A328622 In primorial base representation of n, multiply by 2 all other digits except the least significant, and reduce each such product modulo prime(k) (to get the new digit), where k > 1 is the position of the digit, then convert back to decimal.

0, 1, 4, 5, 2, 3, 12, 13, 16, 17, 14, 15, 24, 25, 28, 29, 26, 27, 6, 7, 10, 11, 8, 9, 18, 19, 22, 23, 20, 21, 60, 61, 64, 65, 62, 63, 72, 73, 76, 77, 74, 75, 84, 85, 88, 89, 86, 87, 66, 67, 70, 71, 68, 69, 78, 79, 82, 83, 80, 81, 120, 121, 124, 125, 122, 123, 132, 133, 136, 137, 134, 135, 144, 145, 148, 149, 146, 147, 126, 127, 130, 131

Offset

0,3

Links

Antti Karttunen, Table of n, a(n) for n = 0..30029

Index entries for sequences that are permutations of the natural numbers

Index entries for sequences related to primorial base

Formula

a(n) = A276085(A328618(A276086(n))).

Example

In primorial base (A049345) 199 is written as "6301" because 6*A002110(3) + 3*A002110(2) + 0*A002110(1) + 1*A002110(0) = 6*30 + 3*6 + 0*2 + 1*1 = 199. Multiplying each digit except the least significant by 2, and then reducing them modulo the corresponding prime leaves us with 2*6 mod 7, 2*3 mod 5, 2*0 mod 3, (with the least significant 1 staying the same), so we get "5101", which is the primorial base expansion of 157, thus a(199) = 157.
For 157, the new "doubled and reduced" expansion is 2*5 mod 7, 2*1 mod 5, 2*0 mod 3 and the trailing 1 stays intact, so we get "3201", which is the primorial base expansion of 103, thus a(157) = 103.

Prog

(PARI)
A002110(n) = prod(i=1, n, prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328618(n) = { my(f = factor(n), m, q); for(k=1, #f~, q = (f[k, 2]\f[k, 1]); m = (f[k, 2]%f[k, 1]); if(m&&(f[k, 1]!=2), f[k, 2] = q*f[k, 1] + ((2*f[k, 2])%f[k, 1]))); factorback(f); };
A328622(n) = A276085(A328618(A276086(n)));

Crossrefs

Cf. A328623 (inverse), and also A289234.

Cf. A002110, A049345, A276085, A276086, A328618.

Sequence in context: A261098 A216252 A335615 * A338248 A328623 A225901

Adjacent sequences: A328619 A328620 A328621 * A328623 A328624 A328625

Keyword

nonn,base

Author

Antti Karttunen, Oct 23 2019

Status

approved