A062996 -id:A062996 - cp's OEIS frontend

A062998 Numbers whose sum of digits is less than or equal to its product of digits.

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, 99, 123, 124, 125, 126, 127, 128, 129, 132, 133, 134, 135

Offset: 1

Henry Bottomley, Jun 29 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A007953, A007954, A034710, A062329, A062996, A062997, A062998, A062999.

Not the same as A037344 (contains 124).

isA062998 := proc(n)
        local dgs,s,p ;
        dgs := convert(n,base,10) ;
        s := add(i,i=dgs) ;
        p := mul(i,i=dgs) ;
        if s <= p then
                true;
        else
                false;
        end if;
end proc:
for n from 2 to 150 do
        if isA062998(n) then
                printf("%d,",n) ;
        end if;
end do:   # R. J. Mathar, Aug 14 2025

Select[Range[100],Total[IntegerDigits[#]]<=Times@@IntegerDigits[#]&] (* Harvey P. Dale, Feb 21 2017 *)

isok(k)={my(d=digits(k)); vecsum(d) <= vecprod(d)} \\ Harry J. Smith, Aug 15 2009

is_A062998(n)={normlp(n=digits(n),1)<=prod(i=1,#n,n[i])} \\ M. F. Hasler, Oct 29 2014

A062329 a(n) = (sum of digits of n) - (product of digits of n).

Original entry on oeis.org

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

Offset: 0

Amarnath Murthy, Jun 21 2001

			a(23) = 2 + 3 - 2*3 = -1.
a(49) = -(4*9) + (4 + 9) = -36 + 13 = -23.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A007953, A007954, A034710, A062996, A062997, A062998, A062999, A070565.

A063543 has a similar graph.

a[n_] := (t = IntegerDigits[n]; Plus @@ t - Times @@ t); Table[ a[n], {n, 0, 75}] (* Robert G. Wilson v *)

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jun 22 2001

Signed version from Henry Bottomley, Jun 29 2001

A062997 Numbers whose sum of digits is strictly greater than its product of digits.

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy, Jun 27 2001

Every multiple of 10 is a term.

			118 is a term as 1 + 1 + 8 = 10, 10 > 8 and 8 = 1 * 1 * 8.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^6.

Cf. A007953, A007954, A034710, A062329, A062996, A062998, A062999.

Select[Range[170], (Plus @@ IntegerDigits[ # ]) > (Times @@ IntegerDigits[ # ]) &] (* Alonso del Arte, May 16 2005 *)

isok(k)={my(d=digits(k)); vecsum(d) > vecprod(d)} \\ Harry J. Smith, Aug 15 2009

Extended by Larry Reeves (larryr(AT)acm.org) and Henry Bottomley, Jun 29 2001

A062999 Numbers whose sum of the digits is strictly less than its product of digits.

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Jun 29 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A007953, A007954, A034710, A062329, A062996, A062997, A062998, A062999.

Select[Range[128], (Plus @@ IntegerDigits[ # ]) < (Times @@ IntegerDigits[ # ]) &] (* Alonso del Arte, May 16 2005 *)

isok(k)={my(d=digits(k)); vecsum(d) < vecprod(d)} \\ Harry J. Smith, Aug 15 2009

A254621 Zerofree numbers having product of digits less than or equal to sum of digits.

Original entry on oeis.org

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

Offset: 1

David A. Corneth, Feb 03 2015

Intersection of A052382 and A062996.

The repunit A002275(k), for k >= 2, appears at position A254622(k-1) + 1. - Wolfdieter Lang, Feb 23 2015

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A052382, A062996, A062997, A062998, A062999, A254622, A002275.

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

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 *)

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

A254622 Number of zerofree positive integers of at most n digits having product of digits less than or equal to sum of digits.

Original entry on oeis.org

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

Offset: 1

David A. Corneth, Feb 03 2015

See A254621 for the numbers with this property. - Wolfdieter Lang, Feb 23 2015

David A. Corneth, Table of n, a(n) for n = 1..2000

Cf. A052382, A062996, A254621.

A274125 Numbers having digits in nondecreasing order and sum of digits greater than or equal to the product of digits.

Original entry on oeis.org

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

Offset: 1

David A. Corneth, Jun 10 2016

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.

David A. Corneth, Table of n, a(n) for n = 1..1000

Cf. A062996, A009994.

upto[nd_] := Sort[FromDigits /@ Reverse /@ Select[ Flatten[ IntegerPartitions[#, nd, Range@ 9] & /@ Range[9 nd], 1], Times @@ # <= Plus @@ # &]]; upto[8] (* Giovanni Resta, Jun 20 2016 *)

is(n) = my(d=digits(n)); prod(i=1, #d, d[i]) <= vecsum(d) && vecsort(d) == d

A274126 Numbers with digits larger than 1 sorted by product of digits minus sum of digits, then by size.

Original entry on oeis.org

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

Offset: 1

David A. Corneth, Jun 10 2016

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.

			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.

David A. Corneth, Table of n, a(n) for n = 1..10696
David A. Corneth, PARI program

Cf. A011540, A037344, A062329, A062996, A254621, A261370, A274125.

PARI
```
See program in link "PARI program".
```

cp's OEIS Frontend

A062998 Numbers whose sum of digits is less than or equal to its product of digits.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

A062329 a(n) = (sum of digits of n) - (product of digits of n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A062997 Numbers whose sum of digits is strictly greater than its product of digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A062999 Numbers whose sum of the digits is strictly less than its product of digits.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A254621 Zerofree numbers having product of digits less than or equal to sum of digits.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A254622 Number of zerofree positive integers of at most n digits having product of digits less than or equal to sum of digits.

Views

Author

Keywords

Comments

Links

Crossrefs

A274125 Numbers having digits in nondecreasing order and sum of digits greater than or equal to the product of digits.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A274126 Numbers with digits larger than 1 sorted by product of digits minus sum of digits, then by size.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs