A324106 Multiplicative with a(p^e) = A005940(p^e).

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 25, 18, 15, 16, 11, 14, 21, 20, 27, 30, 45, 24, 49, 50, 75, 36, 125, 30, 81, 32, 45, 22, 45, 28, 55, 42, 75, 40, 77, 54, 105, 60, 35, 90, 135, 48, 121, 98, 33, 100, 245, 150, 75, 72, 63, 250, 375, 60, 625, 162, 63, 64, 125, 90, 39, 44, 135, 90, 99, 56, 91, 110, 147, 84, 135, 150, 189, 80, 143, 154, 231, 108, 55

Offset: 1

Author

Antti Karttunen, Feb 15 2019

Keywords

• nonn
• mult

Comments

Question: are there any other numbers n besides 1 and those in A070776, for which a(n) = A005940(n)? At least not below 2^25. This is probably easy to prove.

Examples

			For n = 85 = 5*17, a(85) = A005940(5) * A005940(17) = 5*11 = 55. Note that A005940(5) is obtained from the binary expansion of 5-1 = 4, which is "100", and A005940(17) is obtained from the binary expansion of 17-1 = 16, which is "1000".
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Michael De Vlieger, Fan style binary tree of a(n), n = 1..2^12, color coded to show the smallest values in the range r = (2^r - 1)..2^(r+1) in blue and highlighting the largest with red.
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A005940, A070776, A324107 (fixed points), A324108, A324109.

Programs

Mathematica

nn = 128; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Array[Times @@ Map[a, Power @@@ FactorInteger[#]] &, nn] (* Michael De Vlieger, Sep 18 2022 *)

PARI

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A324106(n) = { my(f=factor(n)); prod(i=1, #f~, A005940(f[i,1]^f[i,2])); };