A033476 -id:A033476 - cp's OEIS frontend

A088178 Sequence of distinct products b(n)*b(n+1), n=1,2,3,..., of the terms b(n) of A088177.

1, 2, 4, 6, 3, 5, 10, 8, 12, 9, 15, 20, 16, 24, 18, 21, 7, 11, 22, 14, 28, 32, 40, 25, 30, 36, 42, 35, 45, 27, 33, 44, 48, 60, 50, 70, 49, 56, 64, 72, 54, 66, 55, 65, 13, 17, 34, 26, 39, 51, 68, 52, 78, 84, 98, 63, 81, 90, 80, 88, 77, 91, 104, 96, 108, 99, 110, 100, 120, 132

Offset: 1

John W. Layman, Sep 22 2003

This is a permutation of the natural numbers (see the following comments).
Comments from Thomas Ordowski, Aug 24 2014 to Sep 07 2014: (Start)
If a(n) is a prime then a(m) > a(n) for m > n.
Conjecture: the term a(n) is a prime if and only if every number < a(n) belongs to the set {a(1), a(2), ..., a(n-1)}.
The numbers in A033476 appear in increasing order.
It seems that the squarethe terms in s of the natural numbers also appear in increasing order, but A087811 are not strictly increasing.
Lemma: the sequence a(n) is a permutation of all natural numbers iff b(n) = 1 for infinitely many n, where b(n) = A088177(n), because after every b(n) = 1 is the smallest missing number in the sequence a(n).
Theorem: the sequence a(n) is a permutation of the natural numbers. Proof: see my note to A088177.
At most two consecutive terms can form a decreasing subsequence.
(End)
An equivalent definition. At step n, choose a(n) to be the smallest unused multiple of the auxiliary number r, which is initially 1 and is changed to a(n)/r after each step. - Ivan Neretin, May 04 2015
Considered as a permutation of the positive integers, there are finite cycles (1), (2), (3, 4, 6, 5), (8), (11, 18, 15), (52), and probably others. The cycle containing 7, on the other hand, is ( ..., 85, 46, 17, 7, 10, 9, 12, 20, 14, 24, 25, 30, 27, 42, 66, 99, 160, 308, 343, 430, 517, 902, ... ), and may be infinite. The inverse permutation is A341492. - N. J. A. Sloane, Oct 19 2021

Ivan Neretin, Table of n, a(n) for n = 1..10000 (first 1000 terms from Michael De Vlieger)

Cf. A088177, A341492, A348437-A348439.

Records: A348442, A348443.

a088177[n_Integer] := Module[{t = {1, 1}}, Do[AppendTo[t, 1]; While[Length[Union[Most[t]*Rest[t]]] < i - 1, t[[-1]]++], {i, 3, n}]; t]; a088178[n_Integer] := Last[a088177[n]]*Last[a088177[n + 1]]; a088178 /@ Range[120] (* Michael De Vlieger, Aug 30 2014, based on T. D. Noe's script at A088177 *)

from itertools import islice
def A088178(): # generator of terms
    yield 1
    p, a = {1}, 1
    while True:
        n, na = 1, a
        while na in p:
            n += 1
            na += a
        p.add(na)
        a = n
        yield na
A088178_list = list(islice(A088178(),20)) # Chai Wah Wu, Oct 21 2021

a(n) = A088177(n)* A088177(n+1).

a(m) < a(n)^2 for m < n. - Thomas Ordowski, Sep 02 2014

Edited by N. J. A. Sloane, Oct 18 2021

A089995 Products of pairs of distinct, non-consecutive primes.

Original entry on oeis.org

10, 14, 21, 22, 26, 33, 34, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205, 206, 209, 213

Offset: 1

Reinhard Zumkeller, Nov 19 2003

nonn

m belongs to the sequence iff m=A001358(k) for some k and A089994(k)>0.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006881, A001358, A084126, A084127, A089994, A033476.

With[{nn = 213}, TakeWhile[#, # <= nn &] &@ Union@ Flatten@ Table[Function[p, p Prime@ Range[n + 2, Prime@ PrimePi[nn/p]]]@ Prime@ n, {n, Sqrt@ nn}]] (* Michael De Vlieger, Feb 02 2017 *)

A089994 Number of primes between factors of n-th semiprime.

Original entry on oeis.org

0, 0, 0, 1, 2, 0, 1, 3, 0, 4, 2, 5, 0, 6, 3, 7, 0, 4, 1, 5, 8, 9, 2, 6, 10, 0, 11, 3, 12, 7, 1, 8, 13, 4, 14, 9, 5, 15, 2, 0, 16, 10, 11, 3, 17, 12, 18, 0, 6, 19, 7, 20, 13, 4, 21, 0, 14, 22, 15, 8, 1, 23, 16, 24, 5, 9, 25, 2, 17, 26, 10, 6, 27, 18, 0, 28, 11, 19, 1, 20, 3, 29, 7, 30

Offset: 1

Reinhard Zumkeller, Nov 19 2003

nonn

a(m)=0 iff m in A033476; a(m)>0 iff m in A089995.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001358, A033476, A089995, A084126, A084127, A000720.

pbfs[n_]:=Module[{f=PrimePi/@Transpose[FactorInteger[n]][[1]]}, Max[ 0,Last[f]-First[f]-1]]; pbfs/@Select[Range[300],PrimeOmega[#]==2&] (* Harvey P. Dale, Apr 09 2012 *)

A057602 a(1)=2, a(n+1) is the smallest integer > a(n) such that the smallest prime factor of a(n+1) is the largest prime factor of a(n).

Original entry on oeis.org

2, 4, 6, 9, 15, 25, 35, 49, 77, 121, 143, 169, 221, 289, 323, 361, 437, 529, 667, 841, 899, 961, 1147, 1369, 1517, 1681, 1763, 1849, 2021, 2209, 2491, 2809, 3127, 3481, 3599, 3721, 4087, 4489, 4757, 5041, 5183, 5329, 5767, 6241, 6557, 6889, 7387, 7921

Offset: 1

G. L. Honaker, Jr., Oct 07 2000

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040.

A033476.

Module[{nn=30,pr,ev,od},pr=Prime[Range[nn]];ev=pr^2;od=Times @@@ Partition[ pr,2,1];Join[{2},Riffle[ev,od]]] (* Harvey P. Dale, Mar 02 2015 *)

Even numbered terms are squares of successive primes. Odd numbered terms are the product of two successive primes and are the square root of the product of the previous term and the next term - Jud McCranie, Oct 07 2000

Additional terms from Jud McCranie, Oct 07 2000

cp's OEIS Frontend

A088178 Sequence of distinct products b(n)*b(n+1), n=1,2,3,..., of the terms b(n) of A088177.

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A089995 Products of pairs of distinct, non-consecutive primes.

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A089994 Number of primes between factors of n-th semiprime.

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Programs

Mathematica

A057602 a(1)=2, a(n+1) is the smallest integer > a(n) such that the smallest prime factor of a(n+1) is the largest prime factor of a(n).

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Author

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Mathematica

Formula

Extensions