A358169 -id:A358169 - cp's OEIS frontend

A355536 Irregular triangle read by rows where row n lists the differences between adjacent prime indices of n; if n is prime, row n is empty.

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

Offset: 2

Gus Wiseman, Jul 12 2022

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 version where zero is prepended to the prime indices is A287352.
One could argue that row n = 1 is empty, but adding it changes only the offset, not the data.

			Triangle begins (showing n, prime indices, differences*):
   2:    (1)       .
   3:    (2)       .
   4:   (1,1)      0
   5:    (3)       .
   6:   (1,2)      1
   7:    (4)       .
   8:  (1,1,1)    0 0
   9:   (2,2)      0
  10:   (1,3)      2
  11:    (5)       .
  12:  (1,1,2)    0 1
  13:    (6)       .
  14:   (1,4)      3
  15:   (2,3)      1
  16: (1,1,1,1)  0 0 0

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

Row-lengths are A001222 minus one.
The prime indices are A112798, sum A056239.
Row-sums are A243055.
Constant rows have indices A325328.
The Heinz numbers of the rows plus one are A325352.
Strict rows have indices A325368.
Row minima are A355524.
Row maxima are A286470, also A355526.
An adjusted version is A358169, reverse A355534.
Cf. A066312, A124010, A129654, A243056, A287352, A325394, A355523, A355528, A355531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Differences[primeMS[n]],{n,2,100}]

A253566 Permutation of natural numbers: a(n) = A243071(A122111(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 7, 5, 12, 16, 14, 32, 24, 10, 15, 64, 13, 128, 28, 20, 48, 256, 30, 9, 96, 11, 56, 512, 26, 1024, 31, 40, 192, 18, 29, 2048, 384, 80, 60, 4096, 52, 8192, 112, 22, 768, 16384, 62, 17, 25, 160, 224, 32768, 27, 36, 120, 320, 1536, 65536, 58, 131072, 3072, 44, 63, 72, 104, 262144, 448, 640, 50, 524288, 61, 1048576, 6144, 21

Offset: 1

Antti Karttunen, Jan 03 2015

nonn

Note the indexing: domain starts from one, while the range includes also zero. See also comments in A253564.

The a(n)-th composition in standard order (graded reverse-lexicographic, A066099) is one plus the first differences of the weakly increasing sequence of prime indices of n with 1 prepended. See formula for a simplification. The triangular form is A358169. The inverse is A253565. Not prepending 1 gives A358171. For Heinz numbers instead of standard compositions we have A325351 (without prepending A325352). - Gus Wiseman, Dec 23 2022

			From _Gus Wiseman_, Dec 23 2022: (Start)
This represents the following bijection between partitions and compositions. The reversed prime indices of n together with the a(n)-th composition in standard order are:
   1:        () -> ()
   2:       (1) -> (1)
   3:       (2) -> (2)
   4:     (1,1) -> (1,1)
   5:       (3) -> (3)
   6:     (2,1) -> (1,2)
   7:       (4) -> (4)
   8:   (1,1,1) -> (1,1,1)
   9:     (2,2) -> (2,1)
  10:     (3,1) -> (1,3)
  11:       (5) -> (5)
  12:   (2,1,1) -> (1,1,2)
  13:       (6) -> (6)
  14:     (4,1) -> (1,4)
  15:     (3,2) -> (2,2)
  16: (1,1,1,1) -> (1,1,1,1)
(End)

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A253565.
Cf. A122111, A243071, A253564, A054429, A336120, A336125.
Applying A000120 gives A001222.
A reverse version is A156552, inverse essentially A005940.
The inverse is A253565, triangular form A242628.
The triangular form is A358169.
A048793 gives partial sums of reversed standard comps, Heinz number A019565.
A066099 lists standard compositions, lengths A000120, sums A070939.
A112798 list prime indices, sum A056239.
A358134 gives partial sums of standard compositions, Heinz number A358170.
Cf. A029837, A241916, A243055, A287352, A325390, A355534, A358171, A359042.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
stcinv/@Table[Differences[Prepend[primeMS[n],1]]+1,{n,100}] (* Gus Wiseman, Dec 23 2022 *)

(define (A253566 n) (A243071 (A122111 n)))

a(n) = A243071(A122111(n)).
As a composition of other permutations:
a(n) = A054429(A253564(n)).
a(n) = A336120(n) + A336125(n). - Antti Karttunen, Jul 18 2020
If 2n = Product_{i=1..k} prime(x_i) then a(n) = Sum_{i=1..k-1} 2^(x_k-x_{k-i}+i-1). - Gus Wiseman, Dec 23 2022

A359397 Squarefree numbers with weakly decreasing first differences of 0-prepended prime indices.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 15, 17, 19, 21, 23, 29, 30, 31, 35, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 105, 107, 109, 113, 119, 127, 131, 133, 137, 139, 143, 149, 151, 157, 163, 167, 173, 179, 181, 187, 191, 193, 197

Offset: 1

Gus Wiseman, Dec 31 2022

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.

			715 has prime indices {3,5,6}, with first differences (2,1), which are weakly decreasing, so 715 is in the sequence.

This is the squarefree case of A325362.
These are the sorted Heinz numbers of rows of A359361.
A005117 lists squarefree numbers.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A355536 lists first differences of prime indices, 0-prepended A287352.
A358136 lists partial sums of prime indices, row sums A318283.
Cf. A000009, A000720, A001221, A253566, A261079, A304818, A358137, A358169.

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

Intersection of A325362 and A005117.

A358171 The a(n)-th composition in standard order (A066099) is the first differences plus one of the prime indices of n (A112798).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 1, 4, 0, 6, 0, 8, 2, 7, 0, 5, 0, 12, 4, 16, 0, 14, 1, 32, 3, 24, 0, 10, 0, 15, 8, 64, 2, 13, 0, 128, 16, 28, 0, 20, 0, 48, 6, 256, 0, 30, 1, 9, 32, 96, 0, 11, 4, 56, 64, 512, 0, 26, 0, 1024, 12, 31, 8, 40, 0, 192, 128, 18, 0, 29, 0

Offset: 1

Gus Wiseman, Dec 21 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

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 prime indices of 36 are {1,1,2,2}, with first differences plus one (1,2,1), which is the 13th composition in standard order, so a(36) = 13.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
Prepend 1 to indices: A253566 (cf. A358169), inverse A253565 (cf. A242628).
Taking Heinz number instead of standard composition number gives A325352.
These compositions minus one are listed by A355536, sums A243055.
A001222 counts prime indices, distinct A001221.
A066099 lists standard compositions, lengths A000120, sums A070939.
A112798 lists prime indices, sum A056239.
A355534 = augmented diffs. of rev. prime indices, Heinz numbers A325351.
Cf. A000720, A005940, A055932, A057335, A066312, A286470, A287352, A325390.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Differences[primeMS[n]]+1],{n,100}]

cp's OEIS Frontend

A355536 Irregular triangle read by rows where row n lists the differences between adjacent prime indices of n; if n is prime, row n is empty.

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A253566 Permutation of natural numbers: a(n) = A243071(A122111(n)).

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A359397 Squarefree numbers with weakly decreasing first differences of 0-prepended prime indices.

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A358171 The a(n)-th composition in standard order (A066099) is the first differences plus one of the prime indices of n (A112798).

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