A182318 -id:A182318 - cp's OEIS frontend

A287958 Table read by antidiagonals: T(n, k) = least recursive multiple of n and k; n > 0 and k > 0.

1, 2, 2, 3, 2, 3, 4, 6, 6, 4, 5, 4, 3, 4, 5, 6, 10, 12, 12, 10, 6, 7, 6, 15, 4, 15, 6, 7, 8, 14, 6, 20, 20, 6, 14, 8, 9, 8, 21, 12, 5, 12, 21, 8, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 10, 9, 64, 35, 6, 35, 64, 9, 10, 11, 12, 22, 30, 36, 40, 42, 42, 40

Offset: 1

Rémy Sigrist, Jun 03 2017

We say that m is a recursive multiple of d iff d is a recursive divisor of m (as described in A282446).
More informally, the prime tower factorization of T(n, k) is the union of the prime tower factorizations of n and k (the prime tower factorization of a number is defined in A182318).
This sequence has connections with the classical LCM (A003990).
For any i > 0, j > 0 and k > 0:
- A007947(T(i, j)) = A007947(lcm(i, j)),
- T(i, j) >= 1,
- T(i, j) >= max(i, j),
- T(i, j) >= lcm(i, j),
- T(i, 1) = i,
- T(i, i) = i,
- T(i, j) = T(j, i) (the sequence is commutative),
- T(i, T(j, k)) = T(T(i, j), k) (the sequence is associative),
- T(i, i*j) >= i*j,
- if gcd(i, j) = 1 then T(i, j) = i*j.
See also A287957 for the GCD equivalent.

			Table starts:
n\k|     1   2   3   4   5   6   7   8   9  10
---+-----------------------------------------------
1  |     1   2   3   4   5   6   7   8   9  10  ...
2  |     2   2   6   4  10   6  14   8  18  10  ...
3  |     3   6   3  12  15   6  21  24   9  30  ...
4  |     4   4  12   4  20  12  28  64  36  20  ...
5  |     5  10  15  20   5  30  35  40  45  10  ...
6  |     6   6   6  12  30   6  42  24  18  30  ...
7  |     7  14  21  28  35  42   7  56  63  70  ...
8  |     8   8  24  64  40  24  56   8  72  40  ...
9  |     9  18   9  36  45  18  63  72   9  90  ...
10 |    10  10  30  20  10  30  70  40  90  10  ...
...
T(4, 8) = T(2^2, 2^3) = 2^(2*3) = 2^6 = 64.

Rémy Sigrist, First 100 antidiagonals of array, flattened
Rémy Sigrist, Illustration of the first terms

Cf. A003990, A007947, A182318, A282446, A287957.

T(n,k) = if (n*k==0, return (max(n,k))); my (g=factor(lcm(n,k))); return (prod(i=1, #g~, g[i,1]^T(valuation(n, g[i,1]), valuation(k, g[i,1]))))

A336965 a(n) is the product of the distinct prime numbers appearing in the prime tower factorization of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 6, 6, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 10, 26, 3, 14, 29, 30, 31, 10, 33, 34, 35, 6, 37, 38, 39, 30, 41, 42, 43, 22, 30, 46, 47, 6, 14, 10, 51, 26, 53, 6, 55, 42, 57, 58, 59, 30, 61, 62, 42, 6, 65, 66, 67, 34, 69, 70

Offset: 1

Rémy Sigrist, Aug 09 2020

nonn

The prime tower factorization of a number is defined in A182318.

For any n > 0, a(n) is the product of the terms in n-th row of A336964.

			A001221(a(n)) = A115588(n) for any n > 1.
a(n) = A007947(A279513(n)).
a(n) = n iff n is squarefree (A005117).

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A001221, A005117, A007947, A115588, A182318, A336964.

a(n) = { my (f=factor(n), v=vecprod(f[,1]~)); for (k=1, #f~, v=lcm(v, a(f[k,2]))); v }

A376218 Odd exponentially odd numbers.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 149

Offset: 1

Amiram Eldar, Sep 16 2024

First differs from its subsequence A182318 at n = 8318: a(8318) = 19683 = 3^9 = 3^(3^2) is not a term of A182318.
Numbers whose prime factorization contains only odd primes and odd exponents.
Numbers whose sum of coreful divisors (A057723) is odd (a coreful divisor d of a number k is a divisor that is divisible by every prime that divides k, see also A307958).
The even exponentially odd numbers are numbers of the form 2^k * m, where k is odd and m is a term of this sequence.
The asymptotic density of this sequence is (3/5) * Product_{p prime} (1 - 1/(p*(p+1))) = (3/5) * A065463 = 0.42266532... .

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

Intersection of A005408 and A268335.
Cf. A057723, A065463, A182318, A307958.
Other numbers with an odd sum of divisors: A000079 (unitary divisors), A028982 (all divisors), A069562 (non-unitary divisors), A357014 (exponential divisors).

Select[Range[1, 150, 2], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]

is(k) = k % 2 && vecprod(factor(k)[,2]) % 2;

A182337 List of positive integers whose prime tower factorization, as defined in comments, does not contain the prime 3.

Original entry on oeis.org

1, 2, 4, 5, 7, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 58, 59, 61, 62, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103, 106

Offset: 1

Patrick Devlin, Apr 25 2012

nonn

The prime tower factorization of a number can be recursively defined as follows:
(0) The prime tower factorization of 1 is itself
(1) To find the prime tower factorization of an integer n>1, let n = p1^e1 * p2^e2 * ... * pk^ek be the usual prime factorization of n. Then the prime tower factorization is given by p1^(f1) * p2^(f2) * ... * pk^(fk), where fi is the prime tower factorization of ei.
As an alternative definition, let I(n) be the indicator function for the set of positive integers whose prime tower factorization does not contain a 3. Then I(n) is the multiplicative function satisfying I(p^k) = I(k) for p prime not equal to 3, and I(3^k) = 0.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Patrick Devlin and Edinah Gnang, Primes Appearing in Prime Tower Factorization, arXiv:1204.5251 [math.NT], 2012-2014.

Cf. A182318.

# The integer n is in this sequence if and only if
# containsPrimeInTower(3, n) returns false
containsPrimeInTower:=proc(q, n) local i, L, currentExponent; option remember;
if n <= 1 then return false: end if;
if type(n/q, integer) then return true: end if;
L := ifactors(n)[2];
for i to nops(L) do currentExponent := L[i][2];
  if containsPrimeInTower(q, currentExponent) then return true: end if
end do;
return false:
end proc:
select(x-> not containsPrimeInTower(3,x), [$1..120])[];

indic[1] = 1; indic[n_] := indic[n] = Switch[f = FactorInteger[n], {{3, }}, 0, {{, }}, indic[f[[1, 2]] ], , Times @@ (indic /@ (Power @@@ f))]; Select[Range[120], indic[#] == 1&] (* Jean-François Alcover, Feb 25 2018 *)

A284696 Numbers of the form p^^k, with p prime and k>=0, where ^^ denotes tetration.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 11, 13, 16, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257

Offset: 1

Rémy Sigrist, Apr 01 2017

nonn

Also numbers n such that A284694(n)=A284695(n).
Also numbers with no distinct prime numbers in their prime tower factorization (see A182318 for the definition of the prime tower factorization of a number).
Also numbers n such that A279513(n) is a power of prime (A000961).
This sequence is the union of 1, the prime numbers (A000040), and A275211.

			16 = 2^^3 is in the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Wikipedia, Tetration

Cf. A000040, A000961, A182318, A275211, A279513, A284694, A284695.

A284763 Numbers n such that A279513(n) is squarefree.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 49, 51, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89

Offset: 1

Rémy Sigrist, Apr 02 2017

nonn

Also numbers with no duplicate prime number in their prime tower factorization (see A182318 for the definition of the prime tower factorization of a number).
This sequence contains the squarefree numbers (A005117); 8 = 2^3 is the first term in this sequence that is not squarefree.
All terms belong to A144146; 81 = 3^2^2 is the first term of A144146 that is not in this sequence.

			8 = 2^3 belongs to this sequence.
24 = 3*2^3 does not belong to this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A005117, A144146, A182318.

a279513(n) =  my (f=factor(n)); prod(i=1, #f~, f[i, 1]*a279513(f[i, 2]));
isok(n) = issquarefree(a279513(n)); \\ Michel Marcus, Apr 08 2017

A287620 a(n) = product, with multiplicity, of the prime numbers appearing at leaf positions in the prime tower factorization of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 3, 2, 10, 11, 6, 13, 14, 15, 2, 17, 4, 19, 10, 21, 22, 23, 9, 2, 26, 3, 14, 29, 30, 31, 5, 33, 34, 35, 4, 37, 38, 39, 15, 41, 42, 43, 22, 10, 46, 47, 6, 2, 4, 51, 26, 53, 6, 55, 21, 57, 58, 59, 30, 61, 62, 14, 6, 65, 66, 67, 34, 69, 70, 71

Offset: 1

Rémy Sigrist, May 28 2017

The prime tower factorization of a number is defined in A182318.
a(n) <= n.
a(n) = n iff n is squarefree (A005117).
a(n) is noncomposite iff n belongs to A164336.
This sequence is surjective; see A287621 for the least k such that a(k) = n.
For n>1, A001222(a(n)) = A064372(n).

			See illustration of the first terms in Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of the first terms

Cf. A001222, A005117, A064372, A164336, A182318, A287621.

a(n) = my (f=factor(n)); return (prod(i=1, #f~, if (f[i,2]==1, f[i,1], a(f[i,2]))))

Multiplicative with:
- a(p) = p for any prime p,
- a(p^k) = a(k) for any prime p and k > 1.

A308993 Multiplicative with a(p) = 1 and a(p^e) = p^a(e) for any e > 1 and prime number p.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 2, 5, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 4, 7, 5, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 5, 2, 1, 1, 1, 4, 9, 1, 1, 2, 1, 1, 1

Offset: 1

Rémy Sigrist, Jul 04 2019

To compute a(n): remove every prime number at leaf position in the prime tower factorization of n (the prime tower factorization of a number is defined in A182318).

			See Links sections.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of first terms

Cf. A004709, A005117, A182318, A185102.

a(n) = my (f=factor(n)); prod (i=1, #f~, f[i,1]^if (f[i,2]==1, 0, a(f[i,2])))

a(n) = 1 iff n is squarefree.
a^k(n) = 1 for any k >= A185102(n) (where a^k denotes the k-th iterate of a).
a(n)^2 <= n with equality iff n is the square of some cubefree number (n = A004709(k)^2 for some k > 0).

A336964 Irregular triangle in which first row is 1, n-th row (n > 1) lists distinct prime numbers in the prime tower factorization of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 3, 7, 2, 3, 2, 3, 2, 5, 11, 2, 3, 13, 2, 7, 3, 5, 2, 17, 2, 3, 19, 2, 5, 3, 7, 2, 11, 23, 2, 3, 2, 5, 2, 13, 3, 2, 7, 29, 2, 3, 5, 31, 2, 5, 3, 11, 2, 17, 5, 7, 2, 3, 37, 2, 19, 3, 13, 2, 3, 5, 41, 2, 3, 7, 43, 2, 11, 2, 3, 5, 2, 23, 47, 2, 3

Offset: 1

Rémy Sigrist, Aug 09 2020

The prime tower factorization of a number is defined in A182318.

The n-th row includes the n-th row of A027748.

			Triangle begins:
     1    [1]
     2    [2]
     3    [3]
     4    [2]
     5    [5]
     6    [2, 3]
     7    [7]
     8    [2, 3]
     9    [2, 3]
    10    [2, 5]
    11    [11]
    12    [2, 3]
    13    [13]
    14    [2, 7]
    15    [3, 5]

Rémy Sigrist, Table of n, a(n) for n = 1..10001 (rows for n = 1..4045)

Cf. A027748, A115588 (row lengths), A182318, A336965.

row(n) = { my (f=factor(n), p=f[,1]~); for (k=1, #f~, if (f[k,2]>1, p=concat(p, row(f[k,2])));); if (#p==0, [1], Set(p)) }

A338668 a(n) is the rightmost prime number in prime tower factorization of n; a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 3, 2, 5, 11, 3, 13, 7, 5, 2, 17, 2, 19, 5, 7, 11, 23, 3, 2, 13, 3, 7, 29, 5, 31, 5, 11, 17, 7, 2, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 3, 2, 2, 17, 13, 53, 3, 11, 7, 19, 29, 59, 5, 61, 31, 7, 3, 13, 11, 67, 17, 23, 7, 71, 2, 73, 37, 2

Offset: 1

Rémy Sigrist, Apr 23 2021

nonn

The prime tower factorization of a number is defined in A182318.

			See Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of initial terms

Cf. A006530, A182318, A338669.

a(n) = if (n==1, 1, my (f=factor(n), w=#f~); if (f[w,2]==1, f[w,1], a(f[w,2])))

a(n) <= n with equality iff n = 1 or n is a prime number.

a(n) = a(A053585(n)).

cp's OEIS Frontend

A287958 Table read by antidiagonals: T(n, k) = least recursive multiple of n and k; n > 0 and k > 0.

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A336965 a(n) is the product of the distinct prime numbers appearing in the prime tower factorization of n.

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A376218 Odd exponentially odd numbers.

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A182337 List of positive integers whose prime tower factorization, as defined in comments, does not contain the prime 3.

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Maple

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A284696 Numbers of the form p^^k, with p prime and k>=0, where ^^ denotes tetration.

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A284763 Numbers n such that A279513(n) is squarefree.

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A287620 a(n) = product, with multiplicity, of the prime numbers appearing at leaf positions in the prime tower factorization of n.

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Formula

A308993 Multiplicative with a(p) = 1 and a(p^e) = p^a(e) for any e > 1 and prime number p.

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