sum of prime indices is divisible by their product of prime indices.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 29, 30, 31, 32, 37, 40, 41, 43, 47, 48, 53, 59, 61, 64, 67, 71, 73, 79, 83, 84, 89, 97, 101, 103, 107, 108, 109, 112, 113, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 173, 179, 181, 191, 192, 193

Offset: 1

Gus Wiseman, Jun 10 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.

Also Heinz numbers of the integer partitions counted by A057567. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  23: {9}
  29: {10}
  30: {1,2,3}
  31: {11}
  32: {1,1,1,1,1}
  37: {12}

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

One and positions of ones in A326153.
Cf. A003963, A056239, A057567, A057568, A112798, A301987.
Cf. A325037, A325042, A325044, A326150, A326151, A326154, A326156, A326158.

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

A326156 Number of nonempty subsets of {1..n} whose product is divisible by their sum.

Original entry on oeis.org

0, 1, 2, 4, 5, 10, 19, 34, 64, 129, 267, 541, 1104, 2253, 4694, 9804, 18894, 38539, 76063, 155241, 311938, 636120, 1299869, 2653853, 5183363, 10272289, 20958448, 40945577, 81745769, 167048919, 329598054, 671038751, 1301431524, 2618590422, 5305742557, 10582105199, 20660489585, 42075929255, 85443680451, 172057673225, 338513788818

Offset: 0

Gus Wiseman, Jun 11 2019

nonn

			The a(1) = 1 through a(6) = 19 subsets:
  {1}  {1}  {1}      {1}      {1}          {1}
       {2}  {2}      {2}      {2}          {2}
            {3}      {3}      {3}          {3}
            {1,2,3}  {4}      {4}          {4}
                     {1,2,3}  {5}          {5}
                              {1,2,3}      {6}
                              {1,4,5}      {3,6}
                              {2,3,5}      {1,2,3}
                              {3,4,5}      {1,4,5}
                              {1,2,3,4,5}  {2,3,5}
                                           {2,4,6}
                                           {3,4,5}
                                           {4,5,6}
                                           {1,2,3,6}
                                           {1,3,5,6}
                                           {3,4,5,6}
                                           {1,2,3,4,5}
                                           {1,2,3,4,6}
                                           {2,3,4,5,6}

Bert Dobbelaere, C++ program

Cf. A053632, A057567, A057568, A059529, A063865, A301987, A326150, A326151, A326153/A326154, A326155, A326158, A326178, A326179, A326180.

Table[Length[Select[Subsets[Range[n],{1,n}],Divisible[Times@@#,Plus@@#]&]],{n,0,10}]

a(21)-a(30) from Alois P. Heinz, Jun 13 2019

a(31)-a(40) from Bert Dobbelaere, Jun 22 2019

A326153 Numerator of the product of prime indices of n divided by the sum of prime indices of n, n > 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 4, 6, 1, 1, 4, 1, 3, 4, 5, 1, 2, 3, 6, 4, 2, 1, 1, 1, 1, 10, 7, 12, 2, 1, 8, 3, 1, 1, 8, 1, 5, 12, 9, 1, 1, 2, 9, 14, 3, 1, 8, 15, 4, 8, 10, 1, 6, 1, 11, 2, 1, 2, 5, 1, 7, 18, 3, 1, 4, 1, 12, 9, 4, 20, 4, 1, 3, 2, 13, 1, 1

Offset: 2

Gus Wiseman, Jun 09 2019

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 A326153/A326154 begins: 1, 1, 1/2, 1, 2/3, 1, 1/3, 1, 3/4, 1, 1/2, 1, 4/5, 6/5, 1/4, 1, 4/5, 1, 3/5, 4/3, 5/6, 1, 2/5, 3/2, 6/7, 4/3.

Numerator of A003963(n)/A056239(n).
Positions of ones are A326155 without the first term.
Cf. A057567, A057568, A112798, A301987, A325037, A325042, A325044, A326149, A326150, A326151, A326154, A326156, A326158.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Numerator[Times@@primeMS[n]/Plus@@primeMS[n]],{n,2,100}]

A326154 Denominator of the product of prime indices of n divided by the sum of prime indices of n, n > 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 3, 1, 4, 1, 2, 1, 5, 5, 4, 1, 5, 1, 5, 3, 6, 1, 5, 2, 7, 3, 3, 1, 1, 1, 5, 7, 8, 7, 3, 1, 9, 2, 2, 1, 7, 1, 7, 7, 10, 1, 3, 1, 7, 9, 4, 1, 7, 8, 7, 5, 11, 1, 7, 1, 12, 1, 6, 1, 4, 1, 9, 11, 2, 1, 7, 1, 13, 4, 5, 9, 3, 1, 7, 1, 14, 1, 1, 10

Offset: 2

Gus Wiseman, Jun 09 2019

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 A326153/A326154 begins: 1, 1, 1/2, 1, 2/3, 1, 1/3, 1, 3/4, 1, 1/2, 1, 4/5, 6/5, 1/4, 1, 4/5, 1, 3/5, 4/3, 5/6, 1, 2/5, 3/2, 6/7, 4/3.

Denominator of A003963(n)/A056239(n).
Positions of ones are A326149.
Cf. A057567, A057568, A112798, A301987, A325037, A325042, A325044, A326150, A326151, A326153, A326155, A326156, A326158.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Denominator[Times@@primeMS[n]/Plus@@primeMS[n]],{n,2,100}]

A326158 Nonprime squarefree numbers whose product of prime indices is divisible by their sum of prime indices.

Original entry on oeis.org

30, 65, 154, 165, 190, 259, 273, 385, 390, 435, 442, 481, 506, 561, 595, 609, 658, 714, 741, 781, 874, 935, 1001, 1110, 1118, 1173, 1254, 1281, 1330, 1363, 1403, 1430, 1455, 1469, 1495, 1505, 1653, 1691, 1771, 1786, 1794, 1798, 1887, 1958, 2035, 2067, 2139

Offset: 1

Gus Wiseman, Jun 10 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 sequence of terms together with their prime indices begins:
    30: {1,2,3}
    65: {3,6}
   154: {1,4,5}
   165: {2,3,5}
   190: {1,3,8}
   259: {4,12}
   273: {2,4,6}
   385: {3,4,5}
   390: {1,2,3,6}
   435: {2,3,10}
   442: {1,6,7}
   481: {6,12}
   506: {1,5,9}
   561: {2,5,7}
   595: {3,4,7}
   609: {2,4,10}
   658: {1,4,15}
   714: {1,2,4,7}
   741: {2,6,8}
   781: {5,20}

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

Satisfies A056239(a(n))|A003963(a(n)).
The case with squarefree numbers included is A326150.
The case with primes and squarefree numbers included is A326149.
Cf. A057567, A057568, A112798, A301987.
Cf. A325037, A325042, A325044, A326151, A326153/A326154, A326155, A326156.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,1000],!PrimeQ[#]&&SquareFreeQ[#]&&Divisible[Times@@primeMS[#],Plus@@primeMS[#]]&]

A326172 Number of nonempty subsets of {2..n} whose product is divisible by their sum.

Original entry on oeis.org

0, 0, 1, 2, 3, 6, 12, 21, 34, 69, 140, 278, 561, 1144, 2367, 4936, 9503, 19368, 38202, 77911, 156458, 318911, 651462, 1329624, 2596458, 5144833, 10494839, 20500025, 40923643, 83620258, 164982516, 335873558, 651383048, 1310551707, 2655240565, 5295397093, 10338145110, 21052407259, 42748787713, 86078893923, 169349494068

Offset: 0

Gus Wiseman, Jun 11 2019

nonn

			The a(2) = 1 through a(7) = 21 subsets:
  {2}  {2}  {2}  {2}      {2}          {2}
       {3}  {3}  {3}      {3}          {3}
            {4}  {4}      {4}          {4}
                 {5}      {5}          {5}
                 {2,3,5}  {6}          {6}
                 {3,4,5}  {3,6}        {7}
                          {2,3,5}      {3,6}
                          {2,4,6}      {2,3,5}
                          {3,4,5}      {2,4,6}
                          {4,5,6}      {2,5,7}
                          {3,4,5,6}    {3,4,5}
                          {2,3,4,5,6}  {3,4,7}
                                       {3,5,7}
                                       {4,5,6}
                                       {2,3,6,7}
                                       {2,5,6,7}
                                       {3,4,5,6}
                                       {3,5,6,7}
                                       {2,3,4,5,6}
                                       {2,3,4,5,7}
                                       {2,4,5,6,7}

Cf. A053632, A057567, A057568, A059529, A063865, A301987, A325037, A325044, A326150, A326151, A326153/A326154, A326155, A326156, A326158, A326178.

Table[Length[Select[Subsets[Range[2,n],{1,n}],Divisible[Times@@#,Plus@@#]&]],{n,0,10}]

a(21)-a(29) from Alois P. Heinz, Jun 13 2019

a(30)-a(40) from Bert Dobbelaere, Jun 22 2019

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[#]]]&]

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

A379844 Squarefree numbers x such that the product of prime indices of x is a multiple of the sum of prime indices of x.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 154, 157, 163, 165, 167, 173, 179, 181, 190, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241

Offset: 1

Gus Wiseman, Jan 19 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.

Squarefree case of A326149.
For nonprime instead of squarefree we have A326150.
The non-prime case is A326158.
Partitions of this type are counted by A379733, see A379735.
The even case is A379845, counted by A380221.
A003963 multiplies together prime indices.
A005117 lists the squarefree numbers.
A056239 adds up prime indices.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: 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. A000720, A001222, A036844, A112798, A324850, A324851, A324925, A326151, A326153/A326154, A326156, A326157.

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

Satisfies A056239(a(n))|A003963(a(n)).

cp's OEIS Frontend

A326149 Numbers whose product of prime indices is divisible by their sum of prime indices.

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A326155 Positive integers whose sum of prime indices is divisible by their product of prime indices.

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A326156 Number of nonempty subsets of {1..n} whose product is divisible by their sum.

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A326153 Numerator of the product of prime indices of n divided by the sum of prime indices of n, n > 1.

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A326154 Denominator of the product of prime indices of n divided by the sum of prime indices of n, n > 1.

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A326158 Nonprime squarefree numbers whose product of prime indices is divisible by their sum of prime indices.

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A326172 Number of nonempty subsets of {2..n} whose product is divisible by their sum.

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

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