A076526 -id:A076526 - cp's OEIS frontend

A076745 a(n) = the least positive integer k such that b(k) = n, where b(k) (A076526) is defined by b(k) = r * max{e_1,...,e_r} if k = p_1^e_1 ... p_r^e_r is the canonical prime factorization of k.

2, 4, 8, 12, 32, 24, 128, 48, 120, 96, 2048, 192, 8192, 384, 480, 768, 131072, 960, 524288, 3072, 1920, 6144, 8388608, 3840, 36960, 24576, 7680, 13440, 536870912, 15360, 2147483648, 26880, 30720, 393216, 147840, 53760, 137438953472, 1572864, 122880, 107520, 2199023255552

Offset: 1

Joseph L. Pe, Nov 11 2002

			a(12) = 2 * max{1,2} = 4 since 12 = 2^2 * 3^1 and 12 is the least k for which b(k) = 4. Hence a(4) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Carlos Rivera, Puzzle 201: The Arithmetic Function A(n), The Prime Puzzles and Problems Connection.

Cf. A076526.

a[n_] := Min[Table[2^d*Times @@ Prime[Range[2, n/d]], {d, Divisors[n]}]]; Array[a, 50] (* Amiram Eldar, Sep 08 2024 *)

a(n) = {my(f = factor(n), nd = numdiv(f), v = vector(nd), k = 0); fordiv(f, d, k++; v[k] = 2^d * prod(i = 1, n/d-1, prime(i+1))); vecmin(v);} \\ Amiram Eldar, Sep 08 2024

From Amiram Eldar, Sep 08 2024: (Start)
a(n) = Min_{d|n} (2^d * Product_{i=1..n/d-1} prime(i+1)).
a(p) = 2^p for a prime p.
a(2*p) = 3*2^p for a prime p.
a(3*p) = 15*2^p for a prime p > 2. (End)

More terms from Amiram Eldar, Sep 08 2024

A076558 a(n) = r * min(e_1, ..., e_r), where n = p_1^e_1 . .... p_r^e_r is the canonical prime factorization of n, a(1) = 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 4, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 2, 6, 2, 3, 1, 2, 2, 3, 1, 4, 1, 2, 2, 2, 2, 3, 1, 2, 4, 2, 1, 3, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 4

Offset: 1

Joseph L. Pe, Nov 10 2002

Omega(n) >= a(n) for n >= 1, where Omega(n) = the number of prime factors of n, counting multiplicity.

Positions of records are A000079. - David A. Corneth, May 05 2020

Antti Karttunen, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle #201 The Arithmetic Function A(n), The Prime Puzzles and Problems Connection.
Index entries for sequences computed from exponents in factorization of n.

Cf. A000079, A001221, A001222, A051904, A076526.

a[n_] := Module[{pf}, pf = Transpose[FactorInteger[n]]; Length[pf[[1]]]*Min[pf[[2]]]]; Table[a[i] - Boole[i == 1], {i, 100}]
(* Second program: *)
Table[Length[#] Min[#] - Boole[n == 1] &@ FactorInteger[n][[All, -1]], {n, 100}] (* Michael De Vlieger, Jul 12 2017 *)

a(n) = if(n == 1, 0, my(e = factor(n)[, 2]); vecmin(e) * #e); \\ Amiram Eldar, Sep 08 2024

from sympy import factorint
def a(n):
    f=factorint(n)
    l=[f[p] for p in f]
    return 0 if n==1 else len(l)*min(l)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 13 2017

a(n) = A001221(n) * A051904(n). - Antti Karttunen, Jul 12 2017

a(1)=0 prepended by Antti Karttunen, Jul 12 2017

A335603 a(n) = p*q where p is the sequential number (or PrimePi, A000720) of the largest prime divisor of n, and q is the maximal exponent in the canonical representation of n (A051903).

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 3, 4, 3, 5, 4, 6, 4, 3, 4, 7, 4, 8, 6, 4, 5, 9, 6, 6, 6, 6, 8, 10, 3, 11, 5, 5, 7, 4, 4, 12, 8, 6, 9, 13, 4, 14, 10, 6, 9, 15, 8, 8, 6, 7, 12, 16, 6, 5, 12, 8, 10, 17, 6, 18, 11, 8, 6, 6, 5, 19, 14, 9, 4, 20, 6, 21, 12, 6, 16, 5, 6, 22, 12

Offset: 1

Todor Szimeonov, Jun 11 2020

nonn

a(n) is like a real-valued footprint of n.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000720, A006530, A051903.

A076526 is a similar "footprint" of n.

with(numtheory):
a:= n-> pi(max(factorset(n)))*max(0, seq(i[2], i=ifactors(n)[2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 11 2020

a[n_] := PrimePi[(f = FactorInteger[n])[[-1, 1]]] * Max[f[[;; , 2]]]; Array[a, 100] (* Amiram Eldar, Jun 11 2020 *)

a(n) = if (n==1, 0, my(f=factor(n)); primepi(vecmax(f[, 1]))*vecmax(f[, 2])); \\ Michel Marcus, Jun 11 2020

a(n) = A000720(A006530(n)) * A051903(n). - Alois P. Heinz, Jun 11 2020

Edited by N. J. A. Sloane, Jun 15 2020

cp's OEIS Frontend

A076745 a(n) = the least positive integer k such that b(k) = n, where b(k) (A076526) is defined by b(k) = r * max{e_1,...,e_r} if k = p_1^e_1 ... p_r^e_r is the canonical prime factorization of k.

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A076558 a(n) = r * min(e_1, ..., e_r), where n = p_1^e_1 . .... p_r^e_r is the canonical prime factorization of n, a(1) = 0.

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A335603 a(n) = p*q where p is the sequential number (or PrimePi, A000720) of the largest prime divisor of n, and q is the maximal exponent in the canonical representation of n (A051903).

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cp's OEIS Frontend

A076745 a(n) = the least positive integer k such that b(k) = n, where b(k) (A076526) is defined by b(k) = r * max{e_1,...,e_r} if k = p_1^e_1 *...* p_r^e_r is the canonical prime factorization of k.

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Author

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Examples

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Programs

Mathematica

PARI

Formula

Extensions

A076558 a(n) = r * min(e_1, ..., e_r), where n = p_1^e_1 . .... p_r^e_r is the canonical prime factorization of n, a(1) = 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A335603 a(n) = p*q where p is the sequential number (or PrimePi, A000720) of the largest prime divisor of n, and q is the maximal exponent in the canonical representation of n (A051903).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A076745 a(n) = the least positive integer k such that b(k) = n, where b(k) (A076526) is defined by b(k) = r * max{e_1,...,e_r} if k = p_1^e_1 ... p_r^e_r is the canonical prime factorization of k.