A056153 -id:A056153 - cp's OEIS frontend

A085988 Triangle of least prime signatures such that T(1,1)= 1; T(r,j) = 2*T(r,j-1) for j>1 and T(r+1,1) is the smallest value in A025487 not appearing on an earlier row.

1, 2, 4, 6, 12, 24, 8, 16, 32, 64, 30, 60, 120, 240, 480, 36, 72, 144, 288, 576, 1152, 48, 96, 192, 384, 768, 1536, 3072, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 180, 360, 720, 1440, 2880, 5760, 11520, 23040, 46080, 210, 420, 840, 1680, 3360, 6720, 13440, 26880, 53760, 107520

Offset: 1

Alford Arnold, Jul 12 2003

			The table begins:
   1
   2  4
   6 12  24
   8 16  32  64
  30 60 120 240 480
  36 72 144 288 576 1152
  ...

Cf. A025487, A056153, A062515.

isli(n) = if(n==1, return(1)); my(f = factor(n)); f[#f~, 1] == prime(#f~) && vecsort(f[, 2], , 4) == f[, 2]; \\ A025487
findfirst(all, used) = {for (k=1, #all, if (!vecsearch(used, all[k]), return (all[k])););}
tabl(nn) = {all = select(x->isli(x), vector(nn, k, k)); used = []; for (n=1, oo, if (n==1, row = [1], first = findfirst(all, used); if (!first, return); row = vector(n, k, first*2^(k-1))); print(row); used = vecsort(concat(used, row)););} \\ Michel Marcus, Feb 20 2019

More terms from Michel Marcus, Feb 20 2019

A062515 Leading least prime signatures, ordered by forming the product of primorials greater than 2 with multiplicities given by the canonical sequence of partitions.

Original entry on oeis.org

1, 6, 30, 36, 210, 180, 216, 2310, 1260, 900, 1080, 1296, 30030, 13860, 6300, 7560, 5400, 6480, 7776, 510510, 180180, 69300, 83160, 44100, 37800, 45360, 27000, 32400, 38880, 46656, 9699690, 3063060, 900900, 1081080, 485100, 415800, 498960, 264600, 189000

Offset: 0

Alford Arnold, Jul 10 2001

From Jack W Grahl, Jul 06 2018: (Start)
The least prime signatures (A025487) are the smallest numbers with a given 'prime signature'. For example, 24 = 2^3 * 3 is the smallest number consisting of the cube of a prime multiplied by a prime. They can be expressed as products 2^(k1) * 3^(k2) * 5(k3) * ..., where k1 >= k2 >= k3 >= ...
These can also be defined as all products of primorials. Here the primorials (A002110) are the products of the first n primes. So 24 = 2 * 2 * 6.
The leading least prime signatures (A056153) are the least prime signatures k such that k/2 is not a least prime signature. They can be expressed as products 2^(k1) * 3^(k2) * 5(k3) * ..., where k1 = k2 >= k3 >= ... (note the first operator is equality). A056153 lists these in increasing order.
They can also be defined as all products of primorials 6 or greater. This sequence lists the leading least prime signatures in an ordering derived from this definition. The canonical sequence of partitions maps to this sequence under a mapping which sends 1 -> 6, 2 -> 30, 3 -> 210, etc., and then forms the product of these terms. Thus the first few partitions are [], [1], [2], [1,1], [3], [2,1] and so the first terms of this sequence are 1, 6, 30, 6 * 6 = 36, 210, 30 * 6 = 180.
The previous description described this sequence as the 'leading least prime signatures ordered as in A063008'. This was in error. A063008 gives a different ordering of A025487, also based on the canonical sequence of partitions, but the definition is different from this sequence and the terms do not appear in the same order (with the transposition of 216 and 2310 being the first discrepancy). (End)

			Values in A025487 can be generated via powers of two as follows:
1
2
4,6
8,12
16,24,30
32,48,60,36
64,96,120,72
128,192,240,144,210,180,216
a(3) = 36 because we can write [1,1] and associate this exponent vector with 6*6

Jack W Grahl, Table of n, a(n) for n = 0..1000

Cf. A002110, A025487, A056153 and A063008.

import Data.List(inits)
primes :: [Integer]
primes = 2 : 3 : filter (\a -> all (not . divides a) (takeWhile (\x -> x <= a `div` 2) primes)) [4..]
    where
  divides a b = a `mod` b == 0
primorials :: [Integer]
primorials = map product $ inits primes
partitions :: [[Integer]]
partitions = concat $ map (partitions_of_n) [0..]
partitions_of_n :: Integer -> [[Integer]]
partitions_of_n n = partitions_at_most n n
partitions_at_most :: Integer -> Integer -> [[Integer]]
partitions_at_most _ 0 = [[]]
partitions_at_most 0 _ = []
partitions_at_most m n = concat $ map (\k -> map ([k] ++) (partitions_at_most k (n-k))) ( reverse [1..(min m n)])
a062515 :: [Integer]
a062515 = map primorial_signature partitions
    where
  primorial_signature p = product $ map ((drop 1 primorials) !!) (map fromIntegral p)
-- Jack W Grahl, Jul 06 2018

Clarified and extended by Jack W Grahl, Jul 06 2018

A316532 Leading least prime signatures, ordered by the underlying partitions, as in A063008.

Original entry on oeis.org

1, 6, 30, 36, 210, 180, 2310, 216, 900, 1260, 30030, 1080, 6300, 13860, 510510, 1296, 5400, 7560, 44100, 69300, 180180, 9699690, 6480, 27000, 37800, 83160, 485100, 900900, 3063060, 223092870, 7776, 32400, 45360, 189000, 264600, 415800, 1081080, 5336100

Offset: 0

Jack W Grahl, Jul 06 2018

The sequence A063008 gives the least number with each prime signature, ordered by the underlying partition. This sequence is a subsequence which only includes those prime signatures M for which M/2 is not a prime signature, the so-called 'leading' least prime signatures.
This sequence is therefore constructed by taking the partitions first in increasing order of their sum, then in decreasing order of the first term, then decreasing order of the second term, etc. We drop all partitions, except the empty partition, where the first term and the second term are different. Then we map (m1, m2, m3, ..., mk) to 2^m1 * 3^m2 * ... * pk^mk to give the terms of this sequence.
The sequence A062515 had a description which suggested that it had been confused with this sequence. They are the same leading least prime signatures, but in a different order, given by a different construction using integer partitions.

			The first few partitions are [], [1,1], [1,1,1], [2,2], [1,1,1,1]. So the first few terms are 1, 2 * 3 = 6, 2 * 3 * 5 = 30, 2^2 * 3^2 = 36, 2 * 3 * 5 * 7 = 210.

Subsequence of A063008. A re-ordering of A062515, also of A056153. Cf A025487.

primes :: [Integer]
primes = 2 : 3 : filter (\a -> all (not . divides a) (takeWhile (\x -> x <= a `div` 2) primes)) [4..]
divides :: Integer -> Integer -> Bool
divides a b = a `mod` b == 0
partitions :: [[Integer]]
partitions = concat $ map (partitions_of_n) [0..]
partitions_of_n :: Integer -> [[Integer]]
partitions_of_n n = partitions_at_most n n
partitions_at_most :: Integer -> Integer -> [[Integer]]
partitions_at_most _ 0 = [[]]
partitions_at_most 0 _ = []
partitions_at_most m n = concat $ map (\k -> map ([k] ++) (partitions_at_most k (n-k))) ( reverse [1..(min m n)])
prime_signature :: [Integer] -> Integer
prime_signature p = product $ zipWith (^) primes p
seq :: [Integer]
seq = map prime_signature $ filter compare_first_second partitions
    where
  compare_first_second p
        | length p == 0 = True
        | length p == 1 = False
        | otherwise = p!!0 == p!!1

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A085988 Triangle of least prime signatures such that T(1,1)= 1; T(r,j) = 2*T(r,j-1) for j>1 and T(r+1,1) is the smallest value in A025487 not appearing on an earlier row.

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A062515 Leading least prime signatures, ordered by forming the product of primorials greater than 2 with multiplicities given by the canonical sequence of partitions.

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A316532 Leading least prime signatures, ordered by the underlying partitions, as in A063008.

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