cp's OEIS Frontend

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.

%I A384540 #6 Jun 09 2025 10:39:29
%S A384540 4617,29767,255987,395847,631463,1332331,25640947
%N 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.
%C A384540 See A384537 for more information.
%e A384540 In base 11: 3518 = 3^5 * 18 (in decimal: 4617 = 3^5 * 19);
%e A384540 In base 12: 15287 = 15^2 * 87 (in decimal: 29767 = 17^2 * 103);
%e A384540 In base 2: 111110011111110011 = 11^11 * 10011 * 111110011 (in decimal: 255987 = 3^3 * 19 * 499);
%e A384540 In base 362: (3,7,181)_362 = 3^7 * 181.
%e A384540 In base 300: (7,4,263)_300 = 7^4 * 263.
%e A384540 In base 57: 7B4D = 7 * B^4 * D (in decimal: 1332331 = 7 * 11^4 * 13).
%e A384540 In base 1228: (17,4,307)_1228 = 17^4*307.
%o A384540 (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)
%o A384540 isA384540(n) = {
%o A384540 if(issquarefree(n), return(0)); my(f=factor(n), dr);
%o A384540 if(#f~ == 1, return(0));
%o A384540 dr = if(f[#f~, 2] == 1, f[#f~, 1], f[#f~, 2]);
%o A384540 fordiv(n - dr, b, if(b>=2 && F(n, b)==n, return(b))); return(0);
%o A384540 } \\ returns the (smallest) base to which n is a Davis number whenever possible
%Y A384540 Cf. A080670, A195264, A230625, A384537.
%K A384540 nonn,hard,more
%O A384540 1,1
%A A384540 _Jianing Song_, Jun 02 2025