A325389 -id:A325389 - 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

A325394 Heinz numbers of integer partitions whose augmented differences are weakly increasing.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 64, 67, 71, 73, 75, 77, 79, 81, 83, 89, 91, 97, 101, 103, 105, 107, 109, 113, 119, 121, 125, 127, 128, 131, 137, 139, 143, 149, 151, 157, 163, 167

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 A325356.

			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}
   13: {6}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}

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

Cf. A056239, A093641, A112798, A240026, A325351, A325356, A325360, A325362, A325366, A325389, A325395, A325396, A325400.

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

A307824 Heinz numbers of integer partitions whose augmented differences are all equal.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 11, 13, 15, 16, 17, 19, 23, 29, 31, 32, 37, 41, 43, 47, 53, 55, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 105, 107, 109, 113, 119, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227

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 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 A129654.

			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}
   11: {5}
   13: {6}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   37: {12}
   41: {13}
   43: {14}

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

Cf. A049988, A056239, A093641, A112798, A129654, A325327, A325328, A325351, A325359, 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

A325364 Heinz numbers of integer partitions whose differences (with the last part taken to be zero) are weakly decreasing.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 16, 17, 18, 19, 21, 23, 25, 27, 29, 30, 31, 32, 35, 37, 41, 43, 47, 49, 53, 54, 55, 59, 61, 64, 65, 67, 71, 73, 75, 77, 79, 81, 83, 89, 91, 97, 101, 103, 105, 107, 109, 113, 119, 121, 125, 127, 128, 131, 133, 137, 139

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 differences of a sequence are defined as if the sequence were increasing, so for example the differences of (x, y, z) are (y - x, z - y). We adhere to this standard for integer partitions also even though they are always weakly decreasing. 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 A320509.

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

Cf. A056239, A112798, A320348, A320466, A320509, A325327, A325361, A325364, A325367, A325389, A325390, A325397.

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

A325361 Heinz numbers of integer partitions whose differences are weakly decreasing.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 87, 89

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 differences of a sequence are defined as if the sequence were increasing, for example the differences of (x, y, z) are (y - x, z - y). We adhere to this standard for integer partitions also even though they are always weakly decreasing. For example, the differences of (6,3,1) are (-3,-2).
The enumeration of these partitions by sum is given by A320466.

			Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   72: {1,1,1,2,2}
   76: {1,1,8}
   78: {1,2,6}
   80: {1,1,1,1,3}

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

Cf. A056239, A112798, A320466, A320509, A325328, A325352, A325456, A325457, A325360, A325361, A325364, A320466, A325368, A325389.

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

A325395 Heinz numbers of integer partitions whose augmented differences are strictly increasing.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209, 211, 221

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 A325357.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   29: {10}
   31: {11}
   35: {3,4}
   37: {12}
   41: {13}
   43: {14}
   47: {15}
   49: {4,4}

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

Cf. A056239, A093641, A112798, A240027, A325351, A325357, A325366, A325389, A325394, A325396, A325398, A325456, A325460.

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

A325396 Heinz numbers of integer partitions whose augmented differences are strictly decreasing.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 78, 79, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109, 111, 113, 114, 115

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 A325358.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   17: {7}
   19: {8}
   21: {2,4}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   31: {11}
   33: {2,5}
   34: {1,7}

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

A subsequence of A005117.

Cf. A056239, A093641, A112798, A320466, A325351, A325358, A325366, A325389, A325393, A325394, A325395, A325457.

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

A325355 One plus the number of steps applying A325351 (Heinz number of augmented differences of reversed prime indices) to reach a fixed point.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 23 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 fixed points of A325351 are the Heinz numbers of hooks A093641.

			Repeatedly applying A325351 starting with 78 gives 78 -> 66 -> 42 -> 30 -> 18 -> 12, and 12 is a fixed point, so a(78) = 6.

Positions of 2's are A325359.

Cf. A056239, A093641, A112798, A130091, A289509, A325351, A325352, 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)));
A325355(n) = { my(u=A325351(n)); if(u==n,1,1+A325355(u)); }; \\ Antti Karttunen, Nov 16 2019

More terms from Antti Karttunen, Nov 16 2019

A325350 Number of integer partitions of n whose augmented differences are weakly decreasing.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 10, 13, 17, 21, 26, 32, 38, 46, 56, 66, 78, 92, 106, 124, 145, 166, 191, 220, 249, 284, 325, 366, 413, 468, 523, 586, 659, 733, 817, 913, 1011, 1121, 1245, 1373, 1515, 1674, 1838, 2020, 2223, 2433, 2664, 2920, 3184, 3476, 3797, 4129, 4492

Offset: 0

Gus Wiseman, Apr 23 2019

nonn

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 Heinz numbers of these partitions are given by A325389.

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (31)    (32)     (42)      (52)       (53)
             (111)  (211)   (41)     (51)      (61)       (62)
                    (1111)  (311)    (321)     (421)      (71)
                            (2111)   (411)     (511)      (521)
                            (11111)  (3111)    (3211)     (611)
                                     (21111)   (4111)     (4211)
                                     (111111)  (31111)    (5111)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
For example, (4,2,1,1) has augmented differences (3,2,1,1), which are weakly decreasing, so (4,2,1,1) is counted under a(8).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500

Cf. A007294, A098859, A240026, A320466, A320509, A325349, A325353, A325354, A325356, A325357, A325358, A325361, A325364.

Mathematica
```
aug[y_]:=Table[If[i
				
```

G.f.: Sum_{k>=0} x^k / Product_{j=1..k} (1 - x^(j*(j+1)/2)) (conjecture). - Ilya Gutkovskiy, Apr 25 2019

cp's OEIS Frontend

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

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

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A325394 Heinz numbers of integer partitions whose augmented differences are weakly increasing.

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A307824 Heinz numbers of integer partitions whose augmented differences are all equal.

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A325364 Heinz numbers of integer partitions whose differences (with the last part taken to be zero) are weakly decreasing.

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A325361 Heinz numbers of integer partitions whose differences are weakly decreasing.

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A325395 Heinz numbers of integer partitions whose augmented differences are strictly increasing.

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A325396 Heinz numbers of integer partitions whose augmented differences are strictly decreasing.

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