A071364 -id:A071364 - cp's OEIS frontend

A083258 a(n) = gcd(A046523(n), n).

1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 3, 16, 1, 6, 1, 4, 3, 2, 1, 24, 1, 2, 1, 4, 1, 30, 1, 32, 3, 2, 1, 36, 1, 2, 3, 8, 1, 6, 1, 4, 3, 2, 1, 48, 1, 2, 3, 4, 1, 6, 1, 8, 3, 2, 1, 60, 1, 2, 3, 64, 1, 6, 1, 4, 3, 10, 1, 72, 1, 2, 3, 4, 1, 6, 1, 16, 1, 2, 1, 12, 1, 2, 3, 8, 1, 30, 1, 4, 3, 2, 1, 96, 1, 2

Offset: 1

Labos Elemer, May 09 2003

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A046523, A071364, A083255-A083260, A083261.

Table[GCD[n, Times @@ MapIndexed[Prime[First[#2]]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]], {n, 98}] (* Michael De Vlieger, May 21 2017 *)

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A083258(n) = gcd(n,A046523(n)); \\ Antti Karttunen, May 21 2017

A083261 a(n) = gcd(A046523(n+1), A046523(n)).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 6, 6, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 2, 6, 6, 6, 2, 2, 6, 6, 2, 2, 2, 2, 12, 6, 2, 2, 4, 4, 6, 6, 2, 2, 6, 6, 6, 6, 2, 2, 2, 2, 6, 4, 2, 6, 2, 2, 6, 6, 2, 2, 2, 2, 6, 12, 6, 6, 2, 2, 16, 2, 2, 2, 6, 6, 6, 6, 2, 2, 6, 6, 6, 6, 6, 6, 2, 2, 12, 12, 2, 2, 2, 2

Offset: 1

Labos Elemer, May 09 2003

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A046523, A071364, A083255, A083256, A083257, A083258, A083259, A083260, A286255.

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A083261(n) = gcd(A046523(n),A046523(1+n)); \\ From Antti Karttunen, May 21 2017

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

Gus Wiseman, Apr 18 2020

nonn

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

			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.

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.

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

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).

A335286 n is the a(n)-th positive integer having its sequence of exponents in canonical prime factorization.

Original entry on oeis.org

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

Offset: 1

David A. Corneth, May 30 2020

			a(14) = 3 as 14 has prime signature [1, 1] and it's the third positive integer having that prime signature, after 6 and 10.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A046523, A064839, A071364, A120426, A254524.

p:= proc() 0 end:
a:= proc(n) option remember; local t; a(n-1); t:=
      (l-> mul(ithprime(i)^l[i][2], i=1..nops(l)
       ))(sort(ifactors(n)[2])); p(t):= p(t)+1
    end: a(0):=0:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 01 2020

A071364[n_] := If[n == 1, 1, With[{f = FactorInteger[n]}, Times @@ (Prime[Range[Length[f]]]^f[[All, 2]])]];
Module[{b}, b[_] = 0;
a[n_] := With[{t = A071364[n]}, b[t] = b[t] + 1]];
Array[a, 105] (* Jean-François Alcover, Jan 10 2022 *)

first(n) = { my(m = Map(), res = vector(n)); for(i = 1, n, c = factor(i)[,2]; if(mapisdefined(m, c), res[i] = mapget(m, c) + 1; mapput(m, c, res[i]) , res[i] = 1; mapput(m, c, 1) ) ); res }

Ordinal transform of A071364. - Alois P. Heinz, Jun 01 2020

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

A096153 Natural numbers (greater than 1) arranged in rows according to their ordered prime signature. Square array A(n,k) read by descending antidiagonals.

Original entry on oeis.org

2, 3, 4, 5, 9, 6, 7, 25, 10, 8, 11, 49, 14, 27, 12, 13, 121, 15, 125, 20, 16, 17, 169, 21, 343, 28, 81, 18, 19, 289, 22, 1331, 44, 625, 50, 24, 23, 361, 26, 2197, 45, 2401, 75, 40, 30, 29, 529, 33, 4913, 52, 14641, 98, 56, 42, 32, 31, 841, 34, 6859, 63, 28561, 147, 88, 66

Offset: 1

Alford Arnold, Jul 24 2004

The first row is A000040 (the prime numbers) and the first column is A055932 (the Quet prime signatures).

If we restrict the terms to those having ordered prime signatures that are not represented in A025487 (the least prime signature sequence), we get A096011.

			18 = 2^1 * 3^2, so has ordered prime signature (1,2) given by the exponents in the factorization shown. No earlier number has this prime signature, so 18 is placed at the start of the next empty row (row 7). Thus A(7,1) = 18.

For m >= 2, A077462/A335286 essentially give the row/column containing m.
Cf. A071364, A357126.
See the comments for the relationships with A000040, A025487, A055932, A096011.

Edited by Peter Munn, Oct 23 2023

cp's OEIS Frontend

A083258 a(n) = gcd(A046523(n), n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A083261 a(n) = gcd(A046523(n+1), A046523(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A335286 n is the a(n)-th positive integer having its sequence of exponents in canonical prime factorization.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A096153 Natural numbers (greater than 1) arranged in rows according to their ordered prime signature. Square array A(n,k) read by descending antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions