A260213 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.

114, 120, 147, 198, 264, 420, 500, 506, 513, 525, 533, 550, 558, 568, 581, 648, 1102, 1116, 1168, 1302, 1320, 1377, 1680, 1692, 1710, 1720, 1734, 1755, 1771, 1872, 2106, 2132, 2310, 2332, 2380, 2664, 2714, 2736, 2790, 2914, 2940, 3312

Offset

1,1

Comments

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).

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).

Links

Pieter Post and Giovanni Resta, Table of n, a(n) for n = 1..295 (terms < 10^12, first 141 terms from Pieter Post)

Example

114 = (1 + 5)*(14 + 5) and 114 = (20 - 1)*(20 - 14).
1710 = (17 + 28)*(10 + 28) and 1710 = (55 - 17)*(55 - 10).

Prog

(Python)
def pod(n, m, a):
    kk = 1
    while n > 0:
        kk= kk*(n%m+a)
        n =int(n//m)
    return kk
for c in range (1, 10000):
    for a in range (1, 100):
        if c==pod(c, 100, a):
            print (c)
(PARI)
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

Crossrefs

Cf. A055482.

Sequence in context: A294309 A138693 A272303 * A228961 A057440 A113537

Adjacent sequences: A260210 A260211 A260212 * A260214 A260215 A260216

Keyword

base,nonn,less

Author

Pieter Post, Jul 19 2015

Status

approved