A260213 - OEIS

%I #45 Aug 10 2021 16:55:09

%S 114,120,147,198,264,420,500,506,513,525,533,550,558,568,581,648,1102,

%T 1116,1168,1302,1320,1377,1680,1692,1710,1720,1734,1755,1771,1872,

%U 2106,2132,2310,2332,2380,2664,2714,2736,2790,2914,2940,3312

%N Numbers j such that j = (c_1 + k)*(c_2 + k)*...*(c_m + k) for some k > 0 where c_1, c_2, ..., c_m is the centesimal expansion of j.

%C k cannot be larger than 99, because in that case the product of the terms c_i+k is larger than the number j itself. For j up to 10^12, the highest value for k is 71, for 910602 = (71+91)*(71+6)*(71+2).

%C All terms j < 10000 have the following property. j = c_1//c_2, so j = (c_1 + k)*(c_2 + k). Let kk = c_1 + c_2 + k then j = (kk - c_1)*(kk - c_2). For example, 513 = (5 + 14)*(13 + 14), kk = 5 + 13 + 14 = 32, so 513 = (32 - 5)*(32 - 13).

%H Pieter Post and Giovanni Resta, <a href="/A260213/b260213.txt">Table of n, a(n) for n = 1..295</a> (terms < 10^12, first 141 terms from Pieter Post)

%e 114 = (1 + 5)*(14 + 5) and 114 = (20 - 1)*(20 - 14).

%e 1710 = (17 + 28)*(10 + 28) and 1710 = (55 - 17)*(55 - 10).

%o (Python)

%o def pod(n,m,a):

%o     kk = 1

%o     while n > 0:

%o         kk= kk*(n%m+a)

%o         n =int(n//m)

%o     return kk

%o for c in range (1,10000):

%o     for a in range (1,100):

%o         if c==pod(c,100,a):

%o             print (c)

%o (PARI)

%o is(n)=my(d=digits(n,100),t); while((t=vecprod(d))<n, d+=vector(#d,i,1)); t==n && n>99 \\ _Charles R Greathouse IV_, Aug 28 2015

%Y Cf. A055482.

%K base,nonn,less

%O 1,1

%A _Pieter Post_, Jul 19 2015