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 1-3 of 3 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

Views

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

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

Views

Author

Gus Wiseman, Apr 17 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), which is the 27th composition in standard order, so a(12345678) = 27.

Crossrefs

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.

Programs

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

Formula

a(A057335(n)) = n.
A057335(a(n)) = A071364(n).
a(A334031(n))= A059893(n).
A334031(a(n)) = A331580(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 1-3 of 3 results.

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