A161955 TITO2(n): The operation A161594 in binary, digit-reversals carried out in base 2.

1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 1, 17, 9, 19, 5, 21, 11, 23, 3, 19, 13, 27, 7, 29, 15, 31, 1, 57, 17, 49, 9, 37, 19, 33, 5, 41, 21, 43, 11, 45, 23, 47, 3, 35, 19, 51, 13, 53, 27, 65, 7, 105, 29, 59, 15, 61, 31, 63, 1, 59, 57, 67, 17, 117, 49, 71, 9, 73, 37, 105, 19, 109

Offset: 1

Author

Tanya Khovanova, Jun 22 2009

Keywords

• base
• nonn

Comments

The TITO function in binary: Represent n as a product of its prime factors in binary.

Revert the binary digits of each of these factors, then multiply them with the same multiplicities as in n--so the base-2 representation does not affect the exponents in the canonical prime factorization. Reverse the product in binary to get a(n).

Examples

			To calculate TITO2(n=99): 99 = 3^3*11. Prime factors 3 and 11 in binary are 11 and 1011 correspondingly. Reversing those numbers we get 11 and 1101. The product with multiplicities is the binary product of 11*11*1101 = 1110101. Reversing that we get 1010111, which corresponds to 87. Hence a(99) = 87.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..40000
• Tanya Khovanova, Turning Numbers Inside Out

Crossrefs

Cf. A161594.

Programs

Maple

r:= proc(n) local m, t; m, t:=n, 0; while m>0
      do t:=2*t+irem(m, 2, 'm') od; t end:
a:= n-> r(mul(r(i[1])^i[2], i=ifactors(n)[2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 29 2017

Mathematica

reverseBinPower[{n_, k_}] := FromDigits[Reverse[IntegerDigits[n, 2]], 2]^k fBin[n_] := FromDigits[ Reverse[IntegerDigits[ Times @@ Map[reverseBinPower, FactorInteger[n]], 2]], 2] Table[fBin[n], {n, 200}]

Formula

a(n) = A030101(A162742(n)) - R. J. Mathar, Aug 03 2009

Extensions

Edited by R. J. Mathar, Aug 03 2009