A066307 -id:A066307 - 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

A066306 Prime numbers such that sum of digits equals product of digits.

Original entry on oeis.org

2, 3, 5, 7, 2141, 2411, 4211, 11251, 12511, 15121, 21221, 25111, 1112171, 1127111, 1172111, 1271111, 7112111, 11112811, 11128111, 11218111, 12111811, 12118111, 12181111, 18211111, 81111211, 81112111

Offset: 1

Labos Elemer, Dec 13 2001

			2141 = p[323], 2*1*4*1 = 8 = 2+1+4+1.

Chai Wah Wu, Table of n, a(n) for n = 1..8552

Primes from A034710.

Cf. A000040, A061672, A066307.

[NthPrime(n): n in [1..2*10^4] | &+Intseq(NthPrime(n)) eq &*Intseq(NthPrime(n))]; // Vincenzo Librandi, Nov 18 2015

f[n_] := IntegerDigits[ Prime[n]]; Prime[ Select[ Range[ PrimePi[10^10]], Apply[Plus, f[ # ]] == Apply[Times, f[ # ]] & ]]

More terms from Robert G. Wilson v, Dec 27 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

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 &]

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

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

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A066306 Prime numbers such that sum of digits equals product of 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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ARIBAS

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

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A107650 Numbers n such that both numbers n/(d_1*d_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.

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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.