A034710 -id:A034710 - cp's OEIS frontend

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

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

A249335 Numbers n for which the digital sum contains the same distinct digits as the digital product but the digital sum is not equal to the digital product.

Original entry on oeis.org

99, 1137, 1173, 1317, 1371, 1713, 1731, 3117, 3171, 3344, 3434, 3443, 3711, 4334, 4343, 4433, 7113, 7131, 7311, 11558, 11585, 11855, 15158, 15185, 15518, 15581, 15815, 15851, 18155, 18515, 18551, 22334, 22343, 22433, 23234, 23243, 23324, 23342, 23423, 23432

Offset: 1

Jaroslav Krizek, Oct 25 2014

Numbers n such that A007953(n) contains the same distinct digits as A007954(n) but A007953(n) is not equal to A007954(n).

Complement of A034710 with respect to A249334.

			1137 is a member since 1+1+3+7 = 12 and 1*1*3*7 = 21.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..100 from Jaroslav Krizek)

Cf. A034710, A007953, A007954, A249334.

[n: n in [1..10^6] | (&*Intseq(n)) ne (&+Intseq(n)) and Set(Intseq(&*Intseq(n))) eq Set(Intseq(&+Intseq(n)))]

A064158 Integers m such that (x1x2..xk)^(x1+x2+..xk) = (x1+x2+..xk)^(x1x2..xk) where x1x2..xk are the digits of m in base 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 112, 121, 123, 132, 211, 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

Offset: 1

Felice Russo, Sep 14 2001

With the exception of only 112, 121, and 211, each term of this sequence satisfies (sum of digits) equals (product of digits). For 112, 121, and 211, the sum of the digits is 4, the product of the digits is 2, and the terms qualify because 2^4 equals 4^2. - Harvey P. Dale, Sep 30 2011

			22 belongs to the sequence because (2*2)^(2+2) = (2+2)^(2*2).

Harvey P. Dale, Table of n, a(n) for n = 1..179

Cf. A007953, A007954, A034710.

okQ[n_]:=Module[{idn=IntegerDigits[n],t,p}, t= Times@@idn; p=Total[idn];t^p==p^t]; Select[Range[12500],okQ]

Union of {112, 121, 211} and A034710. - Jason Yuen, Aug 27 2025

More terms from Jason Earls, Dec 04 2001

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

A241846 Numbers for which the cube of the sum of the digits is equal to the square of the product of their digits.

Original entry on oeis.org

0, 1, 88, 333, 11248, 11284, 11428, 11482, 11824, 11842, 12148, 12184, 12418, 12481, 12814, 12841, 14128, 14182, 14218, 14281, 14812, 14821, 18124, 18142, 18214, 18241, 18412, 18421, 21148, 21184, 21418, 21481, 21814, 21841, 24118, 24181, 24811, 28114

Offset: 1

Michel Lagneau, Apr 30 2014

Let d_1 d_2... d_q denote the decimal expansion of a number n. The sequence lists the numbers n such that (d_1 + d_2 +...+ d_q)^3 = (d_1 * d_2 *...* d_q)^2.
The sequence is finite and contains 1419 terms because the maximum sum of the digits of a(n) is 16, the maximum product of the digits is 64 with 16^3 = 64^2 and the greatest number of the sequence is 2222221111.
The primitive values of a(n) (numbers whose decimal digits are not a permutation of another number of the sequence) are 0, 1, 88, 333, 11248, 112228, 1111444, 11112244, 111122224, 1111222222.
Nevertheless, the numbers 112228, 1111444, 11112244, 111122224, 1111222222 are not completely independent; for example, a decimal digit 4 of 1111444 becomes 22 and gives the number 11112244.

			333 is in the sequence because (3+3+3)^3 = (3*3*3)^2 = 729.
11248 is in the sequence because (1+1+2+4+8)^3 = (1*1*2*4*8)^2 = 4096.

Michel Lagneau, Table of n, a(n) for n = 1..1419

Cf. A034710, A117720.

Select[Range[30000], (Plus @@ IntegerDigits[ # ]^3) == (Times @@ IntegerDigits[ # ]^2) &]

A248794 a(n) = the smallest number k for which the sum of digits (A007953(k)) and the product of digits (A007954(k)) are both equal to A002473(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11125, 111126, 1111127, 111111135, 11111128, 111111129, 1111111111145, 1111111111137, 111111111111138, 11111111111111155, 11111111111111139, 1111111111111111147, 111111111111111111156, 1111111111111111111148, 1111111111111111111111157

Offset: 1

Jaroslav Krizek, Nov 02 2014

See comment in A034710 (positive numbers for which the sum of digits equals the product of digits).

			For n = 11; a(11) = 111126 because A002473(11) = 12, A007953(111126) = A007954(111126) = 12.

Cf. A002473, A034710, A007953, A007954.

A248794:=func; [A248794(n):n in[A002473(n)]]

Name clarified by Andrew Howroyd, Sep 20 2024

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 &]

A277217 Numbers k for which the sum of digits of sigma(k) = the product of digits of sigma(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 86, 126, 131, 206, 207, 311, 1123, 1213, 2113, 4111, 10921, 12211, 16581, 21121, 21211, 22111, 39660, 51558, 52940, 60812, 61504, 63548, 68822, 81303, 83409, 87081, 87451, 89708, 94523, 97307, 106118, 108527, 110387, 111611, 120831, 160271

Offset: 1

Jaroslav Krizek, Oct 05 2016

Numbers k such that A067342(k) = A277216(k).
Prime terms: 2, 3, 5, 7, 131, 311, 1123, 1213, 2113, 4111, 12211, ...
Corresponding values of sigma(a(n)): 1, 3, 4, 7, 6, 8, 132, 312, 132, 312, 312, 312, 1124, 1214, 2114, ...
Only 196 terms less than 35*10^8. - Robert G. Wilson v, Oct 07 2016
Alternatively, numbers k such that sigma(k) is in A034710. - Charlie Neder, Dec 27 2018

			86 is a term because sigma(86) = 132; sum and product of digits of 132 = 6.

Robert G. Wilson v, Table of n, a(n) for n = 1..196 (first 100 terms from Jaroslav Krizek)

Cf. A000203, A007953, A007954, A034710.

Cf. A067342 (sum of decimal digits of sigma(n)), A277216 (product of decimal digits of sigma(n)).

[n: n in [1..100000] | &+Intseq(SumOfDivisors(n)) eq &*Intseq(SumOfDivisors(n))];

Select[Range@ 200000, Total@ # == Times @@ # &@ IntegerDigits@ DivisorSigma[1, #] &] (* Michael De Vlieger, Oct 06 2016 *)

isok(n) = my(d=digits(sigma(n))); vecprod(d) == vecsum(d); \\ Michel Marcus, Mar 02 2019

A061597 Numbers whose sum of digits s equals its product of digits and s + 1 and s - 1 are both prime.

Original entry on oeis.org

4, 6, 22, 123, 132, 213, 231, 312, 321, 111126, 111162, 111216, 111261, 111612, 111621, 112116, 112161, 112611, 116112, 116121, 116211, 121116, 121161, 121611, 126111, 161112, 161121, 161211, 162111, 211116, 211161, 211611, 216111, 261111, 611112

Offset: 1

Felice Russo, May 22 2001

			22 belong to the sequence because (2*2)+1=5, (2*2)-1=3, (2+2)+1=5, (2+2)-1=3 and 2+2=2*2.

Harry J. Smith, Table of n, a(n) for n = 1..81

Intersection of A034710 and A061596.

isok(k) = my(d=digits(k), s=vecsum(d)); s == vecprod(d) && isprime(s+1) && isprime(s-1) \\ Harry J. Smith, Jul 25 2009

4 and 6 added by Abdul Gaffar Khan, Nov 29 2015

A107650 Numbers n such that both numbers n/(d_1d_2 ...*d_k) and n/(d_1+d_2+ ... +d_k) are prime, where d_1 d_2 ... d_k is the decimal expansion of n.

Original entry on oeis.org

11133, 11331, 13131, 31113, 112116, 121116, 13111212, 111311115, 11114112112, 111212112112, 1111111711311, 1111171111113, 11111111112611112, 11111111121161112, 11111112111161112, 11111119111131111, 11111131111119111, 11111139111111111, 11111193111111111, 11111211161111112, 11111611111211112, 11116111112111112, 11116111211111112

Offset: 1

Farideh Firoozbakht, May 21 2005

For n in this sequence, let prime p = n/(d_1*d_2* ...*d_k) so that n = d_1*d_2* ...*d_k * p. Then n/(d_1+d_2+ ... +d_k) equals either p or some prime dividing d_1*d_2* ...*d_k, that is 2, 3, 5, or 7. The latter case never takes place and thus n/(d_1*d_2* ...*d_k) = n/(d_1+d_2+ ... +d_k) is the same prime. So this sequence is a subsequence of both A034710 and A066307. - Max Alekseyev, Aug 19 2013

			111311115 is in the sequence because
111311115/(1*1*1*3*1*1*1*1*5) and 111311115/(1+1+1+3+1+1+1+1+5)
are prime(since 1*1*1*3*1*1*1*1*5=1+1+1+3+1+1+1+1+5, the primes are equal).

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1717 from Max Alekseyev).

Do[h = IntegerDigits[m]; l = Length[h]; If[Min[h] > 0 && PrimeQ[m/Sum[h[[k]], {k, l}]] && PrimeQ[m/Product[ h[[k]], {k, l}]], Print[m]], {m, 265000000}]

a(9)-a(10) from Sean A. Irvine, Nov 28 2010

Terms a(11) onward from Max Alekseyev, Aug 20 2013

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

ARIBAS

Mathematica

PARI

A249335 Numbers n for which the digital sum contains the same distinct digits as the digital product but the digital sum is not equal to the digital product.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

A064158 Integers m such that (x1*x2*..xk)^(x1+x2+..xk) = (x1+x2+..xk)^(x1*x2*..xk) where x1x2..xk are the digits of m in base 10.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

ARIBAS

Mathematica

PARI

A241846 Numbers for which the cube of the sum of the digits is equal to the square of the product of their digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A248794 a(n) = the smallest number k for which the sum of digits (A007953(k)) and the product of digits (A007954(k)) are both equal to A002473(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A277217 Numbers k for which the sum of digits of sigma(k) = the product of digits of sigma(k).

Views

Author

Keywords

A064158 Integers m such that (x1x2..xk)^(x1+x2+..xk) = (x1+x2+..xk)^(x1x2..xk) where x1x2..xk are the digits of m in base 10.

A107650 Numbers n such that both numbers n/(d_1d_2 ...*d_k) and n/(d_1+d_2+ ... +d_k) are prime, where d_1 d_2 ... d_k is the decimal expansion of n.