A384540 Numbers in A384537 that are not prime powers: composite numbers, not being prime powers, that are equal to the concatenation of the primes and exponents in their prime factorizations in some bases.

4617, 29767, 255987, 395847, 631463, 1332331, 25640947

Offset

1,1

Comments

See A384537 for more information.

Links

Table of n, a(n) for n=1..7.

Example

In base 11: 3518 = 3^5 * 18 (in decimal: 4617 = 3^5 * 19);
In base 12: 15287 = 15^2 * 87 (in decimal: 29767 = 17^2 * 103);
In base 2: 111110011111110011 = 11^11 * 10011 * 111110011 (in decimal: 255987 = 3^3 * 19 * 499);
In base 362: (3,7,181)_362 = 3^7 * 181.
In base 300: (7,4,263)_300 = 7^4 * 263.
In base 57: 7B4D = 7 * B^4 * D (in decimal: 1332331 = 7 * 11^4 * 13).
In base 1228: (17,4,307)_1228 = 17^4*307.

Prog

(PARI) F(n, b) = my(f=factor(n), d=[]); for(i=1, #f~, d=concat(d, digits(f[i, 1], b)); if(f[i, 2]>1, d=concat(d, digits(f[i, 2], b)))); fromdigits(d, b)
isA384540(n) = {
if(issquarefree(n), return(0)); my(f=factor(n), dr);
if(#f~ == 1, return(0));
dr = if(f[#f~, 2] == 1, f[#f~, 1], f[#f~, 2]);
fordiv(n - dr, b, if(b>=2 && F(n, b)==n, return(b))); return(0);
} \\ returns the (smallest) base to which n is a Davis number whenever possible

Crossrefs

Cf. A080670, A195264, A230625, A384537.

Sequence in context: A265926 A365483 A260054 * A253115 A189983 A295431

Adjacent sequences: A384537 A384538 A384539 * A384541 A384542 A384543

Keyword

nonn,hard,more

Author

Jianing Song, Jun 02 2025

Status

approved