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.

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A070841 Repdigits (A010785), excluding repunits (A002275), ordered by product of digits (A007954).

Original entry on oeis.org

2, 3, 4, 22, 5, 6, 7, 8, 222, 9, 33, 44, 2222, 55, 333, 22222, 66, 77, 88, 444, 222222, 99, 3333, 555, 2222222, 666, 33333, 4444, 22222222, 777, 888, 222222222, 5555, 999, 333333, 44444, 2222222222, 6666, 22222222222, 3333333, 7777, 55555, 8888
Offset: 1

Views

Author

Amarnath Murthy, May 12 2002

Keywords

Crossrefs

Cf. A070840.

Extensions

Edited and extended by Ray Chandler, Feb 10 2009

A134586 a(n) = a(n-1) + A007954(n-1) if a(n-1) is odd, otherwise a(n) = a(n-1) / 2.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 7, 14, 7, 16, 8, 4, 2, 1, 5, 10, 5, 12, 6, 3, 3, 5, 9, 15, 23, 33, 45, 59, 75, 93, 93, 96, 48, 24, 12, 6, 3, 24, 12, 6, 3, 7, 15, 27, 43, 63, 87, 115, 147, 183, 183, 188, 94, 47, 67, 92, 46, 23, 63, 108, 54, 27, 39, 57, 81, 111, 147, 189, 237, 291, 291, 298, 149
Offset: 1

Views

Author

Ctibor O. Zizka, Jan 24 2008

Keywords

Links

Crossrefs

Cf. A007954.

Programs

  • Mathematica
    nxt[{n_,a_}]:={n+1,If[EvenQ[a],a/2,a+Times@@IntegerDigits[n+1]]}; Transpose[ NestList[nxt,{0,1},80]][[2]] (* Harvey P. Dale, Sep 11 2015 *)
  • PARI
    lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, va[n] = if (va[n-1] % 2, va[n-1] + vecprod(digits(n-1)), va[n-1]/2);); va;} \\ Michel Marcus, Nov 28 2020

Extensions

Name corrected by Michel Marcus, Nov 28 2020

A243219 Smallest n-digit integer x such that x + A007954(x) has n+1 digits, where A007954(x) is the product of decimal digits of x.

Original entry on oeis.org

5, 59, 599, 6799, 68899, 689999, 6999999, 77899999, 779999999, 7889999999, 78999999999, 799999999999, 8689999999999, 86999999999999, 878999999999999, 8799999999999999, 88899999999999999, 889999999999999999, 8989999999999999999, 89999999999999999999
Offset: 1

Views

Author

Michel Marcus, Jun 01 2014

Keywords

Comments

The related sequence with x the largest n-digit number such that x + A007954(x) also has n digits would be 4, 90, 990, 9990, 99990, ..., etc.
a(n) <= 9*R_n for all n. Further, floor(a(n+1)/10) >= a(n) for all n. - Derek Orr, Jun 02 2014

Examples

			5 is the smallest integer with 1 digit such that 5 + A007954(5) has 2 digits, with result 5 + 5 = 10, hence a(1)=5.

Crossrefs

Cf. A007954, A242945, A243218.

Programs

  • PARI
    DP(n)= my(d = digits(n)); prod(i=1, #d, d[i]);
a(n) = {for (i=10^(n-1), 10^n-1, if (i + DP(i) >= 10^n, return(i)););}

Extensions

a(10)-a(20) from Derek Orr, Jun 02 2014

A334534 Numbers k such that (k-p)*(k+p) contains k as a substring, where p > 0 and p = A007954(k) is the product of digits of k.

Original entry on oeis.org

25, 28, 128, 225, 293, 678, 725, 742, 749, 4225, 4421, 6225, 8926, 72225, 617371, 1985525, 3679518, 4381824, 6816771, 8572645, 9721317, 43872768, 54639413, 758873243, 5895396725, 7796276839, 8881527332, 9458237492, 9594769255, 9949621217, 25214163187, 31987487294
Offset: 1

Views

Author

Scott R. Shannon, May 05 2020

Keywords

Examples

			25 is a term as p = 2*5 = 10 and (25-10)*(25+10) = 525 which contains '25' as a substring.
8926 is a term as p = 8*9*2*6 = 864 and (8926-864)*(8926+864) = 78926980 which contains '8926' as a substring.

Links

Crossrefs

Cf. A007954, A063108, A028842.

Programs

  • PARI
    isokp(dx, d) = {if (!#setintersect(Set(dx), Set(d)), return (0)); for (i=1, #dx - #d + 1, if (vector(#d, k, dx[k+i-1]) == d, return(1)););}
isokd(x, d, n) = {if (x==n, return (1)); my(dx = digits(x)); if (#dx < #d, return (0)); isokp(dx, d);}
isok(n) = {my(d = digits(n), p = vecprod(d)); if (p>0, isokd((n-p)*(n+p), d, n));} \\ Michel Marcus, May 07 2020

Extensions

More terms from Giovanni Resta, May 07 2020

A339757 a(n) is the number of Zuckerman numbers k for which k/A007954(k) = n, where A007954(k) is the product of the decimal digits of k.

Original entry on oeis.org

9, 1, 2, 1, 1, 1, 1, 2, 3, 0, 1, 2, 3, 1, 0, 0, 1, 3, 2, 0, 1, 2, 3, 0, 0, 0, 1, 2, 2, 0
Offset: 1

Views

Author

Michel Marcus, Apr 04 2021

Keywords

Comments

The indices of 0's are A342593.

Examples

			The integers k=1 to 9 are the Zuckerman numbers that satisfy k/A007954(k)=1, so a(1)=9.

Crossrefs

Cf. A007954 (product of decimal digits), A007602 (Zuckerman numbers), A056770.
Cf. A288069 (Zuckerman quotients), A342593 (Zuckerman non-quotients).
Cf. A343036, A343050.

A341634 Smallest prime whose product of digits (A007954) is the n-th 7-smooth number = A002473(n), with a(0) = 101.

Original entry on oeis.org

101, 11, 2, 3, 41, 5, 23, 7, 181, 19, 251, 43, 127, 53, 281, 29, 541, 37, 83, 11551, 139, 47, 523, 1481, 157, 149, 12451, 67, 59, 283, 11177, 2551, 239, 1187, 1453, 79, 881, 257, 89, 1553, 2851, 199, 347, 563, 1483, 277, 14551, 1753, 269, 827, 853, 15551, 367
Offset: 0

Views

Author

Bernard Schott, Feb 16 2021

Keywords

Comments

For n>=1, equals A107698 without the zeros.
101 is the smallest prime with the digit 0, so A007954(101) = 0 but as 0 is not a 7-smooth number, it is chosen a(0) = 101.

Examples

			83 is prime, A007954(83) = 8*3 = 24 that is the 18th 7-smooth number, and as no prime < 83 has a product of digits = 24, a(18) = 83.

Links

Crossrefs

Cf. A002473, A004954, A007954, A053666, A107698, A335331.

Programs

  • Mathematica
    pod[n_] := Times @@ IntegerDigits[n]; seq[max_] := Module[{sm7 = Join[{0}, Select[Range[max], Max[FactorInteger[#][[;; , 1]]] <= 7 &]], m, s, n, c, i, ind}, m = Length[sm7]; s = Table[0, {m}]; n = 1; c = 0; While[c < m, n = NextPrime[n]; i = pod[n]; If[MemberQ[sm7, i], ind = Position[sm7, i][[1, 1]]]; If[s[[ind]] == 0, c++; s[[ind]] = n]]; s]; seq[150] (* Amiram Eldar, Feb 16 2021 *)

Formula

a(n) = A107698(A002473(n)) for n>=1. - Amiram Eldar, Feb 17 2021

Extensions

More terms from Amiram Eldar, Feb 16 2021

A343050 Zuckerman numbers (A007602) ordered by increasing value of k/A007954(k) where A007954(k) is the product of the decimal digits of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 15, 24, 384, 175, 12, 735, 128, 672, 135, 144, 1575, 11, 1296, 139968, 624, 3276, 1886976, 224, 816, 216, 432, 34992, 1197, 12768, 315, 132, 3168, 115, 6624, 8832, 2916, 1176, 1344, 3915, 739935
Offset: 1

Views

Author

Michel Marcus, Apr 03 2021

Keywords

Comments

a(n) is the Zuckerman number corresponding to A343036(n).

Examples

			As a table, sequence begins:
   1 [1, 2, 3, 4, 5, 6, 7, 8, 9]
   2 [36]
   3 [15, 24]
   4 [384]
   5 [175]
   6 [12]
   7 [735]
   8 [128, 672]
   9 [135, 144, 1575]
  10 []
  11 [11]
  12 [1296, 139968]
  13 [624, 3276, 1886976]
  14 [224]
  15 []
  16 []
  17 [816]
  18 [216, 432, 34992]
  19 [1197, 12768]
  20 []
  21 [315]
  22 [132, 3168]
  23 [115, 6624, 8832]
  24 []
  25 []
  26 []
  27 [2916]
  28 [1176, 1344]
  29 [3915, 739935]
  30 []
  ... where the 1st column is A056770 and the number of terms per rows is A339757.

Crossrefs

Cf. A007954 (product of decimal digits), A007602 (Zuckerman numbers), A056770.
Cf. A288069 (Zuckerman quotients), A342593 (Zuckerman non-quotients), A343036.
Cf. A339757.

Extensions

a(29)-a(45) from David A. Corneth, Apr 03 2021

A376257 a(n) = (n - c(n))*(-1)^c(n), where c(n) is the product of the digits of n (A007954).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 10, -10, 10, -10, 10, -10, 10, -10, 10, -10, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 30, -28, 26, -24, 22, -20, 18, -16, 14, -12, 40, 37, 34, 31, 28, 25, 22, 19, 16, 13, 50, -46, 42, -38, 34, -30, 26, -22, 18, -14, 60, 55, 50, 45, 40, 35, 30, 25, 20, 15, 70, -64, 58
Offset: 1

Views

Author

Stuart Coe, Sep 17 2024

Keywords

Comments

There is an interesting pattern on the graph of the sequence.

Examples

			For n = 129: 1*2*9=18. So a(n) = (129-18)*(-1)^18 = 111.

Links

Crossrefs

Cf. A007954, A070565.

Programs

  • Mathematica
    A376257[n_] := (n - #)*(-1)^# & [Times @@ IntegerDigits[n]];
Array[A376257, 100] (* Paolo Xausa, Dec 10 2024 *)

Formula

a(n) = (n-A007954(n)) * (-1)^A007954(n).

A110057 Number of solutions to x == 0 (mod A007954(x)), x in A002113, 10^(n-1) <= x < 10^n.

Original entry on oeis.org

9, 1, 2, 3, 7, 3, 6, 4, 9, 9, 18, 11, 24, 17, 25, 12, 45, 16, 53, 22
Offset: 1

Views

Author

Jean-Christophe Colin (colinjeanchristophe(AT)yahoo.fr), Sep 04 2005

Keywords

Links

Crossrefs

Cf. A002113, A007954, A117057.

Extensions

Edited by R. J. Mathar, Feb 08 2008
Offset changed to 1 and a(1) inserted by Jinyuan Wang, Jun 15 2022

A226726 Numbers n such that the product of its decimal digits (A007954) is between n/2-1 and n/2+1.

Original entry on oeis.org

1, 19, 27, 36, 289, 379
Offset: 1

Views

Author

José María Grau Ribas, Jun 15 2013

Keywords

Comments

There are no more terms in this sequence. - Charles R Greathouse IV, Jun 15 2013

Crossrefs

Cf. A007954

Programs

  • Mathematica
    j[lis_] := Product[lis[[i]], {i, 1, Length[lis]}];
jj[n_] := j[RealDigits[n][[1]]]; Table[If[n/2 - 1 < j[RealDigits[n][[1]]] < n/2 + 1, n], {n, 1, 1000000}] // Union
Select[Range[400],#/2-1Harvey P. Dale, Oct 22 2024 *)
  • PARI
    is(n)=my(d=digits(n));abs(prod(i=1,#d,d[i])-n/2)<=1
select(is,vector(400,i,i))  \\ Charles R Greathouse IV, Jun 15 2013
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