A057335 -id:A057335 - cp's OEIS frontend

A336321 a(n) = A122111(A225546(n)).

1, 2, 3, 4, 7, 5, 19, 6, 9, 11, 53, 10, 131, 23, 13, 8, 311, 15, 719, 22, 29, 59, 1619, 14, 49, 137, 21, 46, 3671, 17, 8161, 12, 61, 313, 37, 25, 17863, 727, 139, 26, 38873, 31, 84017, 118, 39, 1621, 180503, 20, 361, 77, 317, 274, 386093, 33, 71, 58, 733, 3673, 821641, 34, 1742537, 8167, 87, 18, 151, 67, 3681131, 626, 1627, 41, 7754077, 35, 16290047

Offset: 1

Antti Karttunen and Peter Munn, Jul 17 2020

nonn

A122111 and A225546 are both self-inverse permutations of the positive integers based on prime factorizations, and they share further common properties. For instance, they map the prime numbers to powers of 2: A122111 maps the k-th prime to 2^k, whereas A225546 maps it to 2^2^(k-1).

In composing these permutations, this sequence maps the squarefree numbers, as listed in A019565, to the prime numbers in increasing order; and the list of powers of 2 to the "normal" numbers (A055932), as listed in A057335.

			From _Peter Munn_, Jan 04 2021: (Start)
In this set of examples we consider [a(n)] as a function a(.) with an inverse, a^-1(.).
First, a table showing mapping of the powers of 2:
  n     a^-1(2^n) =    2^n =        a(2^n) =
        A001146(n-1)   A000079(n)   A057335(n)
  0             (1)         1            1
  1               2         2            2
  2               4         4            4
  3              16         8            6
  4             256        16            8
  5           65536        32           12
  6      4294967296        64           18
  ...
Next, a table showing mapping of the squarefree numbers, as listed in A019565 (a lexicographic ordering by prime factors):
  n   a^-1(A019565(n))   A019565(n)      a(A019565(n))   a^2(A019565(n))
      Cf. {A337533}      Cf. {A005117}   = prime(n)      = A033844(n-1)
  0              1               1             (1)               (1)
  1              2               2               2                 2
  2              3               3               3                 3
  3              8               6               5                 7
  4              6               5               7                19
  5             12              10              11                53
  6             18              15              13               131
  7            128              30              17               311
  8              5               7              19               719
  9             24              14              23              1619
  ...
As sets, the above columns are A337533, A005117, A008578, {1} U A033844.
Similarly, we get bijections between sets A000290\{0} -> {1} U A070003; and {1} U A335740 -> A005408 -> A066207.
(End)

A122111 composed with A225546.
Cf. A336322 (inverse permutation).
Other sequences used in a definition of this sequence: A000040, A000188, A019565, A248663, A253550, A253560.
Sequences used to express relationship between terms of this sequence: A003159, A003961, A297002, A334747.
Cf. A057335.
A mapping between the binary tree sequences A334866 and A253563.
Lists of sets (S_1, S_2, ... S_j) related by the bijection defined by the sequence: (A000290\{0}, {1} U A070003), ({1} U A001146, A000079, A055932), ({1} U A335740, A005408, A066207), (A337533, A005117, A008578, {1} U A033844).

a(n) = A122111(A225546(n)).
Alternative definition: (Start)
Write n = m^2 * A019565(j), where m = A000188(n), j = A248663(n).
a(1) = 1; otherwise for m = 1, a(n) = A000040(j), for m > 1, a(n) = A253550^j(A253560(a(m))).
(End)
a(A000040(m)) = A033844(m-1).
a(A001146(m)) = 2^(m+1).
a(2^n) = A057335(n).
a(n^2) = A253560(a(n)).
For n in A003159, a(2n) = b(a(n)), where b(1) = 2, b(n) = A253550(n), n >= 2.
More generally, a(A334747(n)) = b(a(n)).
a(A003961(n)) = A297002(a(n)).
a(A334866(m)) = A253563(m).

A336322 a(n) = A225546(A122111(n)).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 16, 9, 12, 10, 32, 15, 24, 18, 256, 30, 64, 7, 48, 27, 20, 14, 512, 36, 40, 81, 96, 21, 128, 42, 65536, 54, 60, 72, 1024, 35, 120, 45, 768, 70, 192, 105, 80, 162, 28, 210, 131072, 25, 144, 90, 160, 11, 4096, 108, 1536, 135, 56, 22, 2048, 33, 84, 243, 4294967296, 216, 384, 66, 240, 270, 288, 55, 262144, 110, 168, 324, 480, 50

Offset: 1

Antti Karttunen and Peter Munn, Jul 17 2020

nonn

A225546 and A122111 are both self-inverse permutations of the positive integers based on prime factorizations, and they share further common properties. For instance, they map the prime numbers to powers of 2: A225546 maps the k-th prime to 2^2^(k-1), whereas A122111 maps it to 2^k.

In composing these permutations, this sequence maps the list of prime numbers to the squarefree numbers, as listed in A019565; and the "normal" numbers (A055932), as listed in A057335, to ascending powers of 2.

Index entries for sequences that are permutations of the natural numbers

A225546 composed with A122111.
Sorted even bisection: A335738.
Sorted odd bisection (excluding 1): A335740.
Sequences used to express relationship between terms of this sequence: A001222, A003961, A253560, A331590, A350066.
Sequences of sequences (S_1, S_2, ... S_j) with the property a(S_i) = S_{i+1}, or essentially so: (A033844, A000040, A019565), (A057335, A000079, A001146), (A000244, A011764), (A001248, A334110), (A253563, A334866).
The inverse permutation, A336321, lists sequences where the property is weaker (between the sets of terms).

a(A033844(m)) = A000040(m+1). [Offset corrected Peter Munn, Feb 14 2022]
a(A000040(m)) = A019565(m).
a(A057335(m)) = 2^m.
For m >= 1, a(2^m) = A001146(m-1).
a(A253563(m)) = A334866(m).
From Peter Munn, Feb 14 2022: (Start)
a(A253560(n)) = a(n)^2.
For n >= 2, a(A003961(n)) = A331590(a(n), 2^2^(A001222(n)-1)).
a(A350066(n, k)) = A331590(a(n), a(k)).
(End)

A334032 The a(n)-th composition in standard order (graded reverse-lexicographic) is the unsorted prime signature of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 5, 1, 3, 3, 8, 1, 6, 1, 5, 3, 3, 1, 9, 2, 3, 4, 5, 1, 7, 1, 16, 3, 3, 3, 10, 1, 3, 3, 9, 1, 7, 1, 5, 5, 3, 1, 17, 2, 6, 3, 5, 1, 12, 3, 9, 3, 3, 1, 11, 1, 3, 5, 32, 3, 7, 1, 5, 3, 7, 1, 18, 1, 3, 6, 5, 3, 7, 1, 17, 8, 3, 1, 11

Offset: 1

Gus Wiseman, Apr 17 2020

nonn

Unsorted prime signature (A124010) is the sequence of exponents in a number's prime factorization.

The k-th composition in standard order (row k of 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 unsorted prime signature of 12345678 is (1,2,1,1), which is the 27th composition in standard order, so a(12345678) = 27.

Positions of first appearances are A057335 (a partial inverse).
Least number with same prime signature is A071364.
Unsorted prime signature is A124010.
Least number with reversed prime signature is A331580.
Minimal numbers with standard reversed prime signatures are A334031.
The reversed version is A334033.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Aperiodic compositions are A328594.
- Normal compositions are A333217.
- Permutations are A333218.
- Heinz number is A333219.
Cf. A029931, A048793, A052409, A055932, A056239, A112798, A124767, A228351, A233249, A329139, A333220.

stcinv[q_]:=Total[2^Accumulate[Reverse[q]]]/2;
Table[stcinv[Last/@If[n==1,{},FactorInteger[n]]],{n,100}]

a(A057335(n)) = n.
A057335(a(n)) = A071364(n).
a(A334031(n))= A059893(n).
A334031(a(n)) = A331580(n).

A345974 Decimal expansion of Sum_{k>=0} Product_{i=1..k} 1/(prime(i)-1).

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Jun 30 2021

			2.64810175970147102337088414536064714927258700214033932076744547925274...

Peter Cameron, A new constant?, Blog Post, May 23 2021.
Peter Cameron, A new constant?, local copy of Blog Post, with permission.

Cf. A005867, A055932, A057335, A345965.

c:= sum(product(1/(ithprime(i)-1), i=1..k), k=0..infinity):
evalf(c, 140);  # Alois P. Heinz, Jun 30 2021

PARI
```
suminf(k=0,prod(i=1,k,1/(prime(i)-1)))
```

Equals Sum_{k>=0} 1/A005867(k) = Sum_{k>=1} 1/A055932(k) = Sum_{k>=1} 1/A057335(k). - Amiram Eldar, Jun 26 2025

A095209 a(0) = 1, and for n > 0, a(n) = the least multiple of prime(n) such that the geometric mean of a(0) to a(n) is an integer.

Original entry on oeis.org

1, 4, 54, 3750, 504210, 372027810, 144949074270, 209481995953230, 164735296593157290, 401824316553919068810, 2721846739094340967339230, 5095936579799734140259818030, 48850362989361131638352534231610

Offset: 0

Amarnath Murthy, Jun 08 2004

nonn

			(1*4*54*3750)^(1/4) = 30.

Christian Krause, Jamie Morken, et al., A mined LODA assembly source for this sequence
Don Reble, Python program

Cf. A000040, A002110, A020522, A057335, A095210, A307539.

{1}~Join~MapIndexed[Prime[First[#2]]^First[#2]*#1 &, FoldList[Times, Prime@ Range[12]]] (* Michael De Vlieger, Jul 01 2022 *)

A002110(n)=prod(i=1, n, prime(i));
A095209(n) = if(0==n, 1, prime(n)^(n)*A002110(n)); \\ Antti Karttunen, Jun 28 2022

From Antti Karttunen and Peter Munn, May 04 2022: (Start)
The n-th partial product of these terms = A002110(n)^(1+n), i.e., the n-th geometric mean is the n-th power of (n-1)-th primorial.
a(n) = A002110(n) * A307539(n).
a(n) = A057335(A020522(n)). [Found by LODA-miner, follows from the above formulas]
(End)

Edited by Don Reble, Jan 06 2007

Starting offset changed from 1 to 0 and the definition accordingly edited by Antti Karttunen, May 04 2022

A333942 Number of multiset partitions of a multiset whose multiplicities are the parts of the n-th composition in standard order.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 5, 5, 7, 9, 11, 7, 11, 11, 15, 7, 12, 16, 21, 16, 26, 26, 36, 12, 21, 26, 36, 21, 36, 36, 52, 11, 19, 29, 38, 31, 52, 52, 74, 29, 52, 66, 92, 52, 92, 92, 135, 19, 38, 52, 74, 52, 92, 92, 135, 38, 74, 92, 135, 74, 135, 135, 203, 15, 30, 47

Offset: 0

Gus Wiseman, Apr 16 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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 a(1) = 1 through a(11) = 11 multiset partitions:
  {1}  {11}    {12}    {111}      {112}      {122}      {123}
       {1}{1}  {1}{2}  {1}{11}    {1}{12}    {1}{22}    {1}{23}
                       {1}{1}{1}  {2}{11}    {2}{12}    {2}{13}
                                  {1}{1}{2}  {1}{2}{2}  {3}{12}
                                                        {1}{2}{3}
  {1111}        {1112}        {1122}        {1123}
  {1}{111}      {1}{112}      {1}{122}      {1}{123}
  {11}{11}      {11}{12}      {11}{22}      {11}{23}
  {1}{1}{11}    {2}{111}      {12}{12}      {12}{13}
  {1}{1}{1}{1}  {1}{1}{12}    {2}{112}      {2}{113}
                {1}{2}{11}    {1}{1}{22}    {3}{112}
                {1}{1}{1}{2}  {1}{2}{12}    {1}{1}{23}
                              {2}{2}{11}    {1}{2}{13}
                              {1}{1}{2}{2}  {1}{3}{12}
                                            {2}{3}{11}
                                            {1}{1}{2}{3}

The described multiset has A000120 distinct parts.
The sum of the described multiset is A029931.
Multisets of compositions are A034691.
The described multiset is a row of A095684.
Combinatory separations of normal multisets are A269134.
The product of the described multiset is A284001.
The version for prime indices is A318284.
The version counting combinatory separations is A334030.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Length of Lyndon factorization is A329312.
- Dealings are counted by A333939.
- Distinct parts are counted by A334028.
- Length of co-Lyndon factorization is A334029.
Cf. A057335, A065609, A275692, A292884, A318560, A318563, A326774, A333764, A333765, A333940.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ptnToNorm[y_]:=Join@@Table[ConstantArray[i,y[[i]]],{i,Length[y]}];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[facs[Times@@Prime/@ptnToNorm[stc[n]]]],{n,0,30}]

a(n) = A001055(A057335(n)).

A334031 The smallest number whose unsorted prime signature is the reversed n-th composition in standard order.

Original entry on oeis.org

1, 2, 4, 6, 8, 18, 12, 30, 16, 54, 36, 150, 24, 90, 60, 210, 32, 162, 108, 750, 72, 450, 300, 1470, 48, 270, 180, 1050, 120, 630, 420, 2310, 64, 486, 324, 3750, 216, 2250, 1500, 10290, 144, 1350, 900, 7350, 600, 4410, 2940, 25410, 96, 810, 540, 5250, 360, 3150

Offset: 0

Gus Wiseman, Apr 17 2020

nonn

All terms are normal (A055932), meaning their prime indices cover an initial interval of positive integers.
Unsorted prime signature is the sequence of exponents in a number's prime factorization.
The k-th composition in standard order (row k of 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 sequence of terms together with their prime indices begins:
       1: {}
       2: {1}
       4: {1,1}
       6: {1,2}
       8: {1,1,1}
      18: {1,2,2}
      12: {1,1,2}
      30: {1,2,3}
      16: {1,1,1,1}
      54: {1,2,2,2}
      36: {1,1,2,2}
     150: {1,2,3,3}
      24: {1,1,1,2}
      90: {1,2,2,3}
      60: {1,1,2,3}
     210: {1,2,3,4}
      32: {1,1,1,1,1}
     162: {1,2,2,2,2}
For example, the 13th composition in standard order is (1,2,1), and the least number with prime signature (1,2,1) is 90 = 2^1 * 3^2 * 5^1, so a(13) = 90.

The range is A055932.
The non-reversed version is A057335.
Unsorted prime signature is A124010.
Numbers whose prime signature is aperiodic are A329139.
Normal numbers with standard compositions as prime signature are A334032.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Aperiodic compositions are A328594.
- Normal compositions are A333217.
- Heinz number is A333219.
Cf. A029931, A048793, A056239, A112798, A228351, A233249, A272020, A333218, A333220, A334032.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Product[Prime[i]^stc[n][[-i]],{i,DigitCount[n,2,1]}],{n,0,100}]

a(n) = A057335(A059893(n)).

A334033 The a(n)-th composition in standard order (graded reverse-lexicographic) is the reversed unsorted prime signature of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 6, 1, 3, 3, 8, 1, 5, 1, 6, 3, 3, 1, 12, 2, 3, 4, 6, 1, 7, 1, 16, 3, 3, 3, 10, 1, 3, 3, 12, 1, 7, 1, 6, 6, 3, 1, 24, 2, 5, 3, 6, 1, 9, 3, 12, 3, 3, 1, 14, 1, 3, 6, 32, 3, 7, 1, 6, 3, 7, 1, 20, 1, 3, 5, 6, 3, 7, 1, 24, 8, 3, 1

Offset: 1

Gus Wiseman, Apr 18 2020

nonn

Unsorted prime signature (A124010) is the sequence of exponents in a number's prime factorization.

The k-th composition in standard order (row k of 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 unsorted prime signature of 12345678 is (1,2,1,1), whose reverse (1,1,2,1) is the 29th composition in standard order, so a(12345678) = 29.

Positions of first appearances are A334031.
The non-reversed version is A334032.
Unsorted prime signature is A124010.
Least number with reversed prime signature is A331580.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Aperiodic compositions are A328594.
- Normal compositions are A333217.
- Permutations are A333218.
- Heinz number is A333219.
Cf. A029931, A048793, A052409, A055932, A056239, A057335, A071364, A112798, A124767, A228351, A233249, A329139, A333220.

stcinv[q_]:=Total[2^Accumulate[Reverse[q]]]/2;
Table[stcinv[Reverse[Last/@If[n==1,{},FactorInteger[n]]]],{n,100}]

a(A334031(n)) = n.
A334031(a(n)) = A071364(n).
a(A057335(n))= A059893(n).
A057335(a(n)) = A331580(n).

A057334 In A000120, replace each entry k with the k-th prime and replace 0 with 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 5, 2, 3, 3, 5, 3, 5, 5, 7, 2, 3, 3, 5, 3, 5, 5, 7, 3, 5, 5, 7, 5, 7, 7, 11, 2, 3, 3, 5, 3, 5, 5, 7, 3, 5, 5, 7, 5, 7, 7, 11, 3, 5, 5, 7, 5, 7, 7, 11, 5, 7, 7, 11, 7, 11, 11, 13, 2, 3, 3, 5, 3, 5, 5, 7, 3, 5, 5, 7, 5, 7, 7, 11, 3, 5, 5, 7

Offset: 0

Alford Arnold, Aug 27 2000

Michael Gilleland, Some Self-Similar Integer Sequences

Cf. A000040, A000120, A008578, A055932, A057335.

a(n) = if (n==0, 1, prime(hammingweight(n))); \\ Michel Marcus, Feb 08 2014

a(n) = A000040(A000120(n)) for n>0.

a(n) = A008578(1 + A000120(n)). - Alois P. Heinz, Nov 03 2024

More terms from Michel Marcus, Feb 08 2014

A084918 Numbers n >= 1000, such that if prime P divides n, then so does each smaller prime.

Original entry on oeis.org

1024, 1050, 1080, 1152, 1200, 1260, 1296, 1350, 1440, 1458, 1470, 1500, 1536, 1620, 1680, 1728, 1800, 1890, 1920, 1944, 2048, 2100, 2160, 2250, 2304, 2310, 2400, 2430, 2520, 2592, 2700, 2880, 2916, 2940, 3000, 3072, 3150, 3240, 3360, 3456, 3600, 3750

Offset: 0

Alford Arnold, Jul 15 2003

A055932 lists terms below 1000.

Cf. A055932, A056808, A057335, A066099, A071364, A080259, A080404.

espQ[n_]:=Module[{f=FactorInteger[n][[All,1]]},Prime[Range[ PrimePi[ Max[f]]]] == f]; Select[Range[1000,4000],espQ] (* Harvey P. Dale, Mar 09 2019 *)

Edited by Don Reble, Nov 03 2003

cp's OEIS Frontend

A336321 a(n) = A122111(A225546(n)).

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A336322 a(n) = A225546(A122111(n)).

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A334032 The a(n)-th composition in standard order (graded reverse-lexicographic) is the unsorted prime signature of n.

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A345974 Decimal expansion of Sum_{k>=0} Product_{i=1..k} 1/(prime(i)-1).

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Maple

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A095209 a(0) = 1, and for n > 0, a(n) = the least multiple of prime(n) such that the geometric mean of a(0) to a(n) is an integer.

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A333942 Number of multiset partitions of a multiset whose multiplicities are the parts of the n-th composition in standard order.

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A334031 The smallest number whose unsorted prime signature is the reversed n-th composition in standard order.

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A334033 The a(n)-th composition in standard order (graded reverse-lexicographic) is the reversed unsorted prime signature of n.

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