cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A049101 Numbers m such that m divides (product of digits of m) * (sum of digits of m).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 18, 24, 45, 48, 135, 144, 288, 378, 476, 756, 864, 1575, 39366, 69984, 139968
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Next term if it exists is greater than 4*10^7. - Michel ten Voorde
Sequence is finite and bounded above by 10^84, since if 10^k <= n < 10^(k+1) (product of digits of n)*(sum of digits of n) <= k*9^(k+2) which is less than 10^k for k >= 84. - Henry Bottomley, May 18 2000
Numbers with a zero digit are not permitted. - Harvey P. Dale, Jul 16 2011
No further terms to 2.5*10^9. - Robert G. Wilson v, Jul 17 2011
Sequence is complete. - Giovanni Resta, Mar 20 2013
If product of digits is performed on nonzero digits only, then 1088 is also in the sequence. - Giovanni Resta, Mar 22 2013

Examples

			139968 is in the sequence since it divides (1*3*9*9*6*8) * (1+3+9+9+6+8). - _Giovanni Resta_, Mar 20 2013

Links

Crossrefs

Cf. A038369, A049102, A049105, A049106.

Programs

  • Mathematica
    okQ[n_]:=Module[{idn=IntegerDigits[n]},!MemberQ[idn,0] && Divisible[ (Total[idn]*Times@@idn),n]] (* Harvey P. Dale, Jul 16 2011 *)
(* full sequence *) dig[nD_] := Block[{ric, sol = {}, check}, check[mu_, minN_] := Block[{di = DigitCount@minN, k = 1, r}, While[(r = mu/k) >= minN, If[IntegerQ[r] && DigitCount[r] == di, AppendTo[sol, r]]; k++]]; ric[n_, prod_, sum_, lastd_, cnt_] := Block[{t}, If[cnt == nD, check[prod*sum, n], Do[t = nD - cnt - 1; If[n*10^(t+1) <= d*prod*9^t*(sum + d + 9*t), ric[10*n + d, d*prod, d + sum, d, cnt + 1], Break[]], {d, 9, lastd, -1}]]]; ric[0, 1, 0, 1, 0]; Print["nDig=", nD, " sol=", sol = Sort@sol]; sol]; Flatten[dig /@ Range[84]] (* Giovanni Resta, Mar 20 2013 *)

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

Original entry on oeis.org

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

Views

Author

Olivier Gérard

Keywords

Comments

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)

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    Select[ Range[10^6], IntegerQ[ # /(Apply[ Times, IntegerDigits[ # ]] * Apply[ Plus, IntegerDigits[ # ]] ) ] & ]
  • PARI
    isok(n) = my(d=digits(n)); vecprod(d) && (n % (vecsum(d)*vecprod(d)) == 0); \\ Michel Marcus, Jan 23 2019

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

Views

Author

Labos Elemer, Dec 13 2001

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

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

Extensions

Edited by Jon E. Schoenfield, Jul 09 2018

A049105 Ratio from A049101.

Original entry on oeis.org

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

Views

Author

Olivier Gérard

Keywords

Crossrefs

Cf. A049101, A049102, A049106.

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

Views

Author

Labos Elemer and Klaus Brockhaus, Dec 13 2001

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • ARIBAS
    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);
  • Mathematica
    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}]
  • PARI
    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

Views

Author

Labos Elemer and Klaus Brockhaus, Dec 13 2001

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • ARIBAS
    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);
  • Mathematica
    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 *)
  • PARI
    isok(k) = {my(d=digits(k)); k > vecprod(d) * vecsum(d)} \\ Harry J. Smith, Feb 10 2010
Showing 1-6 of 6 results.

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