A049101 -id:A049101 - cp's OEIS frontend

A049105 Ratio from A049101.

1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 4, 2, 4, 8, 1, 1, 8, 8, 6, 5, 4, 2, 2, 8, 3

Offset: 1

A049102 Positive numbers n such that n is a multiple of (product of digits of n) * (sum of digits of n).

1, 12, 111, 112, 135, 144, 216, 432, 2112, 11112, 11115, 11232, 12312, 13824, 14112, 21112, 23112, 27216, 31212, 41112, 81216, 93312, 111132, 122112, 124416, 131112, 132192, 186624, 212112, 221112, 221184, 222912, 239112, 248832, 311472, 316224

Offset: 1

Olivier Gérard

Empirically, it looks as if every number of the form (10^3^n-1)/9 has this property. - David W. Wilson, Dec 12 2001
From David A. Corneth, Jan 23 2019: (Start)
Indeed, (10^3^n-1)/9 is in the sequence. It has digital sum times product of digits equal to 3^n.
Proof: (10^3^0-1)/9 = (10^1-1)/9 = 1 is in the sequence.
If (10^3^k-1)/9 is in the sequence then (10^3^(k + 1)-1)/9 = ((10^3^k-1)/9) * (10^(2*3^k) + 10^(3^k) + 1) = 3 * m * ((10^3^k-1)/9) for some m. This number is divisible by 3 * 3^k = 3^(k + 1) so (10^3^(k+1) - 1)/9 is in the sequence and so (10^3^n - 1) / 9 is in the sequence from which it follows that the sequence is infinite. (End)

			432 is a term because: 4*3*2=24, 4+3+2=9, 24*9=216 and 432/216 = 2.

David A. Corneth, Table of n, a(n) for n = 1..19637 (first 100 terms from Vincenzo Librandi, terms < 10^14)
David A. Corneth, a(n) = [product of digits of a(n)] * [sum of digits of a(n)] * [some factor]

Cf. A038369, A049101, A049105, A049106, A052382.

Select[ Range[10^6], IntegerQ[ # /(Apply[ Times, IntegerDigits[ # ]] * Apply[ Plus, IntegerDigits[ # ]] ) ] & ]

isok(n) = my(d=digits(n)); vecprod(d) && (n % (vecsum(d)*vecprod(d)) == 0); \\ Michel Marcus, Jan 23 2019

A049106 Ratio from A049102.

Original entry on oeis.org

1, 2, 37, 14, 1, 1, 2, 2, 88, 926, 247, 104, 114, 4, 196, 754, 214, 9, 289, 571, 47, 32, 2058, 1696, 36, 2428, 68, 3, 2946, 3071, 96, 86, 123, 3, 103, 61, 113, 119, 138, 97, 797, 41153, 30867, 5672, 30892, 644, 1146, 31142, 289, 41893, 499, 896, 109, 33642

Offset: 1

Olivier Gérard

Cf. A049101, A049102, A049105.

function a049106(a,b: integer); var n,k,j,p,r,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; r := p*k; if r > 0 then if n mod r = 0 then write(n div r,","); end; end; end; end; a049106(0,1220000);

More terms from Klaus Brockhaus, Dec 13 2001

A066308 a(n) = (sum of digits of n) * (product of digits of n).

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 0, 6, 16, 30, 48, 70, 96, 126, 160, 198, 0, 12, 30, 54, 84, 120, 162, 210, 264, 324, 0, 20, 48, 84, 128, 180, 240, 308, 384, 468, 0, 30, 70, 120, 180, 250, 330, 420, 520, 630, 0, 42, 96, 162, 240, 330

Offset: 1

Labos Elemer, Dec 13 2001

a(n) can be greater than, less than, or equal to n; see Example section.

			For n = 12, a(12) = (1 + 2)*(1*2) = 3*2 = 6 < n;
for n = 19, a(19) = (1 + 9)*(1*9) = 90 > n;
for n = 135, a(135) =(1 + 3 + 5)*(1*3*5) = 135 = n.

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

Cf. A007953, A007954.
Cf. A038369 (fixed points), A034710, A061672.
Cf. A049101, A049102, A049103, A049104, A049105, A049106.

asum[x_] := Apply[Plus, IntegerDigits[x]] apro[x_] := Apply[Times, IntegerDigits[x]] a[n]=asum[n]*apro[n]
sdpd[n_]:=Module[{idn=IntegerDigits[n]},Total[idn]Times@@idn]; Array[ sdpd,70] (* Harvey P. Dale, Dec 31 2011 *)

a(n) = my(d = digits(n)); vecsum(d) * vecprod(d); \\ Michel Marcus, Feb 24 2017

Edited by Jon E. Schoenfield, Jul 09 2018

A023651 Numbers k such that (product of digits of k) * (sum of digits of k) = 2k.

Original entry on oeis.org

0, 2, 15, 24, 1575, 39366

Offset: 1

Jason Earls, Dec 11 2001

Except for k = 0, this sequence is a subsequence of A049101. - Jason Yuen, Feb 26 2024

Cf. A007953, A007954, A049101.

Cf. A038364, A038369, A062237, A066282.

Do[ If[ 2n == Apply[ Times, IntegerDigits[n]] Apply[ Plus, IntegerDigits[n]], Print[n]], {n, 0, 10^7} ]

isok(n) = if(n, factorback(digits(n)), 0) * sumdigits(n) == 2*n \\ Mohammed Yaseen, Jul 22 2022

from math import prod
def s(n): return sum(map(int, str(n)))
def p(n): return prod(map(int, str(n)))
for n in range(0, 10**6):
  if p(n)*s(n)==2*n:
    print(n) # Mohammed Yaseen, Jul 22 2022

Offset corrected by Arkadiusz Wesolowski, Oct 17 2012

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

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

A212499 Numbers k that divide the product of digits of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 301

Offset: 1

Arkadiusz Wesolowski, May 19 2012

Every integer except zero divides zero.

A050720(2*n) is the number of terms of length n for n >= 2.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000

Cf. A011540, A049101, A050720.

Union[Range[9], Select[Range[10, 301], DigitCount[#, 10, 0] > 0 &]]
Select[Range[301], Divisible[Product[i, {i, IntegerDigits[#]}], #] &]

a(n+9) = A011540(n+1).

A066156 a(n) is the least k>n such that kn = (product of digits of k) (sum of digits of k), or 0 if no such k exists.

Original entry on oeis.org

10, 135, 15, 139968, 18, 756, 476, 0, 48, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Robert G. Wilson v, Dec 13 2001

a(n) = 0 for all n > 8: see A049101. - Robert Israel, Aug 18 2020

Cf. A007953 (sum of digits), A007954 (product of digits).

Cf. A049101.

Do[ k = n + 1; While[ k*n != Apply[Times, IntegerDigits[k]] Apply[Plus, IntegerDigits[k]], k++ ]; Print[k], {n, 0, 10} ]

Edited by Robert Israel, Aug 18 2020

cp's OEIS Frontend

A049105 Ratio from A049101.

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A049102 Positive numbers n such that n is a multiple of (product of digits of n) * (sum of digits of n).

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A049106 Ratio from A049102.

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A066308 a(n) = (sum of digits of n) * (product of digits of n).

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A023651 Numbers k such that (product of digits of k) * (sum of digits of k) = 2k.

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

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

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A212499 Numbers k that divide the product of digits of k.

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A066156 a(n) is the least k>n such that kn = (product of digits of k) (sum of digits of k), or 0 if no such k exists.