A295417 Self-inverse permutation of natural numbers: in prime factorization of n replace each positive prime exponent e with max + min - e, where max = A051903(n) and min = A051904(n).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 13, 14, 15, 16, 17, 12, 19, 50, 21, 22, 23, 54, 25, 26, 27, 98, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 250, 41, 42, 43, 242, 75, 46, 47, 162, 49, 20, 51, 338, 53, 24, 55, 686, 57, 58, 59, 450, 61, 62, 147, 64, 65, 66

Offset: 1

Author

Rémy Sigrist, Nov 22 2017

Keywords

• nonn

Comments

This sequence was inspired by A293448.

This sequence first differs from A293448 at n = 42: a(42) = 42 whereas A293448(42) = 70.

a(A293448(n)) = A293448(a(n)) for any n > 0.

a(n) = n iff n belongs to A072774.

f(n) = f(a(n)) for any n > 0 and f in { A001221, A006530, A007947, A020639, A051903, A051904 }.

The lines visible in the logarithmic scatterplot of the sequence seems to correspond to integer sets where the function A062760 is constant (see logarithmic scatterplot in Links section).

Examples

			For n = 1620:
- 1620 = 2^2 * 3^4 * 5,
- A051903(1620) = 4 and A051904(1620) = 1,
- a(1620) = 2^(4+1-2) * 3^(4+1-4) * 5^(4+1-1) = 2^3 * 3 * 5^4 = 15000.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Colored logarithmic scatterplot of the first 1000000 terms (where the color is function of A062760(n))
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A001221, A006530, A007947, A020639, A051903, A051904, A062760, A072774 (fixed points), A293448.

Programs

PARI

a(n) = { my(f=factor(n)); if(#f~<=1, return(n), my(mi=vecmin(f[,2]), ma=vecmax(f[,2])); return(prod(i=1, #f~, f[i,1]^(ma+mi-f[i,2])))) }

Formula

a(n) = A007947(n)^(A051903(n) + A051904(n)) / n.