A062998 -id:A062998 - cp's OEIS frontend

A061672 Smallest positive number formed by a set of digits whose product = sum of the digits.

1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 123, 1124, 11125, 11133, 11222, 111126, 1111127, 1111134, 11111128, 11111223, 111111129, 111111135, 1111111144, 11111111136, 11111111224, 111111112222, 1111111111137, 1111111111145, 1111111111233

Offset: 1

Amarnath Murthy, Jun 26 2001

From M. F. Hasler, Oct 29 2014: (Start)
This is the subsequence of terms of A034710 with digits in nondecreasing order, which is meant by "smallest": For example, 132 also has sum of digits = product of digits, but is already "represented" by 123. The word "set" in the definition actually means "multiset".
The sequence is infinite: for any number N whose digits form a nondecreasing sequence whose sum of digits S is not larger than the product of digits P (i.e., N in A062998), a term of the sequence is obtained by prefixing N with P-S digits '1'. (End)

			1124 is a term since 1 + 1 + 2 + 4 = 1*1*2*4 = 8.

Chai Wah Wu, Table of n, a(n) for n = 1..667 (terms n=1..134 from T. D. Noe).
Sean A. Irvine, Java program (github)

Cf. A034710, A249334.

is_A061672(n)={vecsort(n=digits(n))==n && normlp(n,1)==prod(i=1,#n,n[i])} \\ M. F. Hasler, Oct 29 2014

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jun 27 2001
Corrected by Franklin T. Adams-Watters, Oct 25 2006
Further corrections from T. D. Noe, Oct 12 2007

A062996 Numbers whose sum of digits is greater than or equal to its product of digits.

Original entry on oeis.org

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

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[150],Total[IntegerDigits[#]]>=Times@@IntegerDigits[#]&] (* Harvey P. Dale, Sep 27 2023 *)

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

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

A037344 Numbers whose base-2 and base-10 expansions have no digits in common.

Original entry on oeis.org

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, 222

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Not the same as A062998 or A387032.

isA037344 := proc(n)
    local dgs10,dgs2 ;
    dgs10 := convert(convert(n,base,10),set) ;
    dgs2 := convert(convert(n,base,2),set) ;
    if dgs10 intersect dgs2 = {} then
        true ;
    else
        false ;
    end if;
end proc:
A037344 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA037344(a) then
                return a;
            end if;
        end do;
    end if;
end proc:
seq(A037344(n),n=1..200) ; # R. J. Mathar, Aug 14 2025

Select[Range[100],Intersection[IntegerDigits[#,2],IntegerDigits[#,10]]=={}&] (* Vincenzo Librandi, Jun 06 2012 *)

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

A387032 Numbers k with digits different from 0 and 1.

Original entry on oeis.org

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, 222

Offset: 1

David A. Corneth, Aug 13 2025

A062998 contains numbers like 123, 124, 125,.. which are not in this sequence. - R. J. Mathar, Aug 14 2025

A037344 contains numbers like 2047 and 4095 which are not in this sequence. - R. J. Mathar, Aug 14 2025

			2 is in the sequence since it does not contain 0 nor 1.
12 is not in the sequence since it has digit 1.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Intersection of A052382 and A052383.

Cf. A037344, A172424.

isA387032 := proc(n)
    local d ;
    for d in convert(n,base,10) do
        if d <=1 then
            return false;
        end if;
    end do:
    true ;
end proc:
A387032 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA387032(a) then
                return a;
            end if;
        end do;
    end if;
end proc:
seq(A387032(n),n=1..200) ; # R. J. Mathar, Aug 14 2025

Select[Range[222], Total@ DigitCount[#, 10, {0, 1}] == 0 &] (* Michael De Vlieger, Aug 13 2025 *)

is(n) = if(n <= 0, return(0)); Set(digits(n))[1] >= 2

def ok(n): return {"0","1"} & set(str(n)) == set()
print([k for k in range(223) if ok(k)]) # Michael S. Branicky, Aug 13 2025

def A387032(n):
    m = ((k:=7*n+1).bit_length()-1)//3
    return sum((2+((k-(1<<3*m))//(7<<3*j)&7))*10**j for j in range(m)) # Chai Wah Wu, Aug 13 2025

A249443 Numbers with digits in nondecreasing order and digital sum not larger than the product of the digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 39, 44, 45, 46, 47, 48, 49, 55, 56, 57, 58, 59, 66, 67, 68, 69, 77, 78, 79, 88, 89, 99, 123, 124, 125, 126, 127, 128, 129, 133, 134, 135, 136, 137, 138, 139, 144, 145, 146, 147, 148, 149, 155, 156, 157, 158

Offset: 1

M. F. Hasler, Oct 29 2014

Intersection of A009994 and A062998.
Except for the initial 0, a subsequence of the zeroless numbers A052382.
The nonzero terms of this sequence correspond to a term of A061672 obtained by concatenation with A002275(A007954(a(n))-A007953(a(n))).

Cf. A007953, A007954, A034710, A061672, A249334.

is(n)={vecsort(n=digits(n))==n && normlp(n,1)<=prod(i=1,#n,n[i])}

cp's OEIS Frontend

A061672 Smallest positive number formed by a set of digits whose product = sum of the digits.

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A062996 Numbers whose sum of digits is greater than or equal to its product of digits.

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Mathematica

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

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

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Mathematica

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

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Mathematica

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A037344 Numbers whose base-2 and base-10 expansions have no digits in common.

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Maple

Mathematica

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

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Maple

Mathematica

PARI

A387032 Numbers k with digits different from 0 and 1.

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