cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A071364 Smallest number with same sequence of exponents in canonical prime factorization as n.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 12, 2, 6, 6, 16, 2, 18, 2, 12, 6, 6, 2, 24, 4, 6, 8, 12, 2, 30, 2, 32, 6, 6, 6, 36, 2, 6, 6, 24, 2, 30, 2, 12, 12, 6, 2, 48, 4, 18, 6, 12, 2, 54, 6, 24, 6, 6, 2, 60, 2, 6, 12, 64, 6, 30, 2, 12, 6, 30, 2, 72, 2, 6, 18, 12, 6, 30, 2, 48, 16, 6, 2, 60, 6, 6, 6, 24
Offset: 1

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Author

Reinhard Zumkeller, May 21 2002

Keywords

Comments

A046523(a(n))=A046523(n); A046523(n)<=a(n)<=n; A001221(a(n))=A001221(n), A001222(a(n))=A001222(n); A020639(a(n))=2, A006530(a(n))=A000040(A001221(n))<=A006530(n); A000005(a(n))=A000005(n);
a(a(n))=a(n); a(n)=2^k iff n=p^k, p prime, k>0 (A000961); if n>1 is not a prime power, then a(n) mod 6 = 0; range of values = A055932, as distinct prime factors of a(n) are consecutive: a(n)=n iff n=A055932(k) for some k;
a(A003586(n))=A003586(n).

Examples

			a(105875) = a(5*5*5*7*11*11) = 2*2*2*3*5*5 = 600.

Links

Crossrefs

Cf. A071365, A071366.
Cf. A000040.
Cf. A085079, A089247.
The range is A055932.
The reversed version is A331580.
Unsorted prime signature is A124010.
Numbers whose prime signature is aperiodic are A329139.
Cf. A056239, A112798, A233249, A333217, A333219, A334032, A334033.

Programs

  • Haskell
    a071364 = product . zipWith (^) a000040_list . a124010_row
-- Reinhard Zumkeller, Feb 19 2012
  • Mathematica
    Table[ e = Last /@ FactorInteger[n]; Product[Prime[i]^e[[i]], {i, Length[e]}], {n, 88}] (* Ray Chandler, Sep 23 2005 *)
  • PARI
    a(n) = f = factor(n); for (i=1, #f~, f[i,1] = prime(i)); factorback(f); \\ Michel Marcus, Jun 13 2014
  • Python
    from math import prod
from sympy import prime, factorint
def A071364(n): return prod(prime(i+1)**p[1] for i,p in enumerate(sorted(factorint(n).items()))) # Chai Wah Wu, Sep 16 2022

Formula

In prime factorization of n, replace least prime by 2, next least by 3, etc.
a(n) = product(A000040(k)^A124010(k): k=1..A001221(n)). - Reinhard Zumkeller, Apr 27 2013

Extensions

Extended by Ray Chandler, Sep 23 2005

A057335 a(0) = 1, and for n > 0, a(n) = A000040(A000120(n)) * a(floor(n/2)); essentially sequence A055932 generated using A000120, hence sorted by number of factors.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 18, 30, 16, 24, 36, 60, 54, 90, 150, 210, 32, 48, 72, 120, 108, 180, 300, 420, 162, 270, 450, 630, 750, 1050, 1470, 2310, 64, 96, 144, 240, 216, 360, 600, 840, 324, 540, 900, 1260, 1500, 2100, 2940, 4620, 486, 810, 1350, 1890, 2250, 3150, 4410
Offset: 0

Views

Author

Alford Arnold, Aug 27 2000

Keywords

Comments

Note that for n>0 the prime divisors of a(n) are consecutive primes starting with 2. All of the least prime signatures (A025487) are included; with the other values forming A056808.
Using the formula, terms of b(n)= a(n)/A057334(n) are: 1, 1, 2, 2, 4, 4, 6, 6, 8, ..., indeed a(n) repeated. - Michel Marcus, Feb 09 2014
a(n) is the unique normal number whose unsorted prime signature is the k-th composition in standard order (graded reverse-lexicographic). This composition (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. A number is normal if its prime indices cover an initial interval of positive integers. Unsorted prime signature is the sequence of exponents in a number's prime factorization. - Gus Wiseman, Apr 19 2020

Examples

			From _Gus Wiseman_, Apr 19 2020: (Start)
The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      4: {1,1}
      6: {1,2}
      8: {1,1,1}
     12: {1,1,2}
     18: {1,2,2}
     30: {1,2,3}
     16: {1,1,1,1}
     24: {1,1,1,2}
     36: {1,1,2,2}
     60: {1,1,2,3}
     54: {1,2,2,2}
     90: {1,2,2,3}
    150: {1,2,3,3}
    210: {1,2,3,4}
     32: {1,1,1,1,1}
     48: {1,1,1,1,2}
For example, the 27th composition in standard order is (1,2,1,1), and the normal number with prime signature (1,2,1,1) is 630 = 2*3*3*5*7, so a(27) = 630.
(End)

Links

Crossrefs

Cf. A000120, A057334, A055932 and A056808.
Cf. A324939.
Unsorted prime signature is A124010.
Numbers whose prime signature is aperiodic are A329139.
The reversed version is A334031.
A partial inverse is 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.
- Permutations are A333218.
- Heinz number is A333219.
Cf. A005867, A029931, A048793, A052409, A056239, A066099, A112798, A124767, A228351, A233249, A333220, A345974.
Related to A019565 via A122111 and to A000079 via A336321.

Programs

  • Mathematica
    Table[Times @@ Map[If[# == 0, 1, Prime@ #] &, Accumulate@ IntegerDigits[n, 2]], {n, 0, 54}] (* Michael De Vlieger, May 23 2017 *)
  • PARI
    mg(n) = if (n==0, 1, prime(hammingweight(n))); \\ A057334
lista(nn) = {my(v = vector(nn)); v[1] = 1; for (i=2, nn, v[i] = mg(i-1)*v[(i+1)\2];); v;} \\ Michel Marcus, Feb 09 2014
  • PARI
    A057335(n) = if(0==n,1,prime(hammingweight(n))*A057335(n\2)); \\ Antti Karttunen, Jul 20 2020

Formula

a(n) = A057334(n) * a (repeated).
A334032(a(n)) = n; a(A334032(n)) = A071364(n). - Gus Wiseman, Apr 19 2020
a(n) = A122111(A019565(n)); A019565(n) = A122111(a(n)). - Peter Munn, Jul 18 2020
a(n) = A336321(2^n). - Peter Munn, Mar 04 2022
Sum_{n>=0} 1/a(n) = Sum_{n>=0} 1/A005867(n) = 2.648101... (A345974). - Amiram Eldar, Jun 26 2025

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003
New primary name from Antti Karttunen, Jul 20 2020

A331580 Smallest number whose unsorted prime signature is the reversed unsorted prime signature of n.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 18, 2, 6, 6, 16, 2, 12, 2, 18, 6, 6, 2, 54, 4, 6, 8, 18, 2, 30, 2, 32, 6, 6, 6, 36, 2, 6, 6, 54, 2, 30, 2, 18, 18, 6, 2, 162, 4, 12, 6, 18, 2, 24, 6, 54, 6, 6, 2, 150, 2, 6, 18, 64, 6, 30, 2, 18, 6, 30, 2, 108, 2, 6, 12, 18, 6
Offset: 1

Views

Author

Gus Wiseman, Apr 18 2020

Keywords

Comments

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

Examples

			The prime signature of 12345678 = 2*3*3*47*14593 is (1,2,1,1), and the least number with prime signature (1,1,2,1) is 1050 = 2*3*5*5*7, so a(12345678) = 1050.

Crossrefs

The range is A055932.
The non-reversed version is A071364.
Unsorted prime signature is A124010.
Numbers whose prime signature is aperiodic are A329139.
Cf. A056239, A112798, A233249, A333217, A333219, A334032, A334033.

Programs

  • Mathematica
    ptnToNorm[y_]:=Join@@Table[ConstantArray[i,y[[i]]],{i,Length[y]}];
Table[Times@@Prime/@ptnToNorm[Reverse[Last/@If[n==1,{},FactorInteger[n]]]],{n,100}]

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

Views

Author

Gus Wiseman, Apr 17 2020

Keywords

Comments

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.

Examples

			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.

Crossrefs

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.

Programs

  • Mathematica
    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}]

Formula

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

Views

Author

Gus Wiseman, Apr 18 2020

Keywords

Comments

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.

Examples

			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.

Crossrefs

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.

Programs

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

Formula

a(A334031(n)) = n.
A334031(a(n)) = A071364(n).
a(A057335(n))= A059893(n).
A057335(a(n)) = A331580(n).
Showing 1-5 of 5 results.

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