

A114457


Smallest k > 0 such that abs(S(k)P(k)k) equals n, where S(k) is the sum and P(k) is the product of decimal digits of k or 0 if no such k exists.


1



1, 13, 2, 219, 724, 1285, 3, 23, 7789816, 11, 10, 2891, 4, 127, 226, 15, 3248, 163, 52, 31, 5, 33, 262, 12857, 24, 325, 16, 243, 38428, 617, 6, 68177, 172, 0, 62, 2275, 272, 22577, 118, 17, 40, 43, 7, 1339, 136, 25, 154, 143, 128, 125599, 34, 5619, 352, 1483
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OFFSET

0,2


COMMENTS

a(33) > 2*10^9; then sequence continues 62, 2275, 272, 22577, 118, 17, 40, 43, 7, 1339, 136, 25, 154, 143, 128, 125599, 34, 5619, 352, 1483, 18, 145, 8, 15457, 173, 14963, 60, 1727, 517, 1197, 1787456, 235, 642, 53, 116, ...  Robert G. Wilson v, Dec 14 2005
a(33) = 0. Modification of David W. Wilson's proof for A038369 shows that if a(33) > 0, then a(33) has at most 84 digits. This allows an exhaustive search of numbers of the form 2^a*3^b*5^c*7^d which shows that no such number exists. Other values of n for which a(n) is currently unknown and may be equal to 0 (based on analysis of numbers with at most 20 digits) are: 69, 111, 127, 146, 168, 172, 233, 243, 249, 273, 279, 281, 316, 327, 372, 533, 557, 579, 587, 621, 623, 647, 649, 676, 683, 713, 721, 816, 819, 821, 827, 861, 872, 917, 926, 927, 928, 939, 983, 996, 999, ...  Chai Wah Wu, Nov 22 2015
a(69) = a(111) = 0. To compute a(111), numbers of at most 85 digits were checked.  Chai Wah Wu, Dec 04 2015


LINKS



MATHEMATICA

f[n_] := Block[{k = 1}, While[id = IntegerDigits@k; Abs[(Plus @@ id)(Times @@ id)  k] != n, k++ ]; k]; Table[ f[n], {n, 0, 54}] (* Robert G. Wilson v, Dec 14 2005 *)


PROG

(C++) unsigned long long f(int n = 33) { for (unsigned long long i = 0;; i++) { unsigned long long copy = i, prod = 1, sum = 0; while (i) { sum += i%10; prod *= i%10; i/=10; } if (abs(sum * prod  i == n) { return i; } } } // Floris M. Velleman, Dec 17 2014
(PARI) f(k) = my(d=digits(k)); abs(sum(j=1, #d, d[j])*prod(j=1, #d, d[j])  k);
a(n) = {k = 1; while(f(k) != n, k++); k; } \\ Michel Marcus, Jan 02 2015


CROSSREFS



KEYWORD

nonn,base


AUTHOR



EXTENSIONS

Added a(33), edited definition and verified a(34)a(68) by Chai Wah Wu, Nov 22 2015


STATUS

approved



