A066306 -id:A066306 - cp's OEIS frontend

A034710 Positive numbers for which the sum of digits equals the product of digits.

1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 123, 132, 213, 231, 312, 321, 1124, 1142, 1214, 1241, 1412, 1421, 2114, 2141, 2411, 4112, 4121, 4211, 11125, 11133, 11152, 11215, 11222, 11251, 11313, 11331, 11512, 11521, 12115, 12122, 12151, 12212, 12221, 12511

Offset: 1

Erich Friedman

Positive numbers k such that A007953(k) = A007954(k).
If k is a term, the digits of k are solutions of the equation x1*x2*...*xr = x1 + x2 + ... + xr; xi are from [1..9]. Permutations of digits (x1,...,xr) are different numbers k with the same property A007953(k) = A007954(k). For example: x1*x2 = x1 + x2; this equation has only 1 solution, (2,2), which gives the number 22. x1*x2*x3 = x1 + x2 + x3 has a solution (1,2,3), so the numbers 123, 132, 213, 231, 312, 321 have the property. - Ctibor O. Zizka, Mar 04 2008
Subsequence of A249334 (numbers for which the digital sum contains the same distinct digits as the digital product). With {0}, complement of A249335 with respect to A249334. Sequence of corresponding values of A007953(a(n)) = A007954(a(n)): 1, 2, 3, 4, 5, 6, 7, 8, 9, 4, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, ... contains only numbers from A002473. See A248794. - Jaroslav Krizek, Oct 25 2014
There are terms of the sequence ending in any term of A052382. - Robert Israel, Nov 02 2014
The number of digits which are not 1 in a(n) is O(log log a(n)) and tends to infinity as a(n) does. - Robert Dougherty-Bliss, Jun 23 2020

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

Alois P. Heinz, Table of n, a(n) for n = 1..27597 (first 1200 terms from T. D. Noe)

Cf. A002473, A007953, A007954, A052382, A061672, A248794, A249334, A249335.

Cf. A066306 (prime terms), A066307 (nonprimes).

import Data.List (elemIndices)
a034710 n = a034710_list !! (n-1)
a034710_list = elemIndices 0 $ map (\x -> a007953 x - a007954 x) [1..]
-- Reinhard Zumkeller, Mar 19 2011

[n: n in [1..10^6] | &*Intseq(n) eq &+Intseq(n)] // Jaroslav Krizek, Oct 25 2014

Select[Range[12512], (Plus @@ IntegerDigits[ # ]) == (Times @@ IntegerDigits[ # ]) &] (* Alonso del Arte, May 16 2005 *)

is(n)=my(d=digits(n)); vecsum(d)==factorback(d) \\ Charles R Greathouse IV, Feb 06 2017

Corrected by Larry Reeves (larryr(AT)acm.org), Jun 27 2001

Definition changed by N. J. A. Sloane to specifically exclude 0, Sep 22 2007

A066307 Nonprimes whose sum of digits is equal to its product of digits.

Original entry on oeis.org

1, 4, 6, 8, 9, 22, 123, 132, 213, 231, 312, 321, 1124, 1142, 1214, 1241, 1412, 1421, 2114, 4112, 4121, 11125, 11133, 11152, 11215, 11222, 11313, 11331, 11512, 11521, 12115, 12122, 12151, 12212, 12221, 13113, 13131, 13311, 15112, 15211, 21115

Offset: 1

Labos Elemer, Dec 13 2001

			321 = 3*107, 3 + 2 + 1 = 6 = 3*2*1.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..516 from Harry J. Smith)

Composites and 1 from A034710.

Cf. A034710, A066306.

sdpdQ[n_]:=Module[{idn=IntegerDigits[n]},Total[idn]==Times@@idn]; Module[ {upto=25000, cs},cs=Complement[Range[upto],Prime[Range[PrimePi[upto]]]];Select[cs,sdpdQ]] (* Harvey P. Dale, Oct 14 2014 *)

isok(k) = {if(isprime(k), 0, my(d=digits(k)); vecprod(d) == vecsum(d))} \\ Harry J. Smith, Feb 09 2010

A066310 Numbers k such that k < (product of digits of k) * (sum of digits of k).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 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

Offset: 1

Labos Elemer and Klaus Brockhaus, Dec 13 2001

			14 < (1*4)*(1+4) = 20, so 14 is a term of this sequence.
For n=199, (1+9+9)*1*9*9 = 1539 > 199, so 199 is here.

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

Cf. A038369, A049101-A049106, A034710, A061672, A066306-A066309.

function a066311(a,b: integer); var n,k,j,p,d: integer; s: string; begin for n := a to b do s := itoa(n); k := 0; p := 1; for j := 0 to length(s) - 1 do d := atoi(s[j..j]); k := k + d; p := p*d; end; if n < p*k then write(n,","); end; end; end; a066311(0,120);

asum[x_] := Apply[Plus, IntegerDigits[x]] apro[x_] := Apply[Times, IntegerDigits[x]] sz[x_] := asu[x]*apro[x] Do[s=sz[n]; If[Greater[s, n], Print[n]], {n, 1, 200}]

isok(m) = my(d=digits(m)); m < vecprod(d)*vecsum(d); \\ Michel Marcus, Mar 23 2020

A384444 Positive integers k for which the sum of their digits equals the product of their prime digits.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 20, 22, 30, 50, 70, 100, 123, 132, 200, 202, 213, 220, 231, 300, 312, 321, 500, 700, 1000, 1023, 1032, 1203, 1230, 1247, 1274, 1302, 1320, 1356, 1365, 1427, 1472, 1536, 1563, 1635, 1653, 1724, 1742, 2000, 2002, 2013, 2020, 2031, 2103, 2130, 2147

Offset: 1

Felix Huber, Jun 03 2025

Numbers k for which A007953(k) = A384443(k).

If t is a term then t*10^m is also a term for any positive integer m.

			1302 is a term, because 1 + 3 + 0 + 2 = 3*2 = 6.

Felix Huber, Table of n, a(n) for n = 1..10000

Cf. A002110, A006753, A007947, A007954, A066306, A067077, A384443, A384445, A384505.

A384444:=proc(n)
    option remember;
    local k,c;
    if n=1 then
        1
    else
        for k from procname(n-1)+1 do
            c:=convert(k,'base',10);
            if mul(select(isprime,c))=add(c) then
                return k
            fi
        od
    fi;
end proc;
seq(A384444(n),n=1..51);

Select[Range[2147],Total[IntegerDigits[#]]==Times@@Select[IntegerDigits[#],PrimeQ]&] (* James C. McMahon, Jun 20 2025 *)

isok(k) = my(d=digits(k)); vecprod(select(isprime, d)) == vecsum(d); \\ Michel Marcus, Jun 04 2025

A384445 a(n) is the number of multisets of n decimal digits where the sum of the digits equals the product of the prime digits.

Original entry on oeis.org

5, 6, 7, 10, 23, 43, 74, 125, 199, 305, 449, 637, 885, 1216, 1649, 2184, 2852, 3664, 4657, 5863, 7298, 9002, 10993, 13312, 16000, 19084, 22613, 26606, 31120, 36192, 41867, 48220, 55317, 63232, 72022, 81746, 92479, 104282, 117229, 131393, 146843, 163652, 181892

Offset: 1

Felix Huber, Jun 03 2025

			a(3) = 7 because exactly for the 7 multisets with 3 digits {0, 0, 1}, {0, 0, 2}, {0, 0, 3}, {0, 0, 5}, {0, 0, 7}, {0, 2, 2} and {1, 2, 3} their sum equals the product of the prime digits.
a(4) = 10 because exactly for the 10 multisets with 4 digits {0, 0, 0, 1}, {0, 1, 2, 3}, {1, 2, 4, 7}, {1, 3, 5, 6}, {0, 0, 0, 2}, {0, 0, 2, 2}, {0, 0, 0, 3}, {0, 0, 0, 5}, {5, 5, 6, 9} and {0, 0, 0, 7} their sum equals the product of the prime digits.

Felix Huber, Table of n, a(n) for n = 1..200

Cf. A002110, A006753, A007947, A007954, A066306, A067077, A384443, A384444, A384505.

f:=proc(p,n)
    local c,d,i,l,m,r,s,t,u,w,x,y,z;
    m:={0,1,4,6,8,9};
    t:=seq(cat(x,i),i in m);
    y:={l='Union'(t),w='Set'(l),t=~'Atom'};
    d:=(map2(apply,s,{t})=~m) union {s(w)='Set'(s(l))};
    Order:=p+1;
    r:=combstruct:-agfseries(y,d,'unlabeled',z,[[u,s]])[w(z,u)];
    r:=collect(convert(r,'polynom'),[z,u],'recursive');
    c:=coeff(r,z,p);
    coeff(c,u,n)
end proc:
A384445:=proc(n)
    local a,k,m,s,p,j,L;
    a:=1;
        for k from 9*n to 1 by -1 do
            L:=ifactors(k)[2];
            m:=nops(L);
            if m>0 and L[m,1]<=7 then
                p:=n-add(L[j,2],j=1..m);
                s:=k-add(L[j,1]*L[j,2],j=1..m);
                if s=0 and p>=0 then
                    a:=a+1
                elif p>0 and s>0 then
                    a:=a+f(p,s)
                fi
            fi
	od;
	return a
end proc;
seq(A384445(n),n=1..43);

A384505 a(n) is the number of multisets of n positive decimal digits where the sum of the digits equals the product of the prime digits.

Original entry on oeis.org

5, 1, 1, 3, 13, 20, 31, 51, 74, 106, 144, 188, 248, 331, 433, 535, 668, 812, 993, 1206, 1435, 1704, 1991, 2319, 2688, 3084, 3529, 3993, 4514, 5072, 5675, 6353, 7097, 7915, 8790, 9724, 10733, 11803, 12947, 14164, 15450, 16809, 18240, 19757, 21374, 23073, 24876, 26759

Offset: 1

Felix Huber, Jun 11 2025

			a(1) = 5 because exactly for the 5 multisets with 1 digits {1}, {2}, {3}, {5}, and {7} their sum equals the product of the prime digits.
a(2) = 1 because only for 1 multiset with 2 positive digits {2, 2} their sum equals the product of the prime digits: 2 + 2 = 2*2 = 4.
a(3) = 1 because only for 1 multiset with 3 positive digits {1, 2, 3} their sum equals the product of the prime digits: 1 + 2 + 3 = 2*3 = 6.
a(4) = 3 because exactly for the 3 multisets with 4 digits {1, 2, 4, 7}, {1, 3, 5, 6}, and {5, 5, 6, 9} their sum equals the product of the prime digits: 1 + 2 + 4 + 7 = 2 * 7 = 14, 1 + 3 + 5 + 6 = 3*5 = 15, 5 + 5 + 6 + 9 = 5*5 = 25.

Felix Huber, Table of n, a(n) for n = 1..200

Partial sums give A384445.

Cf. A002110, A006753, A007947, A007954, A066306, A067077, A384443, A384444.

f:=proc(p,n)
    local i,l,m,s,t,u,w,x,z;
    m:={1,4,6,8,9};
    t:=seq(cat(x,i),i in m);
    Order:=p+1;
    coeff(coeff(collect(convert(combstruct:-agfseries({l='Union'(t),w='Set'(l),t=~'Atom'},(map2(apply,s,{t})=~m) union {s(w)='Set'(s(l))},'unlabeled',z,[[u,s]])[w(z,u)],'polynom'),[z,u],'recursive'),z,p),u,n)
end proc:
A384505:=proc(n)
    local a,k,m,s,p,j,L;
    if n=1 then
        5
    elif n=2 then
        1
    else
        a:=0;
        for k from 9*n to 1 by -1 do
            L:=ifactors(k)[2];
            m:=nops(L);
            if m>0 and L[m,1]<=7 then
                p:=n-add(L[j,2],j=1..m);
                s:=k-add(L[j,1]*L[j,2],j=1..m);
                if p>0 and s>0 then
                    a:=a+f(p,s)
                fi
            fi
	od;
	return a
	fi;
end proc;
seq(A384505(n),n=1..48);

a(n) = A384445(n) - A384445(n-1) for n > 1.

A066309 Numbers k such that k > (product of digits of k) * (sum of digits of k).

Original entry on oeis.org

10, 11, 12, 13, 20, 21, 22, 30, 31, 32, 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, 124, 125, 126, 130, 131, 132, 133, 134, 140, 141, 142

Offset: 1

Labos Elemer and Klaus Brockhaus, Dec 13 2001

			13 is in the sequence because (1*3)*(1+3) = 3*4 = 12 < 13.
125 is a term because (1*2*5)*(1+2+5) = 10*8 = 80 < 125.

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

Cf. A038369, A049101-A049106, A034710, A061672, A066306, A066307.

function a066312(a,b: integer); var n,k,j,p,d: integer; s: string; begin for n := a to b do s := itoa(n); k := 0; p := 1; for j := 0 to length(s) - 1 do d := atoi(s[j..j]); k := k + d; p := p*d; end; if n > p*k then write(n,","); end; end; end; a066312(0,150);

asum[x_] := Apply[Plus, IntegerDigits[x]] apro[x_] := Apply[Times, IntegerDigits[x]] sz[x_] := asu[x]*apro[x] Do[s=sz[n]; If[Greater[n, s], Print[n]], {n, 1, 1000}]
okQ[n_]:=Module[{idn=IntegerDigits[n]},n> Total[idn]Times@@idn];Select[Range[150],okQ]  (* Harvey P. Dale, Mar 12 2011 *)

isok(k) = {my(d=digits(k)); k > vecprod(d) * vecsum(d)} \\ Harry J. Smith, Feb 10 2010

A272436 Semiprimes such that sum of digits equals product of digits.

Original entry on oeis.org

4, 6, 9, 22, 123, 213, 321, 1142, 1214, 1241, 4121, 11215, 11521, 12115, 12151, 21151, 22121, 51211, 111261, 112611, 116121, 116211, 121161, 162111, 211611, 261111, 621111, 1111217, 1111413, 1111431, 1111721, 1112117, 1117121, 1117211, 1121117, 1121171, 1121711

Offset: 1

K. D. Bajpai, May 06 2016

Intersection of A001358 and A034710.

9 is the only member with digit 9. No member has more than one digit 3 or 6. - Robert Israel, May 06 2016

			1142 appears in the list because 1142 = 2*571 that is semiprime. Also, 1+1+4+2 = 8 = 1*1*4*2.
11215 appears in the list because 1142 = 5*2243 that is semiprime. Also, 1+1+2+1+5 = 10 = 1*1*2*1*5.

Chai Wah Wu, Table of n, a(n) for n = 1..17009 (n = 1..3104 from Robert Israel)

Cf. A001358, A034710, A066306, A066307.

R:= proc(k,d,u,v) option remember;
    if k = 1 then
        if d = v - u then {[d]}
        else {}
        fi
    else
      `union`(seq(map(t -> [op(t),s], procname(k-1,d-s,u+s*k,v*k^s)),s=0..d))
    fi
end proc:
A034710:= proc(d)
  local res, r,  i, t;
  res:= NULL;
  for r in R(9,d,0,1) do
     res:= res, op(map(t -> add(10^(i-1)*t[i],i=1..nops(t)), combinat:-permute([seq(i$r[i],i=1..9)])));
  od:
  sort([res]);
end proc:
map(op, [seq(select(t -> numtheory:-bigomega(t)=2, A034710(i)),i=1..11)]); # Robert Israel, May 06 2016

Select[Range[10000000], (Plus @@ IntegerDigits[#]) == (Times @@ IntegerDigits[#]) && PrimeOmega[#] == 2 &]

A260904 Prime numbers such that sum of digits in base 16 equals product of digits in base 16.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 4673, 5153, 8513, 16673, 17895719, 17899799, 17985809, 18288929, 34697489, 40964369, 118563089, 286337041, 286359841, 293675281, 403775761, 554766721, 554795281, 73303933201, 74109227281, 1172812415521, 1172812443937, 1172812468507

Offset: 1

Chai Wah Wu, Nov 17 2015

			16673_10 = 4121_16 is prime and the sum of digits of 4121 equals the product of digits.

Chai Wah Wu, Table of n, a(n) for n = 1..6023

Cf. A066306.

Select[Prime@ Range[10^7], Total@ # == Times @@ # &@ IntegerDigits[#, 16] &] (* Michael De Vlieger, Nov 18 2015 *)

A264576 Prime numbers such that sum of digits in base 8 equals product of digits in base 8.

Original entry on oeis.org

2, 3, 5, 7, 83, 139, 673, 1289, 2129, 5197, 5449, 6737, 6793, 8849, 12889, 13001, 21577, 38281, 58441, 70537, 90697, 300809, 311897, 398921, 2400851, 2400907, 2429579, 2430601, 2462353, 2658899, 2691659, 2724937, 2925137, 2925193, 2925641, 2957897, 4494929

Offset: 1

Chai Wah Wu, Nov 17 2015

			300809_10 = 1113411_8 is prime and the sum of digits of 1113411 is equal to the product of digits.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A066306.

Select[Prime@ Range[10^6], Total@ # == Times @@ # &@ IntegerDigits[#, 8] &] (* Michael De Vlieger, Nov 18 2015 *)

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A034710 Positive numbers for which the sum of digits equals the product of digits.

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A066307 Nonprimes whose sum of digits is equal to its product of digits.

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A066310 Numbers k such that k < (product of digits of k) * (sum of digits of k).

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A384444 Positive integers k for which the sum of their digits equals the product of their prime digits.

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A384445 a(n) is the number of multisets of n decimal digits where the sum of the digits equals the product of the prime digits.

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A384505 a(n) is the number of multisets of n positive decimal digits where the sum of the digits equals the product of the prime digits.

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A066309 Numbers k such that k > (product of digits of k) * (sum of digits of k).

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A272436 Semiprimes such that sum of digits equals product of digits.

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