A062515 -id:A062515 - 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

A056153 Leading least prime signatures: a(n) is in A025487 but a(n)/2 is not.

Original entry on oeis.org

1, 6, 30, 36, 180, 210, 216, 900, 1080, 1260, 1296, 2310, 5400, 6300, 6480, 7560, 7776, 13860, 27000, 30030, 32400, 37800, 38880, 44100, 45360, 46656, 69300, 83160, 162000, 180180, 189000, 194400, 226800, 233280, 264600, 272160, 279936

Offset: 1

Alford Arnold, Jul 30 2000

nonn

Values of A025487 can be mapped to the numeric partitions. In a similar way, values of a(n) can be mapped to the cyclic partitions by noting that the factorization vector begins (k, k, ...). E.g. 1260 = 2*2*3*3*5*7 yielding the vector (2,2,1,1).

All numbers of the form 2^k1*3^k2*...*p_n^k_n, where k1 = k2 >= ... >= k_n, sorted. - Robert Israel, Feb 20 2019

			36 is a term because 36 is a member of A025487 but 36/2 = 18 is not.
2520 is a member of A025487 as is 2520/2 = 1260, so 2520 is not a term.

Robert Israel, Table of n, a(n) for n = 1..10000 (first 108 terms from Michel Marcus)

Cf. A025487, A062515, A161360.

N:= 10^8: # to get all terms <= N
S:= [seq([i,i,6^i],i=0..floor(log[6](N)))]:
Res:= {seq(s[-1],s=S)}:
r:= 6:
for n from 3 do
  p:= ithprime(n);
  r:= r*p;
  if r > N then break fi;
  S:= map(t ->seq([op(t[1..-2]),i,t[-1]*p^i],i=1..min(t[-2], floor(log[p](N/t[-1])))), S);
  Res:= Res union {seq(s[-1],s=S)};
od:
sort(convert(Res, list)); # Robert Israel, Feb 20 2019

max = 300000; ss = {}; A025487 = Join[{1}, Reap[ Do[s = Sort[FactorInteger[n][[All, 2]]]; If[FreeQ[ss, s], AppendTo[ss, s]; Sow[n]], {n, 2, max}]][[2, 1]]]; Select[A025487, FreeQ[A025487, #/2] &] (* Jean-François Alcover, Jul 11 2012 *)

isli(n) = if(n==1, return(1)); if (frac(n), return (0)); my(f = factor(n)); f[#f~, 1] == prime(#f~) && vecsort(f[, 2], , 4) == f[, 2]; \\ A025487
isok(n) = isli(n) && !isli(n/2); \\ Michel Marcus, Feb 20 2019

Sum_{n>=1} 1/a(n) = A161360 / 2 = 1.247903257029... . - Amiram Eldar, Jul 25 2024

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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A056153 Leading least prime signatures: a(n) is in A025487 but a(n)/2 is not.

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

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