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
Examples
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.
Programs
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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
Comments