A034684 -id:A034684 - cp's OEIS frontend

A304233 If n = Product (p_j^k_j) then a(n) = min{p_j^k_j}*max{p_j^k_j}.

1, 4, 9, 16, 25, 6, 49, 64, 81, 10, 121, 12, 169, 14, 15, 256, 289, 18, 361, 20, 21, 22, 529, 24, 625, 26, 729, 28, 841, 10, 961, 1024, 33, 34, 35, 36, 1369, 38, 39, 40, 1681, 14, 1849, 44, 45, 46, 2209, 48, 2401, 50, 51, 52, 2809, 54, 55, 56, 57, 58, 3481, 15, 3721, 62, 63, 4096, 65

Offset: 1

Ilya Gutkovskiy, May 08 2018

nonn

			a(60) = 15 because 60 = 2^2*3*5, min{2^2,3,5} = 3, max{2^2,3,5} = 5 and 3*5 = 15.

Ilya Gutkovskiy, Logarithmic scatter plot of a(n) up to n=30000
Eric Weisstein's World of Mathematics, Prime Factorization

Cf. A000977 (numbers n such that a(n) < n), A002110, A007774 (fixed points), A034684, A034699, A066048, A100484, A141809.

a[n_] := Min[#[[1]]^#[[2]] & /@FactorInteger[n]] Max[#[[1]]^#[[2]] & /@FactorInteger[n]]; Table[a[n], {n, 65}]

a(n) = A034684(n)*A034699(n).
a(p^k) = p^(2*k) where p is a prime.
a(A002110(k)) = A100484(k).

A381131 If n = (p_1^e_1)(p_2^e_2)(p_3^e_3)*... and min(p_1^e_1,p_2^e_2,...) = p_k^e_k then a(n) = p_k, a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 3, 13, 2, 3, 2, 17, 2, 19, 2, 3, 2, 23, 3, 5, 2, 3, 2, 29, 2, 31, 2, 3, 2, 5, 2, 37, 2, 3, 5, 41, 2, 43, 2, 5, 2, 47, 3, 7, 2, 3, 2, 53, 2, 5, 7, 3, 2, 59, 3, 61, 2, 7, 2, 5, 2, 67, 2, 3, 2, 71, 2, 73, 2, 3, 2, 7, 2, 79, 5

Offset: 1

Ilya Gutkovskiy, Feb 14 2025

nonn

Cf. A020639, A034684, A088387, A381132, A381133.

Table[Exp[MangoldtLambda[Min @@ (#[[1]]^#[[2]] & /@ FactorInteger[n])]], {n, 80}]

a(n) = if (n==1, 1, my(f=factor(n), v=vector(#f~, k, f[k,1]^f[k,2]), m=vecmin(v), i=select(x->(x==m), v, 1)); f[i[1], 1]); \\ Michel Marcus, Feb 19 2025

A381132 If n = (p_1^e_1)(p_2^e_2)(p_3^e_3)*... and min(p_1^e_1,p_2^e_2,...) = p_k^e_k then a(n) = pi(p_k), a(1) = 0.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Feb 14 2025

nonn

Cf. A000720, A034684, A055396, A108230, A381131, A381133.

Table[PrimePi[Exp[MangoldtLambda[Min @@ (#[[1]]^#[[2]] & /@ FactorInteger[n])]]], {n, 80}]

a(n) = if (n==1, 0, my(f=factor(n), v=vector(#f~, k, f[k,1]^f[k,2]), m=vecmin(v), i=select(x->(x==m), v, 1)); primepi(f[i[1], 1])); \\ Michel Marcus, Feb 19 2025

A381133 If n = (p_1^e_1)(p_2^e_2)(p_3^e_3)*... and min(p_1^e_1,p_2^e_2,...) = p_k^e_k then a(n) = e_k, a(1) = 0.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Feb 14 2025

nonn

Cf. A034684, A051904, A088388, A381131, A381132.

Table[PrimeOmega[Min @@ (#[[1]]^#[[2]] & /@ FactorInteger[n])], {n, 100}]

a(n) = if (n==1, 0, my(f=factor(n), v=vector(#f~, k, f[k,1]^f[k,2]), m=vecmin(v), i=select(x->(x==m), v, 1)); f[i[1], 2]); \\ Michel Marcus, Feb 19 2025

A216972 a(4n+2) = 2, otherwise a(n) = n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 2, 7, 8, 9, 2, 11, 12, 13, 2, 15, 16, 17, 2, 19, 20, 21, 2, 23, 24, 25, 2, 27, 28, 29, 2, 31, 32, 33, 2, 35, 36, 37, 2, 39, 40, 41, 2, 43, 44, 45, 2, 47, 48, 49, 2, 51, 52, 53, 2, 55, 56, 57, 2, 59, 60, 61, 2, 63, 64, 65, 2, 67, 68, 69, 2

Offset: 0

Paul Curtz, Sep 21 2012

For n>0, a(n) is the denominator of A214282(n)/(-A214283(n+1)):
1/1, 1/2, 1/3, 3/4, 3/5, 1/2, 3/7, 5/8, 5/9, ...
For n>0, a(n) is the denominator of A214283(n)/A214283(n+1):
0/1, 1/2, 2/3, 3/4, 2/5, 1/2, 4/7, 5/8, 4/9, ...
a(n), first and second differences:
0, 1, 2, 3,  4,  5,  2, 7,  8,  9,  2, 11,  12, ...
1, 1, 1, 1,  1, -3,  5, 1,  1, -7,  9,  1,   1, ...
0, 0, 0, 0, -4,  8, -4, 0, -8, 16, -8,  0, -12, ...

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A028233, A034684, A000265, A026741, A060819, A145979, A214682.

[n mod 4 eq 2 select 2 else n: n in [0..70]]; // Bruno Berselli, Sep 26 2012

a[n_] := If[Mod[n, 4] == 2, 2, n]; Table[a[n], {n, 0, 81}] (* Jean-François Alcover, Sep 25 2012 *)
LinearRecurrence[{0,0,0,2,0,0,0,-1},{0,1,2,3,4,5,2,7},80] (* Harvey P. Dale, Nov 06 2017 *)

makelist(expand(2+(4-(1+(-1)^n)*(1-%i^n))*(n-2)/4), n, 0, 70); /* Bruno Berselli, Sep 26 2012 */

def A216972(n): return 2 if n&3==2 else n # Chai Wah Wu, Jan 31 2024

a(n) = 2*a(n-4) - a(n-8).
a(n+4) - a(n) = 4*A152822(n).
a(2n) + a(2n+1) = |A141124(n)|.
a(4n) + a(4n+1) + a(4n+2) + a(4n+3) = 6*A005408(n) = A017593(n).
G.f.: (x+2*x^2+3*x^3+4*x^4+3*x^5-2*x^6+x^7) / (1-2*x^4+x^8). - Jean-François Alcover, Sep 25 2012
a(n) = 2+(4-(1+(-1)^n)*(1-i^n))*(n-2)/4, where i=sqrt(-1). - Bruno Berselli, Sep 26 2012
a(2n) = 2*|A009531(n)|, a(2n+1) = 2n+1. - Bruno Berselli, Sep 27 2012

A304181 If n = Product (p_j^k_j) then a(n) = min{p_j}^min{k_j}.

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 2, 11, 2, 13, 2, 3, 16, 17, 2, 19, 2, 3, 2, 23, 2, 25, 2, 27, 2, 29, 2, 31, 32, 3, 2, 5, 4, 37, 2, 3, 2, 41, 2, 43, 2, 3, 2, 47, 2, 49, 2, 3, 2, 53, 2, 5, 2, 3, 2, 59, 2, 61, 2, 3, 64, 5, 2, 67, 2, 3, 2, 71, 4, 73, 2, 3, 2, 7, 2, 79, 2, 81, 2, 83, 2, 5

Offset: 1

Ilya Gutkovskiy, May 07 2018

nonn

			a(72) = 4 because 72 = 2^3*3^2, min{2,3} = 2, min{3,2} = 2 and 2^2 = 4.

Eric Weisstein's World of Mathematics, Least Prime Factor
Index entries for sequences computed from exponents in factorization of n

Cf. A000040, A000961 (fixed points), A005117, A020639, A028233, A034684, A051904, A073481, A081811, A304180.

Table[(FactorInteger[n][[1, 1]])^(Min @@ Last /@ FactorInteger[n]), {n, 85}]

a(n) = A020639(n)^A051904(n).
a(p^k) = p^k where p is a prime.
a(A005117(k)) = A073481(k).

cp's OEIS Frontend

A304233 If n = Product (p_j^k_j) then a(n) = min{p_j^k_j}*max{p_j^k_j}.

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A381131 If n = (p_1^e_1)*(p_2^e_2)*(p_3^e_3)*... and min(p_1^e_1,p_2^e_2,...) = p_k^e_k then a(n) = p_k, a(1) = 1.

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A381132 If n = (p_1^e_1)*(p_2^e_2)*(p_3^e_3)*... and min(p_1^e_1,p_2^e_2,...) = p_k^e_k then a(n) = pi(p_k), a(1) = 0.

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A381133 If n = (p_1^e_1)*(p_2^e_2)*(p_3^e_3)*... and min(p_1^e_1,p_2^e_2,...) = p_k^e_k then a(n) = e_k, a(1) = 0.

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A216972 a(4n+2) = 2, otherwise a(n) = n.

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A304181 If n = Product (p_j^k_j) then a(n) = min{p_j}^min{k_j}.

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A381131 If n = (p_1^e_1)(p_2^e_2)(p_3^e_3)*... and min(p_1^e_1,p_2^e_2,...) = p_k^e_k then a(n) = p_k, a(1) = 1.

A381132 If n = (p_1^e_1)(p_2^e_2)(p_3^e_3)*... and min(p_1^e_1,p_2^e_2,...) = p_k^e_k then a(n) = pi(p_k), a(1) = 0.

A381133 If n = (p_1^e_1)(p_2^e_2)(p_3^e_3)*... and min(p_1^e_1,p_2^e_2,...) = p_k^e_k then a(n) = e_k, a(1) = 0.