A381440 -id:A381440 - cp's OEIS frontend

A381434 Numbers appearing only once in A381431 (section-sum partition of prime indices).

1, 2, 3, 4, 8, 9, 10, 14, 15, 16, 20, 22, 27, 28, 32, 33, 35, 40, 44, 45, 50, 55, 56, 64, 75, 77, 80, 81, 88, 98, 99, 100, 112, 128, 130, 135, 160, 170, 175, 176, 182, 190, 195, 196, 200

Offset: 1

Gus Wiseman, Feb 27 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.
The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   20: {1,1,3}
   22: {1,5}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}

In A381431:
- fixed points are A000961, A000005
- conjugate is A048767, fixed points A048768, A217605
- all numbers present are A381432, conjugate A351294
- numbers missing are A381433, conjugate A351295
- numbers appearing only once are A381434 (this), conjugate A381540
- numbers appearing more than once are A381435, conjugate A381541
A000040 lists the primes, differences A001223.
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 section-sum partitions, complement A351293.
A381436 lists section-sum partition of prime indices, conjugate A381440.
Set multipartitions: A050320, A089259, A116540, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A051903, A066328, A116861, A130091, A212166, A239964, A317081, A381437.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Select[Range[100],Count[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]],#]==1&]

The complement is A381433 U A381435.

A381438 Triangle read by rows where T(n>0,k>0) is the number of integer partitions of n whose section-sum partition ends with k.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 2, 1, 0, 2, 3, 1, 0, 0, 3, 4, 1, 2, 0, 0, 4, 7, 2, 1, 0, 0, 0, 5, 9, 4, 1, 2, 0, 0, 0, 6, 13, 4, 4, 1, 0, 0, 0, 0, 8, 18, 6, 3, 2, 3, 0, 0, 0, 0, 10, 26, 9, 5, 2, 2, 0, 0, 0, 0, 0, 12, 32, 12, 8, 4, 2, 4, 0, 0, 0, 0, 0, 15

Offset: 1

Gus Wiseman, Mar 01 2025

The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			Triangle begins:
   1
   1  1
   1  0  2
   2  1  0  2
   3  1  0  0  3
   4  1  2  0  0  4
   7  2  1  0  0  0  5
   9  4  1  2  0  0  0  6
  13  4  4  1  0  0  0  0  8
  18  6  3  2  3  0  0  0  0 10
  26  9  5  2  2  0  0  0  0  0 12
  32 12  8  4  2  4  0  0  0  0  0 15
  47 16 11  4  3  2  0  0  0  0  0  0 18
  60 23 12  8  3  2  5  0  0  0  0  0  0 22
  79 27 20  7  9  4  3  0  0  0  0  0  0  0 27
 Row n = 9 counts the following partitions:
  (711)        (522)    (333)     (441)  .  .  .  .  (9)
  (6111)       (4221)   (3321)                       (81)
  (5211)       (3222)   (32211)                      (72)
  (51111)      (22221)  (222111)                     (63)
  (4311)                                             (621)
  (42111)                                            (54)
  (411111)                                           (531)
  (33111)                                            (432)
  (321111)
  (3111111)
  (2211111)
  (21111111)
  (111111111)

Last column (k=n) is A000009.
Row sums are A000041.
Row sums without the last column (k=n) are A047967.
For first instead of last part we have A116861, rank A066328.
First column (k=1) is A241131 shifted right and starting with 1 instead of 0.
Using Heinz numbers, this statistic is given by A381437.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts section-sum partitions, complement A351293.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Section-sum partition: A381431, A381432, A381433, A381434, A381435, A381436.
Look-and-Say partition: A048767, A351294, A351295, A381440.
Cf. A047966, A051903, A051904, A091602, A181819, A212166.

egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Table[Length[Select[IntegerPartitions[n],k==Last[egs[#]]&]],{n,15},{k,n}]

A381541 Numbers appearing more than once in A048767 (Look-and-Say partition of prime indices).

Original entry on oeis.org

8, 16, 27, 32, 64, 81, 96, 125, 128, 144, 160, 192, 216, 224, 243, 256, 288

Offset: 1

Gus Wiseman, Mar 02 2025

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).
The conjugate of a Look-and-Say partition is a section-sum partition; see A381431, union A381432, count A239455.

			The terms together with their prime indices begin:
    8: {1,1,1}
   16: {1,1,1,1}
   27: {2,2,2}
   32: {1,1,1,1,1}
   64: {1,1,1,1,1,1}
   81: {2,2,2,2}
   96: {1,1,1,1,1,2}
  125: {3,3,3}
  128: {1,1,1,1,1,1,1}
  144: {1,1,1,1,2,2}
  160: {1,1,1,1,1,3}
  192: {1,1,1,1,1,1,2}
  216: {1,1,1,2,2,2}
  224: {1,1,1,1,1,4}
  243: {2,2,2,2,2}
  256: {1,1,1,1,1,1,1,1}
  288: {1,1,1,1,1,2,2}
For example, the term 96 appears in A048767 at positions 44 and 60, with prime indices:
  44: {1,1,5}
  60: {1,1,2,3}

In A048767:
- fixed points are A048768, A217605
- conjugate is A381431, fixed points A000961, A000005
- all numbers present are A351294, conjugate A381432
- numbers missing are A351295, conjugate A381433
- numbers appearing only once are A381540, conjugate A381434
- numbers appearing more than once are A381541 (this), conjugate A381435
A000040 lists the primes.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A381440 lists Look-and-Say partitions of prime indices, conjugate A381436.
Cf. A000720, A001222, A003557, A047966, A051903, A051904, A066328, A071178, A116861, A130091, A239964.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
hls[y_]:=Product[Prime[Count[y,x]]^x,{x,Union[y]}];
Select[Range[100],Count[hls/@IntegerPartitions[Total[prix[#]]],#]>1&]

A381540 Numbers appearing only once in A048767 (Look-and-Say partition of prime indices).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 17, 18, 19, 20, 23, 24, 25, 28, 29, 31, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 79, 80, 83, 88, 89, 92, 97, 98, 99, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117, 121

Offset: 1

Gus Wiseman, Mar 02 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).
The conjugate of a Look-and-Say partition is a section-sum partition; see A381431, union A381432, count A239455.

			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}
  12: {1,1,2}
  13: {6}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  23: {9}
  24: {1,1,1,2}

In A048767:
- fixed points are A048768, A217605
- conjugate is A381431, fixed points A000961, A000005
- all numbers present are A351294, conjugate A381432
- numbers missing are A351295, conjugate A381433
- numbers appearing only once are A381540 (this), conjugate A381434
- numbers appearing more than once are A381541, conjugate A381435
A000040 lists the primes.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A381440 lists Look-and-Say partition of prime indices, conjugate A381436.
Cf. A000720, A001222, A003557, A047966, A051903, A051904, A066328, A071178, A116861, A130091, A239964.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
hls[y_]:=Product[Prime[Count[y,x]]^x,{x,Union[y]}];
Select[Range[100],Count[hls/@IntegerPartitions[Total[prix[#]]],#]==1&]

A382775 Least number appearing n times in A048767 (rank of Look-and-Say partition of prime indices).

Original entry on oeis.org

6, 1, 8, 32, 64, 128, 256, 6144, 512, 27648, 1024, 73728, 2048, 147456, 165888, 4096, 248832, 196608, 8192, 497664, 1119744, 393216, 16384, 2239488

Offset: 0

Gus Wiseman, Apr 11 2025

Also the position of first appearance of n in A382525 (number of times n appears in A048767).
The Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. Hence, the multiplicity of k in the Look-and-Say partition of y is the sum of all parts that appear exactly k times. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			The terms together with their prime indices begin:
       6: {1,2}
       1: {}
       8: {1,1,1}
      32: {1,1,1,1,1}
      64: {1,1,1,1,1,1}
     128: {1,1,1,1,1,1,1}
     256: {1,1,1,1,1,1,1,1}
    6144: {1,1,1,1,1,1,1,1,1,1,1,2}
     512: {1,1,1,1,1,1,1,1,1}
   27648: {1,1,1,1,1,1,1,1,1,1,2,2,2}
    1024: {1,1,1,1,1,1,1,1,1,1}
   73728: {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
    2048: {1,1,1,1,1,1,1,1,1,1,1}
  147456: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
  165888: {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
    4096: {1,1,1,1,1,1,1,1,1,1,1,1}
  248832: {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}

Positions of first appearances in A382525.
The Look-and-Say partition is ranked by A048767, listed by A381440.
Look-and-Say partitions are counted by A239455, complement A351293.
Look-and-Say partitions are ranked by A351294.
Non-Look-and-Say partitions are ranked by A351295, conjugate A381433.
The section-sum partition is ranked by A381431, listed by A381436.
Section-sum partitions are ranked by A381432.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
Cf. A003557, A047966, A048768, A051903, A051904, A066328, A071178, A116861, A130091, A217605, A239964, A381540.

stp[y_]:=Select[Tuples[Select[IntegerPartitions[#], UnsameQ@@#&]&/@y],UnsameQ@@Join@@#&];
z=Table[Length[stp[Last/@FactorInteger[n]]],{n,10000}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[z,k][[1,1]],{k,0,mnrm[z+1]-1}]

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A381434 Numbers appearing only once in A381431 (section-sum partition of prime indices).

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A381438 Triangle read by rows where T(n>0,k>0) is the number of integer partitions of n whose section-sum partition ends with k.

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A381541 Numbers appearing more than once in A048767 (Look-and-Say partition of prime indices).

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A381540 Numbers appearing only once in A048767 (Look-and-Say partition of prime indices).

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A382775 Least number appearing n times in A048767 (rank of Look-and-Say partition of prime indices).

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