A358137 -id:A358137 - cp's OEIS frontend

A019565 The squarefree numbers ordered lexicographically by their prime factorization (with factors written in decreasing order). a(n) = Product_{k in I} prime(k+1), where I is the set of indices of nonzero binary digits in n = Sum_{k in I} 2^k.

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 42, 35, 70, 105, 210, 11, 22, 33, 66, 55, 110, 165, 330, 77, 154, 231, 462, 385, 770, 1155, 2310, 13, 26, 39, 78, 65, 130, 195, 390, 91, 182, 273, 546, 455, 910, 1365, 2730, 143, 286, 429, 858, 715, 1430, 2145, 4290

Offset: 0

Marc LeBrun

A permutation of the squarefree numbers A005117. The missing positive numbers are in A013929. - Alois P. Heinz, Sep 06 2014
From Antti Karttunen, Apr 18 & 19 2017: (Start)
Because a(n) toggles the parity of n there are neither fixed points nor any cycles of odd length.
Conjecture: there are no finite cycles of any length. My grounds for this conjecture: any finite cycle in this sequence, if such cycles exist at all, must have at least one member that occurs somewhere in A285319, the terms that seem already to be quite rare. Moreover, any such a number n should satisfy in addition to A019565(n) < n also that A048675^{k}(n) is squarefree, not just for k=0, 1 but for all k >= 0. As there is on average a probability of only 6/(Pi^2) = 0.6079... that any further term encountered on the trajectory of A048675 is squarefree, the total chance that all of them would be squarefree (which is required from the elements of A019565-cycles) is soon minuscule, especially as A048675 is not very tightly bounded (many trajectories seem to skyrocket, at least initially). I am also assuming that usually there is no significant correlation between the binary expansions of n and A048675(n) (apart from their least significant bits), or, for that matter, between their prime factorizations.
See also the slightly stronger conjecture in A285320, which implies that there would neither be any two-way infinite cycles.
If either of the conjectures is false (there are cycles), then certainly neither sequence A285332 nor its inverse A285331 can be a permutation of natural numbers. (End)
The conjecture made in A087207 (see also A288569) implies the two conjectures mentioned above. A further constraint for cycles is that in any A019565-trajectory which starts from a squarefree number (A005117), every other term is of the form 4k+2, while every other term is of the form 6k+3. - Antti Karttunen, Jun 18 2017
The sequence satisfies the exponential function identity, a(x + y) = a(x) * a(y), whenever x and y do not have a 1-bit in the same position, i.e., when A004198(x,y) = 0. See also A283475. - Antti Karttunen, Oct 31 2019
The above identity becomes unconditional if binary exclusive OR, A003987(.,.), is substituted for addition, and A059897(.,.), a multiplicative equivalent of A003987, is substituted for multiplication. This gives us a(A003987(x,y)) = A059897(a(x), a(y)). - Peter Munn, Nov 18 2019
Also the Heinz number of the binary indices of n, where the Heinz number of a sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k), and a number's binary indices (A048793) are the positions of 1's in its reversed binary expansion. - Gus Wiseman, Dec 28 2022

			5 = 2^2+2^0, e_1 = 2, e_2 = 0, prime(2+1) = prime(3) = 5, prime(0+1) = prime(1) = 2, so a(5) = 5*2 = 10.
From _Philippe Deléham_, Jun 03 2015: (Start)
This sequence regarded as a triangle withs rows of lengths 1, 1, 2, 4, 8, 16, ...:
   1;
   2;
   3,  6;
   5, 10, 15, 30;
   7, 14, 21, 42, 35,  70, 105, 210;
  11, 22, 33, 66, 55, 110, 165, 330, 77, 154, 231, 462, 385, 770, 1155, 2310;
  ...
(End)
From _Peter Munn_, Jun 14 2020: (Start)
The initial terms are shown below, equated with the product of their prime factors to exhibit the lexicographic order. We start with 1, since 1 is factored as the empty product and the empty list is first in lexicographic order.
   n     a(n)
   0     1 = .
   1     2 = 2.
   2     3 = 3.
   3     6 = 3*2.
   4     5 = 5.
   5    10 = 5*2.
   6    15 = 5*3.
   7    30 = 5*3*2.
   8     7 = 7.
   9    14 = 7*2.
  10    21 = 7*3.
  11    42 = 7*3*2.
  12    35 = 7*5.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..8191

Row 1 of A285321.
Equivalent sequences for k-th-power-free numbers: A101278 (k=3), A101942 (k=4), A101943 (k=5), A054842 (k=10).
Cf. A007088, A030308, A000040, A013929, A005117, A103785, A103786, A110765, A064273, A246353, A283475, A283477, A285319, A285331, A285332, A288569, A293442.
Cf. A109162 (iterates).
Cf. also A048675 (a left inverse), A087207, A097248, A260443, A054841.
Cf. A285315 (numbers for which a(n) < n), A285316 (for which a(n) > n).
Cf. A276076, A276086 (analogous sequences for factorial and primorial bases), A334110 (terms squared).
For partial sums see A288570.
A003961, A003987, A004198, A059897, A089913, A331590, A334747 are used to express relationships between sequence terms.
Column 1 of A329332.
Even bisection (which contains the odd terms): A332382.
A160102 composed with A052330, and subsequence of the latter.
Related to A000079 via A225546, to A057335 via A122111, to A008578 via A336322.
Least prime index of a(n) is A001511.
Greatest prime index of a(n) is A029837 or A070939.
Taking prime indices gives A048793, reverse A272020, row sums A029931.
A112798 lists prime indices, length A001222, sum A056239.
Cf. A000120, A000720, A005940, A066099, A358137, A358170.

a019565 n = product $ zipWith (^) a000040_list (a030308_row n)
-- Reinhard Zumkeller, Apr 27 2013

a:= proc(n) local i, m, r; m:=n; r:=1;
      for i while m>0 do if irem(m,2,'m')=1
        then r:=r*ithprime(i) fi od; r
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 06 2014

Do[m=1;o=1;k1=k;While[ k1>0, k2=Mod[k1, 2];If[k2\[Equal]1, m=m*Prime[o]];k1=(k1-k2)/ 2;o=o+1];Print[m], {k, 0, 55}] (* Lei Zhou, Feb 15 2005 *)
Table[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2], {n, 0, 55}]  (* Michael De Vlieger, Aug 27 2016 *)
b[0] := {1}; b[n_] := Flatten[{ b[n - 1], b[n - 1] * Prime[n] }];
  a = b[6] (* Fred Daniel Kline, Jun 26 2017 *)

a(n)=factorback(vecextract(primes(logint(n+!n,2)+1),n))  \\ M. F. Hasler, Mar 26 2011, updated Aug 22 2014, updated Mar 01 2018

from operator import mul
from functools import reduce
from sympy import prime
def A019565(n):
    return reduce(mul,(prime(i+1) for i,v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1
# Chai Wah Wu, Dec 25 2014

(define (A019565 n) (let loop ((n n) (i 1) (p 1)) (cond ((zero? n) p) ((odd? n) (loop (/ (- n 1) 2) (+ 1 i) (* p (A000040 i)))) (else (loop (/ n 2) (+ 1 i) p))))) ;; (Requires only the implementation of A000040 for prime numbers.) - Antti Karttunen, Apr 20 2017

G.f.: Product_{k>=0} (1 + prime(k+1)*x^2^k), where prime(k)=A000040(k). - Ralf Stephan, Jun 20 2003
a(n) = f(n, 1, 1) with f(x, y, z) = if x > 0 then f(floor(x/2), y*prime(z)^(x mod 2), z+1) else y. - Reinhard Zumkeller, Mar 13 2010
For all n >= 0: A048675(a(n)) = n; A013928(a(n)) = A064273(n). - Antti Karttunen, Jul 29 2015
a(n) = a(2^x)*a(2^y)*a(2^z)*... = prime(x+1)*prime(y+1)*prime(z+1)*..., where n = 2^x + 2^y + 2^z + ... - Benedict W. J. Irwin, Jul 24 2016
From Antti Karttunen, Apr 18 2017 and Jun 18 2017: (Start)
a(n) = A097248(A260443(n)), a(A005187(n)) = A283475(n), A108951(a(n)) = A283477(n).
A055396(a(n)) = A001511(n), a(A087207(n)) = A007947(n). (End)
a(2^n - 1) = A002110(n). - Michael De Vlieger, Jul 05 2017
a(n) = A225546(A000079(n)). - Peter Munn, Oct 31 2019
From Peter Munn, Mar 04 2022: (Start)
a(2n) = A003961(a(n)); a(2n+1) = 2*a(2n).
a(x XOR y) = A059897(a(x), a(y)) = A089913(a(x), a(y)), where XOR denotes bitwise exclusive OR (A003987).
a(n+1) = A334747(a(n)).
a(x+y) = A331590(a(x), a(y)).
a(n) = A336322(A008578(n+1)).
(End)

Definition corrected by Klaus-R. Löffler, Aug 20 2014

New name from Peter Munn, Jun 14 2020

A358194 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with partial sums summing to k, where k ranges from n to n(n+1)/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1

Offset: 0

Gus Wiseman, Dec 31 2022

The partial sums of a sequence (a, b, c, ...) are (a, a+b, a+b+c, ...).

			Triangle begins:
  1
  1
  1 1
  1 0 1 1
  1 0 1 1 0 1 1
  1 0 0 1 1 0 1 1 0 1 1
  1 0 0 1 1 1 1 0 1 1 0 1 1 0 1 1
  1 0 0 0 1 1 1 0 1 1 1 1 1 0 1 1 0 1 1 0 1 1
  1 0 0 0 1 1 1 1 0 1 1 1 2 1 0 1 1 1 1 1 0 1 1 0 1 1 0 1 1
For example, the T(15,59) = 5 partitions are: (8,2,2,2,1), (7,3,3,1,1), (6,5,2,1,1), (4,3,2,2,2,2), (3,3,3,3,2,1).

Row sums are A000041.
The version for compositions is A053632.
Row lengths are A152947.
The version for reversed partitions is A264034.
A048793 = partial sums of reversed standard compositions, sum A029931.
A358134 = partial sums of standard compositions, sum A359042.
A358136 = partial sums of prime indices, sum A318283.
A359361 = partial sums of reversed prime indices, sum A304818.
Cf. A000009, A325362, A358137, A359397.

Table[Length[Select[IntegerPartitions[n],Total[Accumulate[#]]==k&]],{n,0,8},{k,n,n*(n+1)/2}]

A358136 Irregular triangle read by rows whose n-th row lists the partial sums of the prime indices of n (row n of A112798).

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 4, 1, 2, 3, 2, 4, 1, 4, 5, 1, 2, 4, 6, 1, 5, 2, 5, 1, 2, 3, 4, 7, 1, 3, 5, 8, 1, 2, 5, 2, 6, 1, 6, 9, 1, 2, 3, 5, 3, 6, 1, 7, 2, 4, 6, 1, 2, 6, 10, 1, 3, 6, 11, 1, 2, 3, 4, 5, 2, 7, 1, 8, 3, 7, 1, 2, 4, 6, 12, 1, 9, 2, 8, 1, 2, 3, 6, 13

Offset: 2

Gus Wiseman, Oct 31 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.

			Triangle begins:
   2: 1
   3: 2
   4: 1 2
   5: 3
   6: 1 3
   7: 4
   8: 1 2 3
   9: 2 4
  10: 1 4
  11: 5
  12: 1 2 4
  13: 6
  14: 1 5
  15: 2 5
  16: 1 2 3 4
  17: 7
  18: 1 3 5

Row-lengths are A001222.
First element in each row is A055396.
Last element in each row is A056239.
Rows are the partial sums of rows of A112798.
Row-sums are A318283.
Sorted Heinz numbers of the rows are A325362.
The version for standard compositions is A358134.
Rows are ranked by A358137.
A000041 counts partitions, strict A000009.
A003963 multiplies prime indices.
A056239 adds up prime indices.
Cf. A000720, A001221, A355536, A358133.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Accumulate[primeMS[n]],{n,30}]

A241916 a(2^k) = 2^k, and for other numbers, if n = 2^e1 * 3^e2 * 5^e3 * ... p_k^e_k, then a(n) = 2^(e_k - 1) * 3^(e_{k-1}) * ... * p_{k-1}^e2 * p_k^(e1+1). Here p_k is the greatest prime factor of n (A006530), and e_k is its exponent (A071178), and the exponents e1, ..., e_{k-1} >= 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 8, 6, 25, 11, 27, 13, 49, 15, 16, 17, 18, 19, 125, 35, 121, 23, 81, 10, 169, 12, 343, 29, 75, 31, 32, 77, 289, 21, 54, 37, 361, 143, 625, 41, 245, 43, 1331, 45, 529, 47, 243, 14, 50, 221, 2197, 53, 36, 55, 2401, 323, 841, 59, 375, 61, 961, 175, 64

Offset: 1

Antti Karttunen, May 03 2014

nonn

For other numbers than the powers of 2 (that are fixed), this permutation reverses the sequence of exponents in the prime factorization of n from the exponent of 2 to that of the largest prime factor, except that the exponents of 2 and the greatest prime factor present are adjusted by one. Note that some of the exponents might be zeros.
Self-inverse permutation of natural numbers, composition of A122111 & A241909 in either order: a(n) = A122111(A241909(n)) = A241909(A122111(n)).
This permutation preserves both bigomega and the (index of) largest prime factor: for all n it holds that A001222(a(n)) = A001222(n) and A006530(a(n)) = A006530(n) [equally: A061395(a(n)) = A061395(n)].
From the above it follows, that this fixes both primes (A000040) and powers of two (A000079), among other numbers.
Even positions from n=4 onward contain only terms of A070003, and the odd positions only the terms of A102750, apart from 1 which is at a(1), and 2 which is at a(2).

Antti Karttunen, Table of n, a(n) for n = 1..8192
A. Karttunen, A few notes on A122111, A241909 & A241916.
Index entries for sequences that are permutations of the natural numbers

A241912 gives the fixed points; A241913 their complement.
Cf. A006530, A137502, A070003, A102750, A278525.
{A000027, A122111, A241909, A241916} form a 4-group.
The sum of prime indices of a(n) is A243503(n).
Even bisection of A358195 = Heinz numbers of rows of A358172.
A112798 lists prime indices, length A001222, sum A056239.
Cf. A005940, A019565, A161511, A246277, A253565, A326844, A356958, A358137.

nn = 65; f[n_] := If[n == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ n]; g[w_List] := Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, w]; Table[If[IntegerQ@ #, n/4, g@ Reverse@(# - Join[{1}, ConstantArray[0, Length@ # - 2], {1}] &@ f@ n)] &@ Log2@ n, {n, 4, 4 nn, 4}] (* Michael De Vlieger, Aug 27 2016 *)

A209229(n) = (n && !bitand(n,n-1));
A241916(n) = if(1==A209229(n), n, my(f = factor(2*n), nbf = #f~, igp = primepi(f[nbf,1]), g = f); for(i=1,nbf,g[i,1] = prime(1+igp-primepi(f[i,1]))); factorback(g)/2); \\ Antti Karttunen, Jul 02 2018

(define (A241916 n) (A122111 (A241909 n)))

a(1)=1, and for n>1, a(n) = A006530(n) * A137502(n)/2.
a(n) = A122111(A241909(n)) = A241909(A122111(n)).
If 2n has prime factorization Product_{i=1..k} prime(x_i), then a(n) = Product_{i=1..k-1} prime(x_k-x_i+1). The opposite version is A000027, even bisection of A246277. - Gus Wiseman, Dec 28 2022

Description clarified by Antti Karttunen, Jul 02 2018

A253565 Permutation of natural numbers: a(0) = 1, a(1) = 2; after which, a(2n) = A253550(a(n)), a(2n+1) = A253560(a(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 6, 8, 7, 25, 15, 27, 10, 18, 12, 16, 11, 49, 35, 125, 21, 75, 45, 81, 14, 50, 30, 54, 20, 36, 24, 32, 13, 121, 77, 343, 55, 245, 175, 625, 33, 147, 105, 375, 63, 225, 135, 243, 22, 98, 70, 250, 42, 150, 90, 162, 28, 100, 60, 108, 40, 72, 48, 64, 17, 169, 143, 1331, 91, 847, 539, 2401, 65, 605, 385, 1715, 275, 1225, 875, 3125, 39

Offset: 0

Antti Karttunen, Jan 03 2015

This sequence can be represented as a binary tree. Each child to the left is obtained by applying A253550 to the parent, and each child to the right is obtained by applying A253560 to the parent:
                                    1
                                    |
                 ...................2...................
                3                                       4
      5......../ \........9                   6......../ \........8
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  7       25         15       27         10       18         12       16
11 49   35  125    21  75   45  81     14  50   30  54     20  36   24  32
etc.
Sequence A253563 is the mirror image of the same tree. Also in binary trees A005940 and A163511 the terms on level of the tree are some permutation of the terms present on the level n of this tree. A252464(n) gives the distance of n from 1 in all these trees. Of these four trees, this is the one where the left child is always smaller than the right child.
Note that the indexing of sequence starts from 0, although its range starts from one.
The term a(n) is the Heinz number of the adjusted partial sums of the n-th composition in standard order, where (1) 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, (2) the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), and (3) we define the adjusted partial sums of a composition to be obtained by subtracting one from all parts, taking partial sums, and adding one back to all parts. See formula for a simplification. A triangular form is A242628. The inverse is A253566. The non-adjusted version is A358170. - Gus Wiseman, Dec 17 2022

			From _Gus Wiseman_, Dec 23 2022: (Start)
This represents the following bijection between compositions and partitions. The n-th composition in standard order together with the reversed prime indices of a(n) are:
   0:        () -> ()
   1:       (1) -> (1)
   2:       (2) -> (2)
   3:     (1,1) -> (1,1)
   4:       (3) -> (3)
   5:     (2,1) -> (2,2)
   6:     (1,2) -> (2,1)
   7:   (1,1,1) -> (1,1,1)
   8:       (4) -> (4)
   9:     (3,1) -> (3,3)
  10:     (2,2) -> (3,2)
  11:   (2,1,1) -> (2,2,2)
  12:     (1,3) -> (3,1)
  13:   (1,2,1) -> (2,2,1)
  14:   (1,1,2) -> (2,1,1)
  15: (1,1,1,1) -> (1,1,1,1)
(End)

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A253566.
Cf. A252737 (row sums), A252738 (row products).
Cf. A122111, A163511, A253550, A253560, A252464, A253563, A054429.
Applying A001222 gives A000120.
A reverse version is A005940.
These are the Heinz numbers of the rows of A242628.
Sum of prime indices of a(n) is A359043, reverse A161511.
A048793 gives partial sums of reversed standard comps, Heinz number A019565.
A066099 lists standard compositions.
A112798 list prime indices, sum A056239.
A358134 gives partial sums of standard compositions, Heinz number A358170.
Cf. A029837, A059893, A070939, A241916, A358135, A358137, A358195, A359042.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Times@@Prime/@#&/@Table[Accumulate[stc[n]-1]+1,{n,0,60}] (* Gus Wiseman, Dec 17 2022 *)

a(0) = 1, a(1) = 2; after which, a(2n) = A253550(a(n)), a(2n+1) = A253560(a(n)).
As a composition of related permutations:
a(n) = A122111(A163511(n)).
a(n) = A253563(A054429(n)).
Other identities and observations. For all n >= 0:
a(2n+1) - a(2n) > 0. [See the comment above.]
If n = 2^(x_1)+...+2^(x_k) then a(n) = Product_{i=1..k} prime(x_k-x_{i-1}-k+i) where x_0 = 0. - Gus Wiseman, Dec 23 2022

A359361 Irregular triangle read by rows whose n-th row lists the partial sums of the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 3, 4, 1, 2, 3, 2, 4, 3, 4, 5, 2, 3, 4, 6, 4, 5, 3, 5, 1, 2, 3, 4, 7, 2, 4, 5, 8, 3, 4, 5, 4, 6, 5, 6, 9, 2, 3, 4, 5, 3, 6, 6, 7, 2, 4, 6, 4, 5, 6, 10, 3, 5, 6, 11, 1, 2, 3, 4, 5, 5, 7, 7, 8, 4, 7, 2, 4, 5, 6, 12, 8, 9, 6, 8, 3, 4, 5, 6, 13

Offset: 2

Gus Wiseman, Dec 30 2022

The partial sums of a sequence (a, b, c, ...) are (a, a+b, a+b+c, ...).

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The partition with Heinz number n is the reversed n-th row of A112798.

			Triangle begins:
   2: 1
   3: 2
   4: 1 2
   5: 3
   6: 2 3
   7: 4
   8: 1 2 3
   9: 2 4
  10: 3 4
  11: 5
  12: 2 3 4
  13: 6
  14: 4 5
  15: 3 5
  16: 1 2 3 4
For example, the integer partition with Heinz number 90 is (3,2,2,1), so row n = 90 is (3,5,7,8).

Row-lengths are A001222.
The version for standard compositions is A048793, non-reversed A358134.
Last element in each row is A056239.
First element in each row is A061395
Rows are the partial sums of rows of A296150.
Row-sums are A304818.
A reverse version is A358136, row sums A318283, Heinz numbers A358137.
The sorted Heinz numbers of rows are A359397.
A000041 counts partitions, strict A000009.
A112798 lists prime indices, product A003963.
A355536 lists differences of prime indices.
Cf. A000720, A001221, A055396, A261079, A325362.

T:= n-> ListTools[PartialSums](sort([seq(numtheory
       [pi](i[1])$i[2], i=ifactors(n)[2])], `>`))[]:
seq(T(n), n=2..50);  # Alois P. Heinz, Jan 01 2023

Table[Accumulate[Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,2,30}]

A359755 Positions of first appearances in the sequence of weighted sums of prime indices (A304818).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 12, 15, 16, 18, 20, 24, 26, 28, 36, 40, 46, 48, 50, 52, 56, 62, 68, 74, 76, 86, 88, 92, 94, 106, 107, 118, 122, 124, 131, 134, 136, 142, 146, 152, 158, 164, 166, 173, 178, 188, 193, 194, 199, 202, 206, 214, 218, 226, 229, 236, 239, 254

Offset: 1

Gus Wiseman, Jan 15 2023

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 weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    6: {1,2}
    7: {4}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}

The version for standard compositions is A089633, zero-based A359756.
Positions of first appearances in A304818, reverse A318283.
The zero-based version is A359675, unsorted A359676.
The reverse zero-based version is A359680, unsorted A359681.
This is the sorted version of A359682, reverse A359679.
The reverse version is A359754.
A053632 counts compositions by weighted sum.
A112798 lists prime indices, length A001222, sum A056239.
A320387 counts multisets by weighted sum, zero-based A359678.
A358136 lists partial sums of prime indices, ranked by A358137, rev A359361.
Cf. A029931, A124757, A243055, A358194, A359497, A359674, A359683.

nn=1000;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ots[y_]:=Sum[i*y[[i]],{i,Length[y]}];
seq=Table[ots[primeMS[n]],{n,1,nn}];
Select[Range[nn],FreeQ[seq[[Range[#-1]]],seq[[#]]]&]

A362616 Numbers in whose prime factorization the greatest factor is the unique mode.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 50, 53, 54, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 89, 97, 98, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 147, 149, 150, 151, 157, 162, 163, 167

Offset: 1

Gus Wiseman, May 05 2023

nonn

First differs from A329131 in lacking 450 and having 1500.

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The factorization of 90 is 2*3*3*5, modes {3}, so 90 is missing.
The factorization of 450 is 2*3*3*5*5, modes {3,5}, so 450 is missing.
The factorization of 900 is 2*2*3*3*5*5, modes {2,3,5}, so 900 is missing.
The factorization of 1500 is 2*2*3*5*5*5, modes {5}, so 1500 is present.
The terms together with their prime indices begin:
     2: {1}          27: {2,2,2}           67: {19}
     3: {2}          29: {10}              71: {20}
     4: {1,1}        31: {11}              73: {21}
     5: {3}          32: {1,1,1,1,1}       75: {2,3,3}
     7: {4}          37: {12}              79: {22}
     8: {1,1,1}      41: {13}              81: {2,2,2,2}
     9: {2,2}        43: {14}              83: {23}
    11: {5}          47: {15}              89: {24}
    13: {6}          49: {4,4}             97: {25}
    16: {1,1,1,1}    50: {1,3,3}           98: {1,4,4}
    17: {7}          53: {16}             101: {26}
    18: {1,2,2}      54: {1,2,2,2}        103: {27}
    19: {8}          59: {17}             107: {28}
    23: {9}          61: {18}             108: {1,1,2,2,2}
    25: {3,3}        64: {1,1,1,1,1,1}    109: {29}

First term with given bigomega is A000079.
For median instead of mode we have A053263.
Partitions of this type are counted by A362612.
A112798 lists prime indices, length A001222, sum A056239.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178 ranks partitions with a unique co-mode, counted by A362610.
A362605 ranks partitions with more than one mode, counted by A362607.
A362606 ranks partitions with more than one co-mode, counted by A362609.
A362614 counts partitions by number of modes, ranked by A362611.
A362615 counts partitions by number of co-modes, ranked by A362613.
Cf. A000040, A002865, A327473, A327476, A358137, A360687.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[100],Commonest[prifacs[#]]=={Max[prifacs[#]]}&]

A359042 Sum of partial sums of the n-th composition in standard order (A066099).

Original entry on oeis.org

0, 1, 2, 3, 3, 5, 4, 6, 4, 7, 6, 9, 5, 8, 7, 10, 5, 9, 8, 12, 7, 11, 10, 14, 6, 10, 9, 13, 8, 12, 11, 15, 6, 11, 10, 15, 9, 14, 13, 18, 8, 13, 12, 17, 11, 16, 15, 20, 7, 12, 11, 16, 10, 15, 14, 19, 9, 14, 13, 18, 12, 17, 16, 21, 7, 13, 12, 18, 11, 17, 16, 22

Offset: 0

Gus Wiseman, Dec 20 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.

			The 29th composition in standard order is (1,1,2,1), with partial sums (1,2,4,5), with sum 12, so a(29) = 12.

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

See link for sequences related to standard compositions.
Each n appears A000009(n) times.
The reverse version is A029931.
Comps counted by this statistic are A053632, ptns A264034, rev ptns A358194.
This is the sum of partial sums of rows of A066099.
The version for Heinz numbers of partitions is A318283, row sums of A358136.
Row sums of A358134.
A011782 counts compositions.
A065120 gives first part of standard compositions, last A001511.
A242628 lists adjusted partial sums, ranked by A253565, row sums A359043.
A358135 gives last minus first of standard compositions.
Cf. A000120, A029837, A070939, A133494, A253566, A358133, A358137.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Total[Accumulate[stc[n]]],{n,0,100}]

A358134 Triangle read by rows whose n-th row lists the partial sums of the n-th composition in standard order (row n of A066099).

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 3, 1, 3, 1, 2, 3, 4, 3, 4, 2, 4, 2, 3, 4, 1, 4, 1, 3, 4, 1, 2, 4, 1, 2, 3, 4, 5, 4, 5, 3, 5, 3, 4, 5, 2, 5, 2, 4, 5, 2, 3, 5, 2, 3, 4, 5, 1, 5, 1, 4, 5, 1, 3, 5, 1, 3, 4, 5, 1, 2, 5, 1, 2, 4, 5, 1, 2, 3, 5, 1, 2, 3, 4, 5, 6, 5, 6, 4, 6, 4, 5

Offset: 1

Gus Wiseman, Oct 31 2022

			Triangle begins:
  1
  2
  1 2
  3
  2 3
  1 3
  1 2 3
  4
  3 4
  2 4
  2 3 4
  1 4
  1 3 4
  1 2 4
  1 2 3 4

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

See link for sequences related to standard compositions.
First element in each row is A065120.
Rows are the partial sums of rows of A066099.
Last element in each row is A070939.
An adjusted version is A242628, ranked by A253565.
The first differences instead of partial sums are A358133.
The version for Heinz numbers of partitions is A358136, ranked by A358137.
Row sums are A359042.
A011782 counts compositions.
A351014 counts distinct runs in standard compositions.
A358135 gives last minus first of standard compositions.
Cf. A000120, A001511, A029837, A029931, A048896, A070939, A133494, A253566, A355536, A357135.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Join@@Table[Accumulate[stc[n]],{n,100}]

cp's OEIS Frontend

A019565 The squarefree numbers ordered lexicographically by their prime factorization (with factors written in decreasing order). a(n) = Product_{k in I} prime(k+1), where I is the set of indices of nonzero binary digits in n = Sum_{k in I} 2^k.

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A358194 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with partial sums summing to k, where k ranges from n to n(n+1)/2.

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A358136 Irregular triangle read by rows whose n-th row lists the partial sums of the prime indices of n (row n of A112798).

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A241916 a(2^k) = 2^k, and for other numbers, if n = 2^e1 * 3^e2 * 5^e3 * ... p_k^e_k, then a(n) = 2^(e_k - 1) * 3^(e_{k-1}) * ... * p_{k-1}^e2 * p_k^(e1+1). Here p_k is the greatest prime factor of n (A006530), and e_k is its exponent (A071178), and the exponents e1, ..., e_{k-1} >= 0.

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A253565 Permutation of natural numbers: a(0) = 1, a(1) = 2; after which, a(2n) = A253550(a(n)), a(2n+1) = A253560(a(n)).

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A359361 Irregular triangle read by rows whose n-th row lists the partial sums of the integer partition with Heinz number n.

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A359755 Positions of first appearances in the sequence of weighted sums of prime indices (A304818).

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A362616 Numbers in whose prime factorization the greatest factor is the unique mode.