A325395 -id:A325395 - cp's OEIS frontend

A325456 Heinz numbers of integer partitions with strictly increasing differences.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87

Offset: 1

Gus Wiseman, May 03 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The enumeration of these partitions by sum is given by A240027.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   6: {1,2}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A056239, A112798, A179269, A240026, A240027, A325328, A325352, A325360, A325361, A325368, A325395, A325398, A325457, A325460.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Less@@Differences[primeptn[#]]&]

A325460 Heinz numbers of integer partitions with strictly increasing differences (with the last part taken to be 0).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 67, 69, 71, 73, 74, 79, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 122, 123, 127, 129, 130, 131

Offset: 1

Gus Wiseman, May 03 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) (with the last part taken to be 0) are (-3,-2,-1).
The enumeration of these partitions by sum is given by A179269.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   17: {7}
   19: {8}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   31: {11}
   33: {2,5}
   34: {1,7}
   37: {12}
   38: {1,8}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

A subsequence of A005117.

Cf. A007294, A056239, A112798, A179269, A325327, A325362, A325364, A325367, A325388, A325390, A325395, A325398, A325456, A325461.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Less@@Differences[Append[primeptn[#],0]]&]

A325359 Numbers of the form p^y * 2^z where p is an odd prime, y >= 2, and z >= 0.

Original entry on oeis.org

9, 18, 25, 27, 36, 49, 50, 54, 72, 81, 98, 100, 108, 121, 125, 144, 162, 169, 196, 200, 216, 242, 243, 250, 288, 289, 324, 338, 343, 361, 392, 400, 432, 484, 486, 500, 529, 576, 578, 625, 648, 676, 686, 722, 729, 784, 800, 841, 864, 961, 968, 972, 1000, 1058

Offset: 1

Gus Wiseman, May 02 2019

nonn

Also Heinz numbers of integer partitions that are not hooks but whose augmented differences are hooks, where the Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k), and a hook is a partition of the form (n,1,1,...,1). The enumeration of these partitions by sum is given by A325459.

			The sequence of terms together with their prime indices begins:
     9: {2,2}
    18: {1,2,2}
    25: {3,3}
    27: {2,2,2}
    36: {1,1,2,2}
    49: {4,4}
    50: {1,3,3}
    54: {1,2,2,2}
    72: {1,1,1,2,2}
    81: {2,2,2,2}
    98: {1,4,4}
   100: {1,1,3,3}
   108: {1,1,2,2,2}
   121: {5,5}
   125: {3,3,3}
   144: {1,1,1,1,2,2}
   162: {1,2,2,2,2}
   169: {6,6}
   196: {1,1,4,4}
   200: {1,1,1,3,3}

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of 2's in A325355.
Numbers n such that n does not belong to A093641 but A325351(n) does.
Cf. A056239, A112798, A136141, A325366, A325389, A325394, A325395, A325396.

N:= 1000: # to get terms <= N
P:= select(isprime, [seq(i,i=3..floor(sqrt(N)),2)]):
B:= map(proc(p) local y;  seq(p^y, y=2..floor(log[p](N))) end proc, P):
sort(map(proc(t) local z;  seq(2^z*t, z=0..ilog2(N/t)) end proc, B)); # Robert Israel, May 03 2019

Select[Range[1000],MatchQ[FactorInteger[2*#],{{2,},{?(#>2&),_?(#>1&)}}]&]

Sum_{n>=1} 1/a(n) = 2 * Sum_{p prime} 1/(p*(p-1)) - 1 = 2 * A136141 - 1 = 0.54631333809959025572... - Amiram Eldar, Sep 30 2020

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A325456 Heinz numbers of integer partitions with strictly increasing differences.

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A325460 Heinz numbers of integer partitions with strictly increasing differences (with the last part taken to be 0).

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A325359 Numbers of the form p^y * 2^z where p is an odd prime, y >= 2, and z >= 0.

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