A325355 -id:A325355 - cp's OEIS frontend

A325351 Heinz number of the augmented differences of the integer partition with Heinz number n.

1, 2, 3, 4, 5, 6, 7, 8, 6, 10, 11, 12, 13, 14, 9, 16, 17, 12, 19, 20, 15, 22, 23, 24, 10, 26, 12, 28, 29, 18, 31, 32, 21, 34, 15, 24, 37, 38, 33, 40, 41, 30, 43, 44, 18, 46, 47, 48, 14, 20, 39, 52, 53, 24, 25, 56, 51, 58, 59, 36, 61, 62, 30, 64, 35, 42, 67, 68, 57, 30, 71, 48, 73, 74, 18, 76, 21, 66, 79, 80, 24, 82, 83, 60, 55, 86, 69, 88, 89, 36, 35

Offset: 1

Gus Wiseman, Apr 23 2019

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

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3). Note that aug preserves length so this sequence preserves omega (number of prime factors counted with multiplicity).

			The partition (3,2,2,1) with Heinz number 90 has augmented differences (2,1,2,1) with Heinz number 36, so a(90) = 36.

Number of appearances of n is A008480(n).

Cf. A056239, A093641 (fixed points), A112798, A325350, A325352, A325355, A325366, A325389, A325394, A325395, A325396.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
aug[y_]:=Table[If[i

augdiffs(n) = { my(diffs=List([]), f=factor(n), prevpi, pi=0, i=#f~); while(i, prevpi=pi; pi = primepi(f[i, 1]); if(prevpi, listput(diffs, 1+(prevpi-pi))); if(f[i, 2]>1, f[i, 2]--, i--)); if(pi, listput(diffs,pi)); Vec(diffs); };
A325351(n) = factorback(apply(prime,augdiffs(n))); \\ Antti Karttunen, Nov 16 2019

More terms from Antti Karttunen, Nov 16 2019

A325366 Heinz numbers of integer partitions whose augmented differences are distinct.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 99, 101, 102, 103

Offset: 1

Gus Wiseman, May 02 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
The enumeration of these partitions by sum is given by A325349.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   17: {7}
   19: {8}
   21: {2,4}
   22: {1,5}
   23: {9}
   25: {3,3}
   26: {1,6}
   29: {10}
   31: {11}

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

Positions of squarefree numbers in A325351.

Cf. A056239, A093641, A112798, A130091, A325349, A325355, A325367, A325368, A325389, A325394, A325395, A325396, A325405.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
aug[y_]:=Table[If[i

A383512 Heinz numbers of conjugate Wilf partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85

Offset: 1

Gus Wiseman, May 13 2025

nonn

First differs from A364347 in having 130 and lacking 110.
First differs from A381432 in lacking 65 and 133.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

			The terms together with their prime indices begin:
     1: {}           17: {7}            35: {3,4}
     2: {1}          19: {8}            37: {12}
     3: {2}          20: {1,1,3}        38: {1,8}
     4: {1,1}        22: {1,5}          39: {2,6}
     5: {3}          23: {9}            40: {1,1,1,3}
     7: {4}          25: {3,3}          41: {13}
     8: {1,1,1}      26: {1,6}          43: {14}
     9: {2,2}        27: {2,2,2}        44: {1,1,5}
    10: {1,3}        28: {1,1,4}        45: {2,2,3}
    11: {5}          29: {10}           46: {1,9}
    13: {6}          31: {11}           47: {15}
    14: {1,4}        32: {1,1,1,1,1}    49: {4,4}
    15: {2,3}        33: {2,5}          50: {1,3,3}
    16: {1,1,1,1}    34: {1,7}          51: {2,7}

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

Partitions of this type are counted by A098859.
The conjugate version is A130091, complement A130092.
Including differences of 0 gives A325367, counted by A325324.
The strict case is A325388, counted by A320348.
The complement is A383513, counted by A336866.
Also requiring distinct multiplicities gives A383532, counted by A383507.
These are the positions of strict rows in A383534, or squarefree numbers in A383535.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A325349 counts partitions with distinct augmented differences, ranks A325366.
A383530 counts partitions that are not Wilf or conjugate Wilf, ranks A383531.
A383709 counts Wilf partitions with distinct augmented differences, ranks A383712.
Cf. A000720, A005117, A048768, A109298, A325325, A325351, A325355, A325368, A381431, A383506.

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], UnsameQ@@DeleteCases[Differences[Prepend[prix[#],0]],0]&]

A383513 Heinz numbers of non conjugate Wilf partitions.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 70, 72, 78, 84, 90, 96, 102, 105, 108, 110, 114, 120, 126, 132, 133, 138, 140, 144, 147, 150, 154, 156, 162, 165, 168, 174, 180, 186, 189, 192, 198, 204, 210, 216, 220, 222, 228, 231, 234, 238, 240, 246

Offset: 1

Gus Wiseman, May 13 2025

nonn

First differs from A381433 in having 65.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

			The terms together with their prime indices begin:
    6: {1,2}
   12: {1,1,2}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   60: {1,1,2,3}
   63: {2,2,4}
   65: {3,6}
   66: {1,2,5}
   70: {1,3,4}
   72: {1,1,1,2,2}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}

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

Partitions of this type are counted by A336866.
The conjugate version is A130092, complement A130091.
Including differences of 0 gives complement of A325367, counted by A325324.
The strict case is the complement of A325388, counted by A320348.
The complement is A383512, counted by A098859.
Also forbidding distinct multiplicities gives A383531, counted by A383530.
These are positions of non-strict rows in A383534, or nonsquarefree numbers in A383535.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A383507 counts partitions that are Wilf and conjugate Wilf, ranks A383532.
A383709 counts Wilf partitions with distinct augmented differences, ranks A383712.
Cf. A000720, A005117, A048768, A238745, A325325, A325351, A325355, A325366, A325368, A381431, A383506.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!UnsameQ@@DeleteCases[Differences[Prepend[prix[#],0]],0]&]

A325407 Nonprime Heinz numbers of multiples of triangular partitions, or of finite arithmetic progressions with offset 0.

Original entry on oeis.org

1, 6, 21, 30, 65, 133, 210, 273, 319, 481, 731, 1007, 1403, 1495, 2059, 2310, 2449, 3293, 4141, 4601, 4921, 5187, 5311, 6943, 8201, 9211, 10921, 12283, 13213, 14993, 15247, 16517, 19847, 22213, 24139, 25853, 28141, 29341, 29539, 30030, 31753, 37211, 40741

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), so these are numbers of the form Product_{k = 1...b} prime(k * c) for some b > 1 and c > 0.

			The sequence of terms together with their prime indices begins:
      1: {}
      6: {1,2}
     21: {2,4}
     30: {1,2,3}
     65: {3,6}
    133: {4,8}
    210: {1,2,3,4}
    273: {2,4,6}
    319: {5,10}
    481: {6,12}
    731: {7,14}
   1007: {8,16}
   1403: {9,18}
   1495: {3,6,9}
   2059: {10,20}
   2310: {1,2,3,4,5}
   2449: {11,22}
   3293: {12,24}
   4141: {13,26}
   4601: {14,28}

Cf. A007294, A007862, A049988, A056239, A112798, A325327, A325328, A325355, A325359, A325367, A325390.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[10000],!PrimeQ[#]&&SameQ@@Differences[Prepend[primeMS[#],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

A383712 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct 0-appended differences.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 20, 23, 25, 28, 29, 31, 37, 41, 43, 44, 45, 47, 49, 50, 52, 53, 59, 61, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 101, 103, 107, 109, 113, 116, 117, 121, 124, 127, 131, 137, 139, 148, 149, 151, 153, 157, 163, 164

Offset: 1

Gus Wiseman, May 15 2025

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Integer partitions with distinct multiplicities are called Wilf partitions.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   20: {1,1,3}
   23: {9}
   25: {3,3}
   28: {1,1,4}
   29: {10}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   44: {1,1,5}
   45: {2,2,3}
   47: {15}
   49: {4,4}
   50: {1,3,3}

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

For just distinct multiplicities we have A130091 (conjugate A383512), counted by A098859.
For just distinct 0-appended differences we have A325367, counted by A325324.
These partitions are counted by A383709.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A383507 counts partitions that are Wilf and conjugate Wilf, ranks A383532.
A383530 counts partitions that are not Wilf or conjugate-Wilf, ranks A383531.
Cf. A000720, A005117, A047966, A238745, A320348, A325325, A325349, A325355, A325366, A325368, A325388, A383506.

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

Equals A130091 /\ A325367.

A325459 Sum of numbers of nontrivial divisors (greater than 1 and less than k) of k for k = 1..n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 3, 5, 6, 8, 8, 12, 12, 14, 16, 19, 19, 23, 23, 27, 29, 31, 31, 37, 38, 40, 42, 46, 46, 52, 52, 56, 58, 60, 62, 69, 69, 71, 73, 79, 79, 85, 85, 89, 93, 95, 95, 103, 104, 108, 110, 114, 114, 120, 122, 128, 130, 132, 132, 142

Offset: 0

Gus Wiseman, May 04 2019

nonn

Also the number of integer partitions of n that are not hooks but whose augmented differences are hooks (original name). The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and otherwise aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

This sequence counts integer partitions with any number of ones and one part > 1 which appears at least twice. The Heinz numbers of these partitions are given by A325359.

			The a(4) = 1 through a(10) = 8 partitions:
  (22)  (221)  (33)    (331)    (44)      (333)      (55)
               (222)   (2221)   (2222)    (441)      (3331)
               (2211)  (22111)  (3311)    (22221)    (4411)
                                (22211)   (33111)    (22222)
                                (221111)  (222111)   (222211)
                                          (2211111)  (331111)
                                                     (2221111)
                                                     (22111111)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A049988, A093641, A325349, A325351, A325355, A325356, A325357, A325358.

Cf. A070824, A006218.

a:= proc(n) option remember; `if`(n<2, 0,
      numtheory[tau](n)-2+a(n-1))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 11 2019

Table[Length[Select[IntegerPartitions[n],MatchQ[#,{x_,y__,1...}/;x>1&&SameQ[x,y]]&]],{n,0,30}]
(* Second program: *)
a[n_] := a[n] = If[n<2, 0, DivisorSigma[0, n] - 2 + a[n-1]];
a /@ Range[0, 100] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

from math import isqrt
def A325459(n): return 0 if n == 0 else (lambda m: 2*(sum(n//k for k in range(1, m+1))-n)+(1-m)*(1+m))(isqrt(n)) # Chai Wah Wu, Oct 07 2021

From M. F. Hasler, Oct 11 2019: (Start)
a(n) = A006218(n) - 2*n + 1, in terms of partial sums of number of divisors.
a(n) = Sum_{k=1..n} A070824(k): partial sums of A070824 = number of nontrivial divisors. (End)

Name changed at the suggestion of Patrick James Smalley-Wall and Luc Rousseau by Gus Wiseman, Oct 11 2019

A325458 Triangle read by rows where T(n,k) is the number of integer partitions of n with largest hook of size k, i.e., with (largest part) + (number of parts) - 1 = k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 1, 4, 0, 0, 0, 0, 2, 5, 0, 0, 0, 0, 2, 3, 6, 0, 0, 0, 0, 0, 4, 4, 7, 0, 0, 0, 0, 0, 3, 6, 5, 8, 0, 0, 0, 0, 0, 1, 6, 8, 6, 9, 0, 0, 0, 0, 0, 0, 6, 9, 10, 7, 10, 0, 0, 0, 0, 0, 0, 2, 11, 12, 12, 8, 11

Offset: 0

Gus Wiseman, May 04 2019

Conjectured to be equal to A049597.

			Triangle begins:
  1
  0  1
  0  0  2
  0  0  0  3
  0  0  0  1  4
  0  0  0  0  2  5
  0  0  0  0  2  3  6
  0  0  0  0  0  4  4  7
  0  0  0  0  0  3  6  5  8
  0  0  0  0  0  1  6  8  6  9
  0  0  0  0  0  0  6  9 10  7 10
  0  0  0  0  0  0  2 11 12 12  8 11
  0  0  0  0  0  0  2  9 16 15 14  9 12
  0  0  0  0  0  0  0  7 16 21 18 16 10 13
  0  0  0  0  0  0  0  4 18 23 26 21 18 11 14
  0  0  0  0  0  0  0  3 12 29 30 31 24 20 12 15
  0  0  0  0  0  0  0  1 12 27 40 37 36 27 22 13 16
  0  0  0  0  0  0  0  0  8 26 42 51 44 41 30 24 14 17
  0  0  0  0  0  0  0  0  6 23 48 57 62 51 46 33 26 15 18
  0  0  0  0  0  0  0  0  2 21 44 70 72 73 58 51 36 28 16 19
Row n = 9 counts the following partitions:
  (333)  (54)     (63)      (72)       (9)
         (432)    (522)     (621)      (81)
         (441)    (531)     (5211)     (711)
         (3222)   (4221)    (42111)    (6111)
         (3321)   (4311)    (321111)   (51111)
         (22221)  (32211)   (2211111)  (411111)
                  (33111)              (3111111)
                  (222111)             (21111111)
                                       (111111111)

Row sums are A000041.
Column sums are 2^(k - 1) for k > 0.
Cf. A008284, A049597, A093641, A116608, A325242, A325349, A325351, A325355, A325359, A325459.

Table[Length[Select[IntegerPartitions[n],If[n==0,k==0,First[#]+Length[#]-1==k]&]],{n,0,19},{k,0,n}]

Franklin T. Adams-Watters has conjectured at A049597 that the k-th column gives the coefficients of the sum of Gaussian polynomials [k,m] for m = 0..k.

cp's OEIS Frontend

A325351 Heinz number of the augmented differences of the integer partition with Heinz number n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A325366 Heinz numbers of integer partitions whose augmented differences are distinct.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A383512 Heinz numbers of conjugate Wilf partitions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A383513 Heinz numbers of non conjugate Wilf partitions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A325407 Nonprime Heinz numbers of multiples of triangular partitions, or of finite arithmetic progressions with offset 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A383712 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct 0-appended differences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A325459 Sum of numbers of nontrivial divisors (greater than 1 and less than k) of k for k = 1..n.

Views

Author

Keywords

Comments