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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A034054 Numbers with multiplicative digital root value 7.

Original entry on oeis.org

7, 17, 71, 117, 171, 711, 1117, 1171, 1711, 7111, 11117, 11171, 11711, 17111, 71111, 111117, 111171, 111711, 117111, 171111, 711111, 1111117, 1111171, 1111711, 1117111, 1171111, 1711111, 7111111, 11111117, 11111171, 11111711, 11117111
Offset: 1

Views

Author

Patrick De Geest, Sep 15 1998

Keywords

Comments

Numbers with one 7, and zero or more 1s. - Daniel Forgues, Oct 09 2011

Links

Crossrefs

Cf. A031347.
Cf. A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056 (numbers having multiplicative digital roots 0-9).

Programs

  • Mathematica
    Sort[Flatten[Table[FromDigits/@Permutations[Join[{7},PadRight[{},n,1]]],{n,0,10}]]] (* Harvey P. Dale, Jul 20 2015 *)
  • PARI
    t(k)=while(k>9, k=prod(i=1, #k=digits(k), k[i])); k
for(n=1, 1e8, if(t(n) == 7, print1(n, ", "))); \\ Altug Alkan, Oct 22 2015

Formula

There are n(n+1)/2 elements up to 10^n, so a(n) is about 10^sqrt(2n).

A034056 Numbers with multiplicative digital root value 9.

Original entry on oeis.org

9, 19, 33, 91, 119, 133, 191, 313, 331, 911, 1119, 1133, 1191, 1313, 1331, 1911, 3113, 3131, 3311, 9111, 11119, 11133, 11191, 11313, 11331, 11911, 13113, 13131, 13311, 19111, 31113, 31131, 31311, 33111, 91111, 111119, 111133, 111191, 111313, 111331
Offset: 1

Views

Author

Patrick De Geest, Sep 15 1998

Keywords

Comments

Numbers with one 9 or two 3s, and zero or more 1s. - Daniel Forgues, Oct 09 2011

Links

Crossrefs

Cf. A031347.
Cf. A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056 (numbers having multiplicative digital roots 0-9).

Programs

  • Mathematica
    Module[{nn=6,ne,te},ne=Union[FromDigits/@Flatten[Permutations/@Table[PadRight[{9},n,1],{n,nn}],1]];te=Rest[Union[FromDigits/@ Flatten[ Permutations/@Table[PadRight[{3,3},n,1],{n,nn}],1]]];Join[ne,te]]//Sort (* Harvey P. Dale, Apr 14 2025 *)

Formula

There are n(n+1)(n+2)/6 elements up to 10^n, so a(n) is about 10^sqrt(6n).

A277061 Numbers with multiplicative digital root > 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 53, 57, 61, 62, 63, 64, 66, 67, 68, 71, 72, 73, 74, 75, 76, 77, 79, 81, 82, 83, 84, 86, 88, 89, 91, 92, 93, 94, 97, 98, 99, 111, 112, 113, 114, 115
Offset: 1

Views

Author

J. Lowell, Sep 26 2016

Keywords

Comments

Question: when will numbers not in this sequence outnumber numbers in this sequence? Up to n = 1249, there are 524 terms, so 525 terms not in this sequence. Up to n = 1522, there are n/2 terms. No n > 1522 has that property. Up to 10^10, only about 1.46% of numbers are a term.
To find how many terms there are up to 10^n, see if A009994(i) is for 2 <= i <= binomial(n + 9, 9). If it is then that adds A047726(A009994(i)) to the total (we don't have to worry about digits 0 in A009994(i) as there aren't any for the specified i). One may put further constraints on i. For example, A009994(i) can't contain an even digit and a 5 in the same number. - David A. Corneth, Sep 27 2016

Examples

			25 is not in this sequence because 2*5 = 10 and 1*0 = 0.

Links

Crossrefs

Cf. A009994, A047726.
Cf. A031347, A034048 (complement).
Cf. A028843 (a subsequence).
Union of A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056 (numbers having multiplicative digital roots 1-9).
Cf. A052382 (a supersequence).

Programs

  • Mathematica
    Select[Range@ 112, FixedPoint[Times @@ IntegerDigits@ # &, #] > 0 &] (* Michael De Vlieger, Sep 26 2016 *)
  • PARI
    is(n) = n=digits(n); while(#n>1,n=digits(prod(i=1,#n,n[i]))); #n>0 \\ David A. Corneth, Sep 27 2016

Extensions

More terms from Michael De Vlieger, Sep 26 2016

A199982 Composite numbers with digital product = 3.

Original entry on oeis.org

1113, 1131, 1311, 3111, 13111, 31111, 111113, 111131, 111311, 1111113, 1111131, 1111311, 1113111, 1131111, 1311111, 3111111, 11111113, 11311111, 13111111, 31111111, 111111311, 111131111, 111311111, 113111111, 311111111, 1111111113, 1111111131, 1111111311
Offset: 1

Views

Author

Jaroslav Krizek, Nov 13 2011

Keywords

Comments

Also composite numbers whose multiplicative digital root is 3. Complement of A107689 with respect to A034050.

Links

Crossrefs

Cf. A107689 (primes with digital product = 3).
Cf. A034050 (numbers with digital product = 3).

Programs

  • Mathematica
    Table[Select[FromDigits[Permutations[PadRight[{3},n,1]]],CompositeQ],{n,4,10}]//Flatten//Sort (* Harvey P. Dale, May 31 2025 *)

Extensions

Incorrect 111113111 removed by Sean A. Irvine, Jan 06 2025
Previous Showing 11-14 of 14 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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