A328625 In primorial base representation of n, multiply all other digits except the least significant with 1+{their right hand side neighbor}, and reduce each modulo prime(k) (with k > 1) to get a new digit for the position k (the least significant digit stays as it is), then convert back to decimal.

0, 1, 2, 5, 4, 3, 6, 7, 14, 17, 22, 21, 12, 13, 26, 29, 10, 9, 18, 19, 8, 11, 28, 27, 24, 25, 20, 23, 16, 15, 30, 31, 32, 35, 34, 33, 66, 67, 74, 77, 82, 81, 102, 103, 116, 119, 100, 99, 138, 139, 128, 131, 148, 147, 174, 175, 170, 173, 166, 165, 60, 61, 62, 65, 64, 63, 126, 127, 134, 137, 142, 141, 192, 193, 206, 209, 190, 189, 48, 49, 38, 41, 58, 57, 114

Offset

0,3

Comments

In primorial base (see A049345) we keep the least significant digit (0 or 1) intact, and replace each digit d(i) left of that (for i >= 2) with a new digit value computed as d(i)*(1+d(i-1)) mod prime(i). a(n) is then the newly constructed primorial expansion converted back to decimal.

Because for all primes p, Z_p is a field (not just a ring), this sequence is a permutation of nonnegative integers, and roughly speaking, offers a kind of analog of A003188 for primorial base system. Note however that it is the digit neighbor on the right (not left) hand side that affects here what will be the new digit at each position.

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(A328624(n)).

For all n, A328620(a(n)) = A328620(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 left of the least significant by 1+{digit one step right}, and reducing modulo the corresponding prime yields 4*6 mod 7, 1*3 mod 5, 2*0 mod 3, (with the least significant 1 staying the same), so we get "3301", which is the primorial base expansion of 109, thus a(199) = 109.

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)); };
A328624(n) = { my(m=1, p=2, e, g=1); while(n, e = (n%p); m *= (p^((g*e)%p)); g = e+1; n = n\p; p = nextprime(1+p)); (m); };
A328625(n) = A276085(A328624(n));

Crossrefs

Cf. A002110, A049345, A276085, A328620, A328624, A328626 (inverse permutation), A328628.

Cf. also A003188, A289234, A328622.

Sequence in context: A128173 A265362 A264989 * A060125 A265361 A328626

Adjacent sequences: A328622 A328623 A328624 * A328626 A328627 A328628

Keyword

nonn

Author

Antti Karttunen, Oct 23 2019

Status

approved