A095904 -id:A095904 - cp's OEIS frontend

A095990 Numbers with ordered prime signature (1,2).

18, 50, 75, 98, 147, 242, 245, 338, 363, 507, 578, 605, 722, 845, 847, 867, 1058, 1083, 1183, 1445, 1587, 1682, 1805, 1859, 1922, 2023, 2523, 2527, 2645, 2738, 2883, 3179, 3362, 3698, 3703, 3757, 3971, 4107, 4205, 4418, 4693, 4805, 5043, 5547, 5618, 5819

Offset: 1

Alford Arnold, Jul 18 2004

Numbers of the form p*q^2 where p and q are primes with p < q.

A054753 contains natural numbers with ordered prime signatures (2,1) and (1,2).

			18 = 2*3*3, 50 = 2*5*5, 75 = 3*5*5, 98 = 2*7*7, 147 = 3*7*7, ...

Enrique Pérez Herrero, Table of n, a(n) for n = 1..3000
OEIS Wiki, Ordered prime signatures

Subsequence of A071365.

Cf. A054753, A095904, A096156.

Take[ Sort[ Flatten[ Table[ Prime[p]*Prime[q]^2, {q, 2, 16}, {p, q - 1}]]], 46] (* Robert G. Wilson v, Jul 23 2004 *)

list(lim)=my(v=List());forprime(q=3, sqrtint(lim\2), forprime(p=2, min(lim\q^2,q-1), listput(v,p*q^2))); Set(v) \\ Charles R Greathouse IV, Feb 26 2014

Edited and extended by Robert G. Wilson v, Jul 23 2004

A179216 Permutation of triangular array of numbers (greater than 1) arranged by prime signature.

Original entry on oeis.org

2, 4, 3, 6, 9, 5, 8, 10, 25, 7, 12, 27, 14, 49, 11, 16, 18, 125, 15, 121, 13, 24, 81, 20, 343, 21, 169, 17, 30, 40, 625, 28, 1331, 22, 289, 19, 32, 42, 54, 2401, 44, 2197, 26, 361, 23, 36, 243, 66, 56, 14641, 45, 4913, 33, 529, 29, 48, 100, 3125, 70, 88, 28561, 50, 6859

Offset: 0

Will Nicholes, Jul 10 2010

The left diagonal is the sequence of primes (A000040). The right diagonal is sequence A025487 (least prime signatures).

Will Nicholes, Iterative mapping of prime signatures.

Cf. A095904, A179217, A179218, A179221, A179222, A179225, A179226, A179228.

a(n) = A061579(A095904(n))

A181087 Partitions of n in the order of increasing smallest numbers of prime signatures.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 2, 4, 1, 3, 1, 1, 1, 5, 2, 2, 1, 4, 1, 1, 2, 6, 2, 3, 1, 5, 1, 1, 3, 7, 2, 4, 1, 2, 2, 1, 6, 1, 1, 1, 1, 3, 3, 1, 1, 4, 8, 2, 5, 1, 2, 3, 1, 7, 1, 1, 1, 2, 3, 4, 1, 1, 5, 9, 2, 6, 1, 2, 4, 1, 8, 1, 1, 1, 3, 3, 5, 2, 2, 2, 1, 1, 6, 10, 1, 3, 3, 2, 7, 1, 1, 2, 2, 4, 4, 1, 2, 5, 1, 9, 1, 1, 1, 4, 3, 6, 2, 2, 3, 1, 1, 7, 11, 1, 3, 4, 2, 8, 1, 1

Offset: 1

Alois P. Heinz, Jan 23 2011

The parts of each partition are listed in increasing order.

			Smallest number with prime signature [1,1,1] is 2^1*3^1*5^1 = 30, the smallest number for [4] is 2^4 = 16, and thus [4] < [1,1,1] in this order.
First partitions in the order of increasing smallest numbers of prime signatures are: [1], [2], [1,1], [3], [1,2], [4], [1,3], [1,1,1], [5], [2,2], [1,4], [1,1,2], [6], [2,3], [1,5], [1,1,3], [7], [2,4], ...
Smallest numbers with these prime signatures are:  2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, ... A025487

Alois P. Heinz, Table of n, a(n) for n = 1..18132

Cf. A036036, A036037, A080576, A025487, A095904.

DeleteDuplicates[Map[Sort[Map[Last, FactorInteger[#]]] &, Range[1000]]] // Grid (* Geoffrey Critzer, Nov 27 2015 *)

def A181087_build(w):
    seen = set()
    a = []
    for n in PositiveIntegers():
        psig = tuple(sorted(m for p,m in factor(n)))
        if psig not in seen:
            a.extend(psig)
            seen.add(psig)
            if len(a) >= w: return a  # D. S. McNeil, Jan 23 2011

A179218 Inverse function to A179216.

Original entry on oeis.org

0, 2, 1, 5, 3, 9, 6, 4, 7, 14, 10, 20, 12, 18, 15, 27, 16, 35, 23, 25, 33, 44, 21, 8, 42, 11, 31, 54, 28, 65, 36, 52, 63, 75, 45, 77, 88, 102, 29, 90, 37, 104, 40, 50, 117, 119, 55, 13, 61, 133, 73, 135, 38, 150, 48, 168, 187, 152, 66, 170, 207, 86, 78, 228, 47, 189, 100, 250

Offset: 2

Will Nicholes, Jul 10 2010

nonn

A permutation of all nonnegative integers.

Will Nicholes, Iterative mapping of prime signatures.

Cf. A095904, A179216, A179217.

A178849 n-th integer having n-th prime-containing prime signature.

Original entry on oeis.org

2, 9, 14, 343, 44, 28561, 135, 110, 6436343, 1225, 464, 228, 4750104241, 3087, 1376, 696, 2488651484819, 15376, 1452, 4288, 1218, 456533, 1968, 3936588805702081, 89888, 2646, 12928, 2070, 810448, 5152, 8594754748609397887, 440896, 7938

Offset: 1

Will Nicholes, Jun 17 2010

nonn

This sequence forms the diagonal of the table (A095904) containing all of the integers >= 2, ordered by prime signature.
The first row of the table is A025487 (starting with n=2),
The first column of the table is the primes (A000040).
The description for this sequence specifies "prime-containing" to exclude the prime signature for 1, which is also excluded from A095904.

			The 5th prime-containing prime signature (i.e. excluding the prime signature for 1) is {1,2}.
The 5th integer to have that prime signature is 44 (see A054753 for a list).
Therefore a(5) is 44.

Cf. A025487, A000040, A095904.

Extended by Ray Chandler, Aug 01 2010

cp's OEIS Frontend

A095990 Numbers with ordered prime signature (1,2).

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A179216 Permutation of triangular array of numbers (greater than 1) arranged by prime signature.

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A181087 Partitions of n in the order of increasing smallest numbers of prime signatures.

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A179218 Inverse function to A179216.

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A178849 n-th integer having n-th prime-containing prime signature.

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