A325374 -id:A325374 - cp's OEIS frontend

A325334 Number of integer partitions of n with adjusted frequency depth 3 whose parts cover an initial interval of positive integers.

0, 0, 0, 1, 0, 0, 2, 0, 0, 1, 1, 0, 2, 0, 0, 2, 0, 0, 2, 0, 1, 2, 0, 0, 2, 0, 0, 1, 1, 0, 4, 0, 0, 1, 0, 0, 3, 0, 0, 1, 1, 0, 3, 0, 0, 3, 0, 0, 2, 0, 1, 1, 0, 0, 2, 1, 1, 1, 0, 0, 4, 0, 0, 2, 0, 0, 3, 0, 0, 1, 1, 0, 3, 0, 0, 2, 0, 0, 3, 0, 1, 1, 0, 0, 4, 0, 0, 1, 0, 0, 5, 1, 0, 1, 0, 0, 2, 0, 0, 1, 1, 0, 2, 0, 0, 4

Offset: 0

Gus Wiseman, May 01 2019

nonn

The adjusted frequency depth of an integer partition (A325280) is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).

The Heinz numbers of these partitions are given by A325374.

			The first 30 terms count the following partitions:
   3: (21)
   6: (321)
   6: (2211)
   9: (222111)
  10: (4321)
  12: (332211)
  12: (22221111)
  15: (54321)
  15: (2222211111)
  18: (333222111)
  18: (222222111111)
  20: (44332211)
  21: (654321)
  21: (22222221111111)
  24: (333322221111)
  24: (2222222211111111)
  27: (222222222111111111)
  28: (7654321)
  30: (5544332211)
  30: (444333222111)
  30: (333332222211111)
  30: (22222222221111111111)

Antti Karttunen, Table of n, a(n) for n = 0..100000

Column k = 3 of A325336.

Cf. A007862, A181819, A182850, A320348, A323014, A325245, A325280, A325326, A325335, A325374.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
unifQ[m_]:=SameQ@@Length/@Split[m];
Table[Length[Select[IntegerPartitions[n],normQ[#]&&!SameQ@@#&&unifQ[#]&]],{n,0,30}]

A007862(n) = sumdiv(n, d, ispolygonal(d, 3));
A325334(n) = if(!n,n,A007862(n)-1); \\ Antti Karttunen, Jan 17 2025

a(n) = A007862(n) - 1.

Data section extended to a(105) by Antti Karttunen, Jan 17 2025

A365308 Powers of primorials P(k)^m, k > 1, m > 1, where P(k) = A002110(k).

Original entry on oeis.org

36, 216, 900, 1296, 7776, 27000, 44100, 46656, 279936, 810000, 1679616, 5336100, 9261000, 10077696, 24300000, 60466176, 362797056, 729000000, 901800900, 1944810000, 2176782336, 12326391000, 13060694016, 21870000000, 78364164096, 260620460100, 408410100000, 470184984576

Offset: 1

Michael De Vlieger, Oct 02 2023

Proper subset of A303606, in turn a proper subset of A286708, in turn a proper subset of A126706.

Numbers in A322793 that are not powers of 2.

			Terms less than 10^4 include P(2)^2 = 36, P(2)^3 = 216, P(2)^4 = 1296, P(2)^5 = 7776, and P(3)^2 = 900.

Michael De Vlieger, Table of n, a(n) for n = 1..4935
Michael De Vlieger, 1024 X 1024 Bitmap showing A322793(n) in black if a power of 2 (i.e., in A000079) else white if in this sequence, n = 1..2^20, arranged from left to right in rows, then from top to bottom.

Cf. A000079, A002110, A100778, A126706, A286708, A303606, A322793, A325374.

nn = 2^39; k = 2; P = 6; Union@ Reap[While[j = 2; While[P^j < nn, Sow[P^j]; j++]; j > 2, k++; P *= Prime[k]] ][[-1, 1]]

Intersection of A100778 and A303606.
This sequence is {A325374 \ {A002110 \ {1,2}}} = {A322793 \ {A000079 \ {1,2}}}.
Sum_{n>=1} 1/a(n) = Sum_{k>=2} 1/(P(k)*(P(k)-1)) = 0.03450573145072369022... . - Amiram Eldar, Mar 10 2024

A325387 Numbers with adjusted frequency depth 4 whose prime indices cover an initial interval of positive integers.

Original entry on oeis.org

12, 18, 24, 48, 54, 72, 96, 108, 144, 162, 192, 288, 324, 360, 384, 432, 486, 540, 576, 600, 648, 720, 768, 864, 972, 1152, 1200, 1260, 1350, 1440, 1458, 1500, 1536, 1620, 1728, 1944, 2100, 2160, 2250, 2304, 2400, 2592, 2880, 2916, 2940, 3072, 3150, 3240, 3456

Offset: 1

Gus Wiseman, May 02 2019

nonn

The adjusted frequency depth of a positive integer n is 0 if n = 1, and otherwise it is 1 plus the number of times one must apply A181819 to reach a prime number, where A181819(k = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of k. For example, 180 has adjusted frequency depth 5 because we have: 180 -> 18 -> 6 -> 4 -> 3.

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 with adjusted frequency depth 4 whose parts cover an initial interval of positive integers. The enumeration of these partitions by sum is given by A325335.

			The sequence of terms together with their prime indices begins:
    12: {1,1,2}
    18: {1,2,2}
    24: {1,1,1,2}
    48: {1,1,1,1,2}
    54: {1,2,2,2}
    72: {1,1,1,2,2}
    96: {1,1,1,1,1,2}
   108: {1,1,2,2,2}
   144: {1,1,1,1,2,2}
   162: {1,2,2,2,2}
   192: {1,1,1,1,1,1,2}
   288: {1,1,1,1,1,2,2}
   324: {1,1,2,2,2,2}
   360: {1,1,1,2,2,3}
   384: {1,1,1,1,1,1,1,2}
   432: {1,1,1,1,2,2,2}
   486: {1,2,2,2,2,2}
   540: {1,1,2,2,2,3}
   576: {1,1,1,1,1,1,2,2}
   600: {1,1,1,2,3,3}

Cf. A055932, A056239, A112798, A181819,, A323014, A325280, A325326, A325335, A325336, A325374.

normQ[n_Integer]:=Or[n==1,PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]]];
fdadj[n_Integer]:=If[n==1,0,Length[NestWhileList[Times@@Prime/@Last/@FactorInteger[#1]&,n,!PrimeQ[#1]&]]];
Select[Range[10000],normQ[#]&&fdadj[#]==4&]

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A325334 Number of integer partitions of n with adjusted frequency depth 3 whose parts cover an initial interval of positive integers.

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A365308 Powers of primorials P(k)^m, k > 1, m > 1, where P(k) = A002110(k).

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A325387 Numbers with adjusted frequency depth 4 whose prime indices cover an initial interval of positive integers.

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