A129598 -id:A129598 - cp's OEIS frontend

A253560 Multiply n by its largest prime factor: a(n) = A006530(n) * n.

1, 4, 9, 8, 25, 18, 49, 16, 27, 50, 121, 36, 169, 98, 75, 32, 289, 54, 361, 100, 147, 242, 529, 72, 125, 338, 81, 196, 841, 150, 961, 64, 363, 578, 245, 108, 1369, 722, 507, 200, 1681, 294, 1849, 484, 225, 1058, 2209, 144, 343, 250, 867, 676, 2809, 162, 605, 392, 1083, 1682, 3481, 300, 3721, 1922, 441

Offset: 1

Antti Karttunen, Jan 03 2015

nonn

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

Essentially the same as A129598, except that here we have a(1) = 1.
Cf. A070003 (same sequence without 1, sorted into ascending order).
Cf. A000040, A006530, A052126, A061395, A122111, A253550, A253561, A253563, A253565.
Differs from A072995 for the first time at n=15, where a(15) = 75, while A072995(15) = 225.

A253560 := proc(n)
    n*A006530(n) # code re-use
end proc:
seq( A253560(n),n=1..80) ; # R. J. Mathar, Jun 27 2024

a[n_] := n FactorInteger[n][[-1, 1]];
Array[a, 100] (* Jean-François Alcover, Feb 15 2021 *)

a(n) = if (n==1, 1, n*vecmax(factor(n)[,1])); \\ Michel Marcus, Feb 15 2021

Scheme
```
(define (A253560 n) (* (A006530 n) n))
```

a(1) = 1; for n > 1, a(n) = A006530(n) * n = A000040(A061395(n)) * n.
Other identities:
a(n) >= A253550(n) for all n >= 1.
a(n) = A129598(n) for all n >= 2.
A052126(a(n)) = n. [A052126 works as an inverse function for this injection.]

A129595 Array A(i,j): A(1,1), A(2,1), A(1,2), A(3,1), A(2,2), A(1,3), ... of elementwise sums of partitions encoded in the prime factorizations of i and j.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 9, 9, 4, 5, 8, 6, 8, 5, 6, 25, 27, 27, 25, 6, 7, 18, 15, 16, 15, 18, 7, 8, 49, 12, 125, 125, 12, 49, 8, 9, 16, 35, 54, 10, 54, 35, 16, 9, 10, 27, 81, 343, 45, 45, 343, 81, 27, 10, 11, 50, 18, 32, 21, 24, 21, 32, 18, 50, 11, 12, 121, 30, 81, 625, 175

Offset: 1

Antti Karttunen, May 01 2007, based on Marc LeBrun's Jan 11 2006 message on SeqFan mailing list

As described by Marc LeBrun, we can map integers 1-to-1 to partitions in a "crazy" order: factor n, take the (finite) tuple of exponents, add 1 to the first, use the rest as successive differences between parts and finally subtract 1 from the last part, thus we get the following partitions (elements in ascending order): 2 -> [1] -> 1, 3 -> [0,1] -> 1+1, 4 -> [2] -> 2, 5 -> [0,0,1] -> 1+1+1, 6 -> [1,1] -> 2+2, 7 -> [0,0,0,1] -> 1+1+1+1, 8 -> [3] -> 3, 9 -> [0,2] -> 1+2, 10 -> [1,0,1] -> 2+2+2, etc.
Inverse process: from a sorted (elements in ascending order) partition of n, subtract 1 from the first part, then take the first differences of parts and add 1 to the last (of differences or the first part if a singular partition) and use them as the exponents for A000040(1), A000040(2), etc. and multiply.
This array is obtained when we encode in such a way the partition obtained as an element-wise sum of two partitions encoded by i and j. The element-wise addition begins from the largest elements of the partitions, continuing towards the smaller elements and if the partitions do not contain the same number of elements, the shorter is prepended with as many zeros as needed to make them of equal length.
On what condition does A(i,j) = i*j ? E.g., A(3,5)=15, A(3,10)=30, A(5,11)=55. However A(3,7)=35 and A(5,7)=21.

			a(54) = A(9,2) = 27 because when we add element-wise partition 1+2 encoded by 9 to a singular partition 1 encoded by 2, we get partition 1+3, which maps to exponent tuple [0,3] and 27 = 2^0 * 3^3.

A. Karttunen, Table of n, a(n) for n = 1..1275

Cf. A129593, A129596, A129597, A129598.

A122111 gives the involution of natural numbers induced when partition conjugation (see A129594) is applied to the same encoding.

A277334 Numbers n, that apart from 2 are all odd and for which n/(largest prime dividing n) is squarefree.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141, 143, 145, 147, 149, 151, 155, 157, 159, 161, 163, 165, 167, 169

Offset: 1

Antti Karttunen, Oct 12 2016

nonn

In other words, after 1 and 2, such odd numbers that only the largest prime factor in their prime factorization may have exponent 1 or 2, while all lesser prime factors occur at most once.

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

Disjoint union of A056911 and A129598(A056911(n)).
Cf. A005117, A008683, A052126.
Cf. A277332 (permutation of this sequence).
Differs from A091377 for the first time at n=36, where a(36)=75, while A091377(36)=77.

with(numtheory): A277334_list := n -> seq(`if`(i=2 or (i::odd and issqrfree(i/ max(factorset(i)))),i,NULL),i=1..n): A277334_list(169); # Peter Luschny, Oct 23 2016

cp's OEIS Frontend

A253560 Multiply n by its largest prime factor: a(n) = A006530(n) * n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Scheme

Formula

A129595 Array A(i,j): A(1,1), A(2,1), A(1,2), A(3,1), A(2,2), A(1,3), ... of elementwise sums of partitions encoded in the prime factorizations of i and j.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A277334 Numbers n, that apart from 2 are all odd and for which n/(largest prime dividing n) is squarefree.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple