A351576 Factorial base expansion of n reinterpreted as a primorial base expansion, then converted back to decimal.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 0

Author

Antti Karttunen, Apr 01 2022

Keywords

• nonn
• base

Examples

			n = 313 has factorial base representation (see A007623) "23001" because 2*5! + 3*4! + 1*1! = 240+72+1 = 313. When this is reinterpreted as a primorial base expansion (see A049345), we obtain 2*A002110(4) + 3*A002110(3) + 1*A002110(0) = 511, therefore a(313) = 511.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..40319
• Index entries for sequences related to factorial base representation.
• Index entries for sequences related to primorial base.

Crossrefs

Cf. A000142, A002110, A007623, A049345, A276076, A276085.

Cf. also A276156.

Programs

Mathematica

a[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; FromDigits[Reverse[s], MixedRadix[Reverse@ Prime@ Range@ Length[s]]]]; Array[a, 100, 0] (* Amiram Eldar, Feb 07 2024 *)

PARI

A002110(n) = prod(i=1,n,prime(i));
A276076(n) = { my(i=0,m=1,f=1,nextf); while((n>0),i=i+1; nextf = (i+1)*f; if((n%nextf),m*=(prime(i)^((n%nextf)/f));n-=(n%nextf));f=nextf); m; };
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A351576(n) = A276085(A276076(n));

Formula

a(n) = A276085(A276076(n)).