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Sum of digits is greater than or equal to product of digits.
+10
9
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 30, 31, 40, 41, 50, 51, 60, 61, 70, 71, 80, 81, 90, 91, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 130, 131, 132
MATHEMATICA
Select[Range[150], Total[IntegerDigits[#]]>=Times@@IntegerDigits[#]&] (* Harvey P. Dale, Sep 27 2023 *)
PROG
(PARI) SumD(x)= { s=0; while (x>9, s+=x-10*(x\10); x\=10); return(s + x) } ProdD(x)= { p=1; while (x>9, p*=(x-10*(x\10)); x\=10); return(p*x) } { n=0; for (a=1, 10^9, if (SumD(a) >= ProdD(a), write("b062996.txt", n++, " ", a); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 15 2009
Numbers with property that sum of digits is less than or equal to product of digits.
+10
9
1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 23, 24, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 39, 42, 43, 44, 45, 46, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 69, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 97, 98
MATHEMATICA
Select[Range[100], Total[IntegerDigits[#]]<=Times@@IntegerDigits[#]&] (* Harvey P. Dale, Feb 21 2017 *)
PROG
(PARI) SumD(x)= { s=0; while (x>9, s+=x-10*(x\10); x\=10); return(s + x) } ProdD(x)= { p=1; while (x>9, p*=(x-10*(x\10)); x\=10); return(p*x) } { n=0; for (a=1, 10^9, if (SumD(a) <= ProdD(a), write("b062998.txt", n++, " ", a); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 15 2009 (PARI) is_ A062998(n)={normlp(n=digits(n), 1)<=prod(i=1, #n, n[i])} \\ M. F. Hasler, Oct 29 2014
CROSSREFS
Not the same as A037344 (contains 124).
a(n) = (sum of digits of n) - (product of digits of n).
+10
8
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, 3, 1, -1, -3, -5, -7, -9, -11, -13, -15, 4, 1, -2, -5, -8, -11, -14, -17, -20, -23, 5, 1, -3, -7, -11, -15, -19, -23, -27, -31, 6, 1, -4, -9, -14, -19, -24, -29, -34, -39, 7, 1, -5, -11, -17, -23, -29, -35, -41, -47, 8, 1, -6, -13, -20, -27
EXAMPLE
a(23) = 2 + 3 - 2*3 = -1.
a(49) = -(4*9) + (4 + 9) = -36 + 13 = -23.
MATHEMATICA
a[n_] := (t = IntegerDigits[n]; Plus @@ t - Times @@ t); Table[ a[n], {n, 0, 75}] (* Robert G. Wilson v *)
EXTENSIONS
Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jun 22 2001
Numbers k with property that the sum of the digits of k is strictly less than the product of the digits of k.
+10
6
23, 24, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 39, 42, 43, 44, 45, 46, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 69, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 97, 98, 99, 124, 125, 126, 127
MATHEMATICA
Select[Range[128], (Plus @@ IntegerDigits[ # ]) < (Times @@ IntegerDigits[ # ]) &] ( Alonso del Arte, May 16 2005)
PROG
(PARI) SumD(x)= { s=0; while (x>9, s+=x-10*(x\10); x\=10); return(s + x) } ProdD(x)= { p=1; while (x>9, p*=(x-10*(x\10)); x\=10); return(p*x) } { n=0; for (a=1, 10^9, if (SumD(a) < ProdD(a), write("b062999.txt", n++, " ", a); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 15 2009
Zerofree numbers having product of digits less than or equal to sum of digits.
+10
4
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 31, 41, 51, 61, 71, 81, 91, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 131, 132, 141, 151, 161, 171, 181, 191, 211, 212, 213, 221, 231, 311, 312, 321, 411
MAPLE
extend:= proc(t, b, d)
local i, j, m, s, p;
p:= t[2];
s:= t[3];
if s = 0 then if b=2 then j:= 3 else j:= 2 fi
else for j from 0 to d-nops(t[1]) while p*b^j <= s + j*b do od
fi:
seq([[op(t[1]), b$i], p*b^i, s+i*b], i=0..j-1);
end proc:
f:= proc(d)
local j, b, Res;
Res:= [seq([[1$j], 1, j], j=0..d)];
for b from 2 to 9 do
Res:= map(extend, Res, b, d)
od:
Res:= map(t -> op(combinat:-permute(t[1])), Res);
subs(0=NULL, sort(map(t -> add(t[i]*10^(i-1), i=1..nops(t)), Res)));
end proc:
MATHEMATICA
m[w_] := Flatten@Table[i, {i, 9}, {w[[i]]}]; a[upd_] := Union@ Flatten@ Table[ FromDigits /@ Flatten[Permutations /@ m /@ Select[ Flatten[ Permutations /@ (IntegerPartitions[d + 9, {9}, Range[d + 1]] - 1), 1], Times @@ (Range[9]^#) <= Total[# Range[9]] &], 1], {d, upd}]; a[12] (* terms with up to 12 digits, Giovanni Resta, May 19 2015 *) zfnQ[n_]:=Module[{idn=IntegerDigits[n]}, FreeQ[idn, 0]&&Times@@idn <= Total[ idn]]; Select[Range[500], zfnQ] (* Harvey P. Dale, Jun 29 2019 *)
PROG
(PARI) is(n)={my(d=digits(n)); my(p=prod(i=1, #d, d[i])); 0 < p && p<=vecsum(d) } \\ David A. Corneth, May 15 2015
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