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Search: a062996 -id:a062996
Displaying 1-8 of 8 results found. page 1
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A062998 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Harry J. Smith, Table of n, a(n) for n=1,...,1000
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
Cf. A007953, A007954, A034710, A062329, A062996, A062997, A062998, A062999.
Not the same as A037344 (contains 124).
KEYWORD
base,nonn,easy
AUTHOR
Henry Bottomley, Jun 29 2001
STATUS
approved
A062329 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,21
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..1000
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 *)
CROSSREFS
Cf. A007953, A007954, A034710, A062996, A062997, A062998, A062999, A070565.
A063543 has a similar graph.
KEYWORD
sign,base,easy,look
AUTHOR
Amarnath Murthy, Jun 21 2001
EXTENSIONS
Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jun 22 2001
Signed version from Henry Bottomley, Jun 29 2001
STATUS
approved
A062997 Sum of digits is strictly greater than product of digits. +10
6
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 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, 130, 131, 140, 141, 150, 151, 160, 161, 170 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Every multiple of 10 is a term.
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..1000
EXAMPLE
118 is a term as 1 + 1 + 8 = 10, 10 > 8 and 8 = 1 * 1 * 8.
MATHEMATICA
Select[Range[170], (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("b062997.txt", n++, " ", a); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 15 2009
CROSSREFS
Cf. A007953, A007954, A034710, A062329, A062996, A062998, A062999.
KEYWORD
base,nonn,easy
AUTHOR
Amarnath Murthy, Jun 27 2001
EXTENSIONS
Extended by Larry Reeves (larryr(AT)acm.org) and Henry Bottomley, Jun 29 2001
STATUS
approved
A062999 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..1000
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
CROSSREFS
Cf. A007953, A007954, A034710, A062329, A062996, A062997, A062998, A062999.
KEYWORD
base,nonn
AUTHOR
Henry Bottomley, Jun 29 2001
STATUS
approved
A254621 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Intersection of A052382 and A062996.
The repunit A002275(k), for k >= 2, appears at position A254622(k-1) + 1. - Wolfdieter Lang, Feb 23 2015
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000
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:
f(5); # Robert Israel, May 19 2015
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
CROSSREFS
Cf. A052382, A062996, A062997, A062998, A062999, A254622, A002275.
KEYWORD
nonn,base
AUTHOR
David A. Corneth, Feb 03 2015
STATUS
approved
A254622 Number of zerofree positive integers of at most n digits having product of digits less than or equal to sum of digits. +10
2
9, 27, 61, 124, 255, 474, 860, 1597, 2726, 4232, 6741, 10391, 16177, 24480, 34926, 47875, 65896, 92255, 139262, 198988, 271642, 357674, 474239, 609442, 773493, 1029477, 1404094, 1857856, 2392385, 3169652, 4059322, 5102419, 6482700, 8041992 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
See A254621 for the numbers with this property. - Wolfdieter Lang, Feb 23 2015
LINKS
David A. Corneth, Table of n, a(n) for n = 1..2000
CROSSREFS
Cf. A052382, A062996, A254621.
KEYWORD
nonn,base
AUTHOR
David A. Corneth, Feb 03 2015
STATUS
approved
A274125 Numbers having digits in nondecreasing order and sum of digits greater than or equal to the product of digits. +10
2
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122, 123, 1111, 1112, 1113, 1114, 1115, 1116, 1117, 1118, 1119, 1122, 1123, 1124, 11111, 11112, 11113, 11114, 11115, 11116, 11117, 11118, 11119, 11122 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Intersection of A062996 and A009994. Permuting the digits of the terms of this sequence gives A254621. Permutations of digits can be found in A261370. The union of A254621 and A011540 is A062996.
LINKS
David A. Corneth, Table of n, a(n) for n = 1..1000
MATHEMATICA
upto[nd_] := Sort[FromDigits /@ Reverse /@ Select[ Flatten[ IntegerPartitions[#, nd, Range@ 9] & /@ Range[9 nd], 1], Times @@ # <= Plus @@ # &]]; upto[8] (* Giovanni Resta, Jun 20 2016 *)
PROG
(PARI) is(n) = my(d=digits(n)); prod(i=1, #d, d[i]) <= vecsum(d) && vecsort(d) == d
CROSSREFS
Cf. A062996, A009994.
KEYWORD
nonn,base
AUTHOR
David A. Corneth, Jun 10 2016
STATUS
approved
A274126 Numbers with digits larger than 1 sorted by product of digits minus sum of digits, then by size. +10
2
2, 3, 4, 5, 6, 7, 8, 9, 22, 23, 24, 222, 25, 33, 26, 27, 34, 223, 28, 29, 35, 44, 224, 2222, 36, 233, 37, 45, 225, 38, 46, 226, 39, 55, 234, 2223, 47, 227, 333, 56, 48, 228, 235, 244, 2224, 22222, 49, 57, 229, 66, 236, 334, 2233, 58, 67, 245, 2225, 237, 59, 68, 335 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Let PS(n) be product of digits of n minus sum of digits of n (=-A062329(n)). Then a(n) is PS(A037344(m)) ordered by PS(n) for values of m such that A037344 has its digits in nondecreasing order. If PS(m) some nonzero term m of A002276 exceed some bound, all positive integers t larger than that term without zeros and ones exceed have a larger value for PS(t).
Prepending -A062329(a(n)) or more ones before a(n) gives terms of A274125.
Permuting digits of A274125 gives A254621. Permutations of digits can be found in A261370. The union of A254621 and A011540 is A062996. The b-file lists terms having PS(n) <= 10^6.
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10696
David A. Corneth, PARI program
EXAMPLE
Suppose we want to order the nondecreasing integers without zeros and ones up to PS(m) = 50. We see that 222222 has PS(222222) = 52, so we only have to check such nondecreasing integers up to 222222. Not all of those must be checked, which is used in the program.
25 is a term. Prepending PS(25) = -A062329(25) = 3 ones before 25 gives 11125, which is a term of A274125. Permuting digits of 11125 gives for example 12511, which is a term of A254621.
PROG
(PARI) See program in link "PARI program".
CROSSREFS
Cf. A011540, A037344, A062329, A062996, A254621, A261370, A274125.
KEYWORD
nonn,base
AUTHOR
David A. Corneth, Jun 10 2016
STATUS
approved
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Last modified August 29 09:12 EDT 2024. Contains 375511 sequences. (Running on oeis4.)