A392879 Let p = A129252(n), if p = 1, then a(n) = n, otherwise a(n) = n * p^((e mod p)-e) * q^floor(e/p), where e is p-adic valuation of n and q is the least prime number greater than p.

1, 2, 3, 3, 5, 6, 7, 6, 9, 10, 11, 9, 13, 14, 15, 9, 17, 18, 19, 15, 21, 22, 23, 18, 25, 26, 5, 21, 29, 30, 31, 18, 33, 34, 35, 27, 37, 38, 39, 30, 41, 42, 43, 33, 45, 46, 47, 27, 49, 50, 51, 39, 53, 10, 55, 42, 57, 58, 59, 45, 61, 62, 63, 27, 65, 66, 67, 51, 69, 70, 71, 54, 73, 74, 75, 57, 77, 78, 79, 45, 15, 82

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..22500

Index entries for sequences related to primorial base.

Formula

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

Example

For n = 12 = 2^2 * 3, the least prime that "overflows" (has an exponent >= the prime itself) is 2, so we divide by 2^(2-(2 mod 2)) and multiply by 3^floor(2/2), i.e., divide by 4 and multiply by 3, to obtain a(12) = 9.
For n = 27 = 3^3, the least prime which overflows is 3, so we divide by 3^(3-(3 mod 3)) and multiply by 5^floor(3/3), i.e., divide by 3^3 and multiply by 5, to obtain a(27) = 5.
For n = 36 = 2^2 * 3^2, the least prime which overflows is 2, so we divide by 2^2 and multiply by 3, to obtain a(36) = 3^3 = 27.
For n = 54 = 2^1 * 3^3, the least prime which overflows is 3, so we divide by 3^(3-(3 mod 3)) and multiply by 5^floor(3/3), i.e., divide by 3^3 and multiply by 5, to obtain a(54) = 2*5 = 10.
For n = 72 = 2^3 * 3^2, the least prime which overflows is 2, so we divide by 2^(3-(3 mod 2)), i.e., by 2^2, and multiply by 3^floor(3/2), i.e., by 3, to obtain a(72) = 54 = 2 * 3^3.

Prog

(PARI)
A129252(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(p)); if(pp > n, return(1))); };
A392879(n) = { my(p=A129252(n), e); if(1==p, return(n), e = valuation(n, p); (n * p^((e%p)-e) * nextprime(1+p)^(e\p))); };

Crossrefs

Cf. A048103 (fixed points), A129252, A276085, A276086, A392878, A392880 (number of iterations needed to reach A392878(n)).

Cf. also A277886 for somewhat analogous sequence for base-2.

Sequence in context: A384249 A392877 A392878 * A255563 A331288 A115350

Adjacent sequences: A392876 A392877 A392878 * A392880 A392881 A392882

Keyword

nonn,easy

Author

Antti Karttunen, Jan 31 2026

Status

approved