A325708 -id:A325708 - cp's OEIS frontend

A325755 Numbers n divisible by their prime shadow A181819(n).

1, 2, 9, 12, 18, 36, 40, 60, 84, 112, 120, 125, 132, 156, 180, 204, 225, 228, 250, 252, 276, 280, 336, 348, 352, 360, 372, 396, 440, 441, 444, 450, 468, 492, 516, 520, 540, 560, 564, 600, 612, 636, 675, 680, 684, 708, 732, 760, 804, 828, 832, 840, 852, 876

Offset: 1

Gus Wiseman, May 19 2019

nonn

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

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 containing their multiset of multiplicities as a submultiset (counted by A325702).

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     9: {2,2}
    12: {1,1,2}
    18: {1,2,2}
    36: {1,1,2,2}
    40: {1,1,1,3}
    60: {1,1,2,3}
    84: {1,1,2,4}
   112: {1,1,1,1,4}
   120: {1,1,1,2,3}
   125: {3,3,3}
   132: {1,1,2,5}
   156: {1,1,2,6}
   180: {1,1,2,2,3}
   204: {1,1,2,7}
   225: {2,2,3,3}
   228: {1,1,2,8}
   250: {1,3,3,3}
   252: {1,1,2,2,4}

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

Cf. A181819, A182857, A290689, A290822, A323014, A324843, A325702, A325706, A325708, A325756.

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Select[Range[100],Divisible[#,red[#]]&]

A325756 A number k belongs to the sequence if k = 1 or k is divisible by its prime shadow A181819(k) and the quotient k/A181819(k) also belongs to the sequence.

Original entry on oeis.org

1, 2, 12, 336, 360, 45696, 52416, 75600, 22665216, 31804416, 42928704, 77792400, 92610000, 164656800, 174636000

Offset: 1

Gus Wiseman, May 19 2019

We define the prime shadow A181819(k) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
     12: {1,1,2}
    336: {1,1,1,1,2,4}
    360: {1,1,1,2,2,3}
  45696: {1,1,1,1,1,1,1,2,4,7}
  52416: {1,1,1,1,1,1,2,2,4,6}
  75600: {1,1,1,1,2,2,2,3,3,4}

Cf. A181819, A182857, A290689, A290822, A323014, A324843, A325702, A325706, A325708, A325755.

red[n_] := If[n == 1, 1, Times @@ Prime /@ Last /@ FactorInteger[n]];
suQ[n_]:=n==1||Divisible[n,red[n]]&&suQ[n/red[n]];
Select[Range[10000],suQ]

ps(n) = my(f=factor(n)); prod(k=1, #f~, prime(f[k, 2])); \\ A181819
isok(k) = {if ((k==1), return(1)); my(p=ps(k)); ((k % p) == 0) && isok(k/p);} \\ Michel Marcus, Jan 09 2021

a(9)-a(15) from Amiram Eldar, Jan 09 2021

A325707 Number of integer partitions of n covering an initial interval of positive integers and containing all of their distinct multiplicities.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 2, 1, 2, 2, 4, 4, 5, 6, 7, 8, 10, 11, 13, 16, 18, 23, 26, 32, 36, 43, 48, 57, 64, 74, 84, 98, 110, 127, 145, 165, 189, 215, 244, 277, 316, 356, 403, 455, 513, 577, 650, 727, 817, 913, 1024, 1143, 1279, 1425, 1592, 1773, 1977, 2198, 2448, 2717

Offset: 0

Gus Wiseman, May 18 2019

nonn

The Heinz numbers of these partitions are given by A325708.

			The initial terms count the following partitions:
   1: (1)
   3: (21)
   4: (211)
   5: (221)
   6: (321)
   6: (2211)
   7: (3211)
   8: (3221)
   8: (32111)
   9: (3321)
   9: (32211)
  10: (4321)
  10: (33211)
  10: (32221)
  10: (322111)
  11: (43211)
  11: (33221)
  11: (332111)
  11: (322211)
  12: (43221)
  12: (432111)
  12: (33321)
  12: (332211)
  12: (3222111)

Cf. A000009 (partitions covering an initial interval), A055932, A109297, A114639, A114640, A290689, A324753, A325702, A325706, A325708.

Table[Length[Select[IntegerPartitions[n],Range[Length[Union[#]]]==Union[#]&&SubsetQ[Sort[#],Sort[Length/@Split[#]]]&]],{n,0,30}]

A325766 Number of integer partitions of n covering an initial interval of positive integers and containing their own multiset of multiplicities (as a submultiset).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 4, 5, 4, 6, 7, 8, 6, 12, 11, 19, 16, 22, 22, 25, 32, 38, 45, 45, 51, 53, 71, 69, 85, 92, 118, 125, 147, 149, 184, 187, 230, 254, 290, 317, 372, 397, 449, 502, 544, 616, 680, 758, 841, 930, 1042, 1151, 1262

Offset: 0

Gus Wiseman, May 19 2019

nonn

The Heinz numbers of these partitions are given by A325767.

			The initial terms count the following partitions:
   1: (1)
   4: (2,1,1)
   5: (2,2,1)
   6: (2,2,1,1)
   7: (3,2,1,1)
   8: (3,2,1,1,1)
   9: (3,2,2,1,1)
  10: (3,2,2,1,1,1)
  11: (3,3,2,2,1)
  11: (3,3,2,1,1,1)
  11: (3,2,2,2,1,1)
  12: (4,3,2,1,1,1)
  13: (4,3,2,2,1,1)
  13: (4,3,2,1,1,1,1)
  13: (3,3,3,2,1,1)
  13: (3,3,2,2,2,1)
  13: (3,3,2,2,1,1,1)
  14: (4,3,2,2,1,1,1)
  14: (3,3,3,2,2,1)
  14: (3,3,3,2,1,1,1)
  14: (3,3,2,2,2,1,1)

Cf. A000009 (partitions covering an initial interval), A055932, A114639, A114640, A290689, A324753, A325702, A325706, A325707, A325708, A325767.

submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap]
Table[Length[Select[IntegerPartitions[n],Range[Length[Union[#]]]==Union[#]&&submultQ[Sort[Length/@Split[#]],Sort[#]]&]],{n,0,30}]

A325767 Heinz numbers of integer partitions covering an initial interval of positive integers and containing their own multiset of multiplicities (as a submultiset).

Original entry on oeis.org

1, 2, 12, 18, 36, 60, 120, 180, 360, 450, 540, 600, 840, 1260, 1350, 1500, 1680, 1800, 2250, 2520, 2700, 3000, 3780, 4200, 4500, 5040, 5400, 5880, 6750, 8400, 9000, 10500, 11340, 11760, 12600, 13500, 15120, 17640, 18480, 18900, 20580, 21000, 22680, 25200

Offset: 1

Gus Wiseman, May 19 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A325766.

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
    12: {1,1,2}
    18: {1,2,2}
    36: {1,1,2,2}
    60: {1,1,2,3}
   120: {1,1,1,2,3}
   180: {1,1,2,2,3}
   360: {1,1,1,2,2,3}
   450: {1,2,2,3,3}
   540: {1,1,2,2,2,3}
   600: {1,1,1,2,3,3}
   840: {1,1,1,2,3,4}
  1260: {1,1,2,2,3,4}
  1350: {1,2,2,2,3,3}
  1500: {1,1,2,3,3,3}
  1680: {1,1,1,1,2,3,4}
  1800: {1,1,1,2,2,3,3}
  2250: {1,2,2,3,3,3}
  2520: {1,1,1,2,2,3,4}

Cf. A000009, A055932, A056239, A112798, A109297, A114640, A290689, A325702, A325708, A325766.

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Select[Range[1000],#==1||Range[PrimeNu[#]]==PrimePi/@First/@FactorInteger[#]&&Divisible[#,red[#]]&]

cp's OEIS Frontend

A325755 Numbers n divisible by their prime shadow A181819(n).

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A325756 A number k belongs to the sequence if k = 1 or k is divisible by its prime shadow A181819(k) and the quotient k/A181819(k) also belongs to the sequence.

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A325707 Number of integer partitions of n covering an initial interval of positive integers and containing all of their distinct multiplicities.

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A325766 Number of integer partitions of n covering an initial interval of positive integers and containing their own multiset of multiplicities (as a submultiset).

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A325767 Heinz numbers of integer partitions covering an initial interval of positive integers and containing their own multiset of multiplicities (as a submultiset).

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