A061013 -id:A061013 - cp's OEIS frontend

A038367 Numbers n with property that (product of digits of n) is divisible by (sum of digits of n).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 22, 30, 36, 40, 44, 50, 60, 63, 66, 70, 80, 88, 90, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 120, 123, 130, 132, 138, 140, 145, 150, 154, 159, 160, 167, 170, 176, 180, 183, 189, 190, 195, 198, 200, 201, 202, 203

Offset: 1

Felice Russo

Equal to the disjoint union of A061013 and A011540 \ {0}. Contains in particular all positive single-digit integers, those with a digit 0, and 22*{1,...,18}. If x is in the sequence, any digit-permutation of x is also in the sequence. - M. F. Hasler, Feb 28 2018

Iain Fox, Table of n, a(n) for n = 1..10000

See A061013 for case where 0 digits are excluded. Cf. A055931.

[0] cat [n: n in [1..250] | IsIntegral(&*Intseq(n)/&+Intseq(n))]; // Bruno Berselli, Feb 09 2016

isA038367 := proc(n)
    if type( A007954(n)/A007953(n),'integer') then
        true;
     else
        false;
    end if;
end proc :
for n from 1 to 500 do
    if isA038367(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jun 30 2020

okQ[n_]:=Module[{idn=IntegerDigits[n]},Divisible[Times@@idn,Total[idn]]]
Select[Range[500],okQ] (* Harvey P. Dale, Nov 24 2010 *)

is(n)=n&&prod(i=1,#n=digits(n),n[i])%vecsum(n)==0 \\ M. F. Hasler, Feb 28 2018

Corrected by Vladeta Jovovic and Larry Reeves (larryr(AT)acm.org), Jun 08 2001

Erroneous 0 term removed by David A. Corneth, Jun 05 2016

A062045 Positive numbers whose product of digits is 12 times their sum.

Original entry on oeis.org

666, 1479, 1497, 1568, 1586, 1658, 1685, 1749, 1794, 1856, 1865, 1947, 1974, 2349, 2394, 2439, 2446, 2464, 2493, 2644, 2934, 2943, 3249, 3294, 3345, 3354, 3429, 3435, 3453, 3492, 3534, 3543, 3924, 3942, 4179, 4197, 4239, 4246, 4264, 4293, 4329, 4335, 4353, 4392

Offset: 1

Amarnath Murthy, Jun 28 2001

			2349 belongs to the sequence as (2*3*4*9)/(2+3+4+9) = 216/18 = 12.

Harry J. Smith, Table of n, a(n) for n = 1..500

Subsequence of A061013 and hence A038367 and A052382.

Cf. A011540, A034710, A062034, A062035, A062036, A062382, A062037, A062384, A062040, A062041, A062043.

okQ[n_]:=Module[{idn=IntegerDigits[n]},Times@@idn/Total[idn]==12]
Select[Range[10000],okQ] (* Harvey P. Dale, Nov 25 2010 *)

isok(n) = my(d=digits(n)); vecprod(d)==12*vecsum(d) \\ Mohammed Yaseen, Sep 12 2022

from math import prod
def ok(n): d = list(map(int, str(n))); return prod(d) == 12*sum(d)
print([k for k in range(1, 4400) if ok(k)]) # Michael S. Branicky, Sep 12 2022

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jul 06 2001
More terms from Harvey P. Dale, Nov 25 2010
Offset corrected by Mohammed Yaseen, Sep 12 2022

A064154 Numbers whose product of digits equals the number of digits times the sum of digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 44, 63, 159, 167, 176, 195, 235, 253, 325, 333, 352, 519, 523, 532, 591, 617, 671, 716, 761, 915, 951, 1247, 1274, 1344, 1427, 1434, 1443, 1472, 1724, 1742, 2147, 2174, 2226, 2262, 2417, 2471, 2622, 2714, 2741, 3144, 3414

Offset: 1

Felice Russo, Sep 14 2001

A subset of A061013.

			36 belongs to the sequence because 3*6 = 18 and 2*(3+6) = 18.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1001 terms from Harry J. Smith)

Cf. A061013, A316147.

Select[Range[0, 3414], (d = IntegerDigits[#]; Times @@ d == Length[d] Plus @@ d) &] (* Giovanni Resta, Jun 25 2018 *)

for(n=0,10000,my(v=digits(n));if(vecprod(v)==#v*vecsum(n),print1(n,", "))) \\ Derek Orr, Sep 09 2018

Offset set to 1 by Giovanni Resta, Jun 25 2018

A126789 a(n) is the smallest number such that the product of its digits is n times the sum of its digits, or 0 if no such number exists.

Original entry on oeis.org

1, 36, 66, 88, 257, 268, 279, 448, 369, 459, 0, 666, 0, 578, 579, 678, 0, 1689, 0, 2558, 789, 0, 0, 1899, 13557, 0, 999, 3477, 0, 2589, 0, 2688, 0, 0, 13578, 3489, 0, 0, 0, 3588, 0, 2799, 0, 0, 4569, 0, 0, 4668, 4677, 5568, 0, 0, 0, 3699, 0, 3789, 0, 0, 0, 4599, 0, 0

Offset: 1

Tanya Khovanova, Feb 19 2007

a(11) = 0. Proof: 11 is a prime number and the product of digits of a number in base 10 can never be a multiple of 11. - Stefan Steinerberger, Jun 07 2007

More generally, a(n) = 0 for all n which are divisible by a prime bigger than 7. This means that the sequence will almost always be 0 (with the set of exceptions having density 0). In each term the digits will be increasing (otherwise we could rearrange the digits so that they form a smaller number with the specified property). If no prime factors of n exceed 7, does this mean that a(n) is not 0? - Stefan Steinerberger, Jun 14 2007

			a(2)=36 because 3*6/(3+6) = 2 and no number smaller than 36 has this property.

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

This sequence is a subsequence of A061013 (Product of digits of n) is divisible by (sum of digits of n), where 0's are not permitted.

for n from 1 to 10 do b:=proc(k) local kk: kk:=convert(k,base,10): if product(kk[j],j=1..nops(kk))=n*sum(kk[j],j=1..nops(kk)) then k else fi end: a[n]:=[seq(b(k),k=1..1000)][1]: od: seq(a[n],n=1..10); # program works only for n from 1 to 10 Emeric Deutsch, Mar 07 2007

a[1] := 1; a[n_] := Module[{}, k = 0; If[FactorInteger[n][[ -1, 1]] < 8, k = 1; While[Times @@ IntegerDigits[k] != n*Plus @@ IntegerDigits[k], k++ ]]; k]; Table[a[i], {i, 1, 80}] (* Stefan Steinerberger, Jun 14 2007 *)

More terms from Emeric Deutsch, Mar 07 2007

More terms from Stefan Steinerberger, Jun 14 2007

A062519 Numbers for which (product of digits) / (sum of digits) is an integer > 1.

Original entry on oeis.org

36, 44, 63, 66, 88, 138, 145, 154, 159, 167, 176, 183, 189, 195, 198, 224, 235, 242, 246, 253, 257, 264, 268, 275, 279, 286, 297, 318, 325, 333, 345, 347, 352, 354, 357, 369, 374, 375, 381, 396, 415, 422, 426, 435, 437, 448, 451, 453, 456, 459, 462, 465

Offset: 1

Amarnath Murthy, Jun 26 2001

			63 is a member as (6*3)/(6+3) = 2 > 1.

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

Sequence A061013 allows product = sum.

pdsdQ[n_]:=Module[{idn=IntegerDigits[n],r},r=Times@@idn/Total[idn];IntegerQ[ r]&&r>1]; Select[Range[500],pdsdQ] (* Harvey P. Dale, Dec 22 2019 *)

ok(n)={my(d=digits(n), p=vecprod(d), s=vecsum(d)); p > s && p%s == 0} \\ Andrew Howroyd, Sep 17 2024

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jun 29 2001

Offset changed by Andrew Howroyd, Sep 17 2024

cp's OEIS Frontend

A038367 Numbers n with property that (product of digits of n) is divisible by (sum of digits of n).

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A062045 Positive numbers whose product of digits is 12 times their sum.

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A064154 Numbers whose product of digits equals the number of digits times the sum of digits.

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A126789 a(n) is the smallest number such that the product of its digits is n times the sum of its digits, or 0 if no such number exists.

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Maple

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Extensions

A062519 Numbers for which (product of digits) / (sum of digits) is an integer > 1.

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Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions