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.

A361809 Fixed points of A181820 and A361808.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 15, 46, 58, 817, 5494, 8502
Offset: 1

Views

Author

Pontus von Brömssen, Mar 25 2023

Keywords

Comments

Numbers k such that the partition with Heinz number k is identical to the partition given by the prime signature of A025487(k).
There are no more terms below 10177058 = A025488(143).

Examples

			4 is a term because the partition with Heinz number 4 = 2^2 = prime(1)^2 is (1,1), which is identical to the partition given by the prime signature of A025487(4) = 6 = 2^1*3^1.
15 is a term because the partition with Heinz number 15 = 3*5 = prime(2)*prime(3) is (2,3), which is identical to the partition given by the prime signature of A025487(15) = 72 = 2^3*3^2.
8502 is a term because the partition with Heinz number 8502 = 2*3*13*109 = prime(1)*prime(2)*prime(6)*prime(29) is (1,2,6,29), which is identical to the partition given by the prime signature of A025487(8502) = 68491306598400 = 2^29*3^6*5^2*7.

Crossrefs

Cf. A025487, A025488, A181820, A361808.

Formula

A181820(a(n)) = A361808(a(n)) = a(n).

A181821 a(n) = smallest integer with factorization as Product p(i)^e(i) such that Product p(e(i)) = n.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 30, 36, 24, 32, 60, 64, 48, 72, 210, 128, 180, 256, 120, 144, 96, 512, 420, 216, 192, 900, 240, 1024, 360, 2048, 2310, 288, 384, 432, 1260, 4096, 768, 576, 840, 8192, 720, 16384, 480, 1800, 1536, 32768, 4620, 1296, 1080, 1152, 960, 65536
Offset: 1

Views

Author

Matthew Vandermast, Dec 07 2010

Keywords

Comments

A permutation of A025487. a(n) is the member m of A025487 such that A181819(m) = n. a(n) is also the member of A025487 whose prime signature is conjugate to the prime signature of A108951(n).
If n = Product_i prime(e(i)) with the e(i) weakly decreasing, then a(n) = Product_i prime(i)^e(i). For example, 90 = prime(3) * prime(2) * prime(2) * prime(1), so a(90) = prime(1)^3 * prime(2)^2 * prime(3)^2 * prime(4)^1 = 12600. - Gus Wiseman, Jan 02 2019

Examples

			The canonical factorization of 24 is 2^3*3^1. Therefore, p(e(i)) = prime(3)*prime(1)(i.e., A000040(3)*A000040(1)), which equals 5*2 = 10. Since 24 is the smallest integer for which p(e(i)) = 10, a(10) = 24.

Links

Crossrefs

Other rearrangements of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181822.
Cf. A046523, A181819, A181820.
Cf. A000040, A001221, A001222, A002110, A056239, A071625, A112798, A118914, A122111, A124859, A182850, A305936, A361808.

Programs

  • Maple
    a:= n-> (l-> mul(ithprime(i)^l[i], i=1..nops(l)))(sort(map(i->
             numtheory[pi](i[1])$i[2], ifactors(n)[2]), `>`)):
seq(a(n), n=1..70);  # Alois P. Heinz, Sep 05 2018
  • Mathematica
    With[{s = Array[If[# == 1, 1, Times @@ Map[Prime@ Last@ # &, FactorInteger@ #]] &, 2^16]}, Array[First@ FirstPosition[s, #] &, LengthWhile[Differences@ Union@ s, # == 1 &]]] (* Michael De Vlieger, Dec 17 2018 *)
Table[Times@@MapIndexed[Prime[#2[[1]]]^#1&,Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,30}] (* Gus Wiseman, Jan 02 2019 *)
  • PARI
    A181821(n) = { my(f=factor(n),p=0,m=1); forstep(i=#f~,1,-1,while(f[i,2], f[i,2]--; m *= (p=nextprime(p+1))^primepi(f[i,1]))); (m); }; \\ Antti Karttunen, Dec 10 2018
  • Python
    from math import prod
from sympy import prime, primepi, factorint
def A181821(n): return prod(prime(i)**e for i, e in enumerate(sorted(map(primepi,factorint(n,multiple=True)),reverse=True),1)) # Chai Wah Wu, Sep 15 2023

Formula

If A108951(n) = Product p(i)^e(i), then a(n) = Product A002110(e(i)). I.e., a(n) = A108951(A181819(A108951(n))).
a(A181819(n)) = A046523(n). - [See also A124859]. Antti Karttunen, Dec 10 2018
a(n) = A025487(A361808(n)). - Pontus von Brömssen, Mar 25 2023
a(n) = A108951(A122111(n)). - Antti Karttunen, Sep 15 2023

Extensions

Definition corrected by Gus Wiseman, Jan 02 2019

A181820 a(1) = 1; for n > 1, if A025487(n) = Product p(i)^e(i), then a(n) = Product p(e(i)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 8, 11, 9, 14, 12, 13, 15, 22, 20, 17, 21, 18, 26, 16, 25, 28, 19, 33, 30, 34, 24, 35, 44, 23, 39, 42, 38, 40, 55, 27, 52, 29, 50, 51, 36, 49, 66, 46, 56, 65, 45, 68, 31, 70, 57, 32, 60, 77, 78, 58, 88, 85, 63, 76, 37, 110, 69, 48, 84, 91, 75, 102, 62, 54, 98, 104, 95
Offset: 1

Views

Author

Matthew Vandermast, Dec 07 2010

Keywords

Comments

A permutation of the positive integers.
The partition given by the prime signature of A025487(n) has Heinz number a(n). - Pontus von Brömssen, Mar 25 2023

Examples

			A025487(8) = 24 = 2^3*3 has the exponents (3,1) in its canonical prime factorization. Accordingly, a(8) = prime(3)*prime(1) (i.e., A000040(3)*A000040(1)), which equals 5*2=10.

Links

Crossrefs

A181815 is another mapping from the members of A025487 to the positive integers. Also see A181819, A181821.
Cf. A000040, A122111, A361808 (inverse), A361809 (fixed points).

Formula

a(n) = A181819(A025487(n)).
a(n) = A122111(A181815(n)).
Showing 1-3 of 3 results.

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