A326153 -id:A326153 - 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

A326151 Numbers whose product of prime indices is twice their sum of prime indices.

Original entry on oeis.org

49, 63, 65, 81, 150, 154, 190, 198, 364, 468, 580, 840, 952, 1080, 1224, 1480, 2128, 2288, 2736, 3440, 5152, 5280, 6624, 8480, 9408, 10816, 12096, 12992, 15552, 16704, 19520, 24960, 26752, 27776, 35712, 44800, 45440, 56576, 57600, 66304, 85248, 101120, 118272

Offset: 1

Gus Wiseman, Jun 09 2019

nonn

The only squarefree terms are 65, 154, and 190. See A326157 for a proof.
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 twice their sum of parts. The enumeration of these partitions by sum is given by A326152.

			The sequence of terms together with their prime indices begins:
     49: {4,4}
     63: {2,2,4}
     65: {3,6}
     81: {2,2,2,2}
    150: {1,2,3,3}
    154: {1,4,5}
    190: {1,3,8}
    198: {1,2,2,5}
    364: {1,1,4,6}
    468: {1,1,2,2,6}
    580: {1,1,3,10}
    840: {1,1,1,2,3,4}
    952: {1,1,1,4,7}
   1080: {1,1,1,2,2,2,3}
   1224: {1,1,1,2,2,7}
   1480: {1,1,1,3,12}
   2128: {1,1,1,1,4,8}
   2288: {1,1,1,1,5,6}
   2736: {1,1,1,1,2,2,8}
   3440: {1,1,1,1,3,14}

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 80 terms from Jinyuan Wang)

Satisfies A003963(a(n)) = 2 * A056239(a(n)).

Cf. A112798, A301987, A325037, A325038, A325044, A326149, A326152, A326153/A326154, A326156, A326157.

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

is(k) = {my(f=factor(k)); for(i=1, #f~, f[i, 1]=primepi(f[i, 1])); factorback(f)==2*sum(i=1, #f~, f[i, 2]*f[i, 1]); } \\ Jinyuan Wang, Jun 27 2020

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}]

A326150 Nonprime numbers whose product of prime indices is divisible by their sum of prime indices.

Original entry on oeis.org

9, 30, 49, 63, 65, 81, 84, 108, 125, 150, 154, 165, 169, 190, 198, 200, 259, 264, 273, 333, 351, 361, 364, 385, 390, 435, 442, 468, 481, 490, 495, 506, 525, 561, 580, 595, 609, 624, 630, 658, 675, 700, 714, 741, 765, 781, 783, 810, 840, 841, 846, 874, 900, 918

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 sequence of terms together with their prime indices begins:
    9: {2,2}
   30: {1,2,3}
   49: {4,4}
   63: {2,2,4}
   65: {3,6}
   81: {2,2,2,2}
   84: {1,1,2,4}
  108: {1,1,2,2,2}
  125: {3,3,3}
  150: {1,2,3,3}
  154: {1,4,5}
  165: {2,3,5}
  169: {6,6}
  190: {1,3,8}
  198: {1,2,2,5}
  200: {1,1,1,3,3}
  259: {4,12}
  264: {1,1,1,2,5}
  273: {2,4,6}
  333: {2,2,12}

Satisfies A056239(a(n))|A003963(a(n)).
The case with primes included is A326149. The squarefree case A326158.
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[#]&&Divisible[Times@@primeMS[#],Plus@@primeMS[#]]&]

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

A326179 Number of subsets of {1..n} containing n whose product is divisible by their sum.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 9, 15, 30, 65, 138, 274, 563, 1149, 2441, 5110, 9090, 19645, 37524, 79178, 156697, 324182, 663749, 1353984, 2529510, 5088926, 10686159, 19987129, 40800192, 85303150, 162549135, 341440697, 630392773, 1317158898, 2687152135, 5276362642, 10078384386, 21415439670, 43367751196, 86613992774, 166456115593

Offset: 0

Gus Wiseman, Jun 13 2019

nonn

			The a(1) = 1 through a(7) = 15 subsets:
  {1}  {2}  {3}      {4}  {5}          {6}          {7}
            {1,2,3}       {1,4,5}      {3,6}        {1,6,7}
                          {2,3,5}      {2,4,6}      {2,5,7}
                          {3,4,5}      {4,5,6}      {3,4,7}
                          {1,2,3,4,5}  {1,2,3,6}    {3,5,7}
                                       {1,3,5,6}    {1,2,4,7}
                                       {3,4,5,6}    {2,3,6,7}
                                       {1,2,3,4,6}  {2,5,6,7}
                                       {2,3,4,5,6}  {3,5,6,7}
                                                    {1,2,5,6,7}
                                                    {1,3,4,5,7}
                                                    {1,3,4,6,7}
                                                    {2,3,4,5,7}
                                                    {2,4,5,6,7}
                                                    {1,2,3,4,5,6,7}

Cf. A053632, A057567, A057568, A059529, A063865, A301987, A326153/A326154, A326156, A326158, A326178, A326180.

Table[Length[Select[Subsets[Range[n],{1,n}],MemberQ[#,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 23 2019

A326180 Number of maximal subsets of {1..n} containing n whose product is divisible by their sum.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 3, 1, 1, 1, 11, 1, 16, 1, 1, 1, 27, 1

Offset: 0

Gus Wiseman, Jun 13 2019

			The a(6) = 3, a(10) = 11, and a(12) = 16 subsets:
  {1,3,5,6}    {1,2,4,5,6,7,10}      {1,2,3,4,5,6,7,8,12}
  {1,2,3,4,6}  {1,2,3,4,5,7,8,10}    {1,3,4,5,6,7,8,10,12}
  {2,3,4,5,6}  {1,2,3,4,6,7,9,10}    {1,3,4,6,7,8,9,10,12}
               {1,2,3,5,6,7,8,10}    {1,3,4,5,6,8,10,11,12}
               {1,2,3,5,7,8,9,10}    {1,2,3,4,5,6,8,9,10,12}
               {1,2,5,6,7,8,9,10}    {1,2,3,4,6,7,8,9,11,12}
               {1,3,4,5,6,7,9,10}    {1,2,3,5,6,7,8,9,10,12}
               {1,3,4,6,7,8,9,10}    {1,2,3,5,6,7,8,9,11,12}
               {1,4,5,6,7,8,9,10}    {1,3,4,5,6,7,8,9,11,12}
               {1,2,3,4,5,6,8,9,10}  {1,2,3,4,6,7,8,10,11,12}
               {2,3,4,5,6,7,8,9,10}  {1,2,3,4,6,8,9,10,11,12}
                                     {1,3,5,6,7,8,9,10,11,12}
                                     {1,2,3,4,5,6,7,9,10,11,12}
                                     {1,2,3,4,5,7,8,9,10,11,12}
                                     {1,2,4,5,6,7,8,9,10,11,12}
                                     {2,3,4,5,6,7,8,9,10,11,12}

Cf. A053632, A057567, A057568, A059529, A060462, A063865, A301987, A326153/A326154, A326156, A326158, A326178, A326179.

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n],{1,n}],MemberQ[#,n]&&Divisible[Times@@#,Plus@@#]&]]],{n,0,10}]

a(A060462(n)) = 1.

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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A326151 Numbers whose product of prime indices is twice their sum of prime indices.

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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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A326150 Nonprime numbers whose product of prime indices is divisible by their sum of prime indices.

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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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A326179 Number of subsets of {1..n} containing n whose product is divisible by their sum.