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 1-3 of 3 results.

A099118 Conjectured number of times that S(k+n) = S(k), where S is the Kempner function A002034.

Original entry on oeis.org

0, 1, 2, 2, 3, 0, 9, 3, 2, 5, 18, 2, 28, 9, 2, 1, 53, 2, 79, 5, 10, 23
Offset: 1

Views

Author

T. D. Noe, Sep 28 2004

Keywords

Comments

Numbers k up to 10^8 have been tested. Tutescu's conjecture is the case n=1.

References

  • L. Tutescu, "On a Conjecture Concerning the Smarandache Function." Abstracts of Papers Presented to the Amer. Math. Soc. 17, 583, 1996.

Links

Crossrefs

Cf. A099119 (greatest k such that S(k) = S(k-n)), A099120 (least m such that n = S(k) = S(k+m)).

Programs

  • Mathematica
    (*See A002034 for the Kempner function*) nMax=22; iMax=10^6; iTab=Table[{}, {nMax}]; cTab=Table[0, {nMax}]; a=Table[Kempner[i], {i, nMax+1}]; Do[If[a[[i]]==a[[i-n]], cTab[[n]]++; AppendTo[iTab[[n]], i]], {n, nMax}, {i, n+1, nMax+1}]; Do[a=RotateLeft[a]; a[[nMax+1]]=Kempner[i]; Do[If[a[[nMax+1]]==a[[nMax-n+1]], cTab[[n]]++; AppendTo[iTab[[n]], i]], {n, nMax}], {i, nMax+2, iMax}]; cTab

A099120 Least m such that n = S(k) = S(k+m) for some k, where S is the Kempner function A002034.

Original entry on oeis.org

3, 4, 5, 2, 7, 32, 27, 8, 11, 26, 13, 48, 19, 4096, 17, 74, 19, 447, 27, 121, 23, 4005, 3125, 169, 177147, 2401, 29, 1203, 31, 134217728, 459, 289, 551, 2684163, 37
Offset: 3

Views

Author

T. D. Noe, Sep 28 2004

Keywords

Comments

Consider the set Sn of d(n!)-d((n-1)!) positive integers k with S(k) = n, where d is the divisor counting function A000005. For each n, a(n) gives the least difference of integers in the set Sn. For prime n, a(n) = n. For n a power of a prime, a(n) = A046021(n), the least k in Sn. The Tutescu conjecture, which states that the equation S(k) = S(k+1) has no solutions, is equivalent to a(n) > 1 for all n.

Examples

			a(6) = 2 because S(k) = 6 for k = 9, 16, 18, 36, 45, 48, 72, 80, 90, 144, 180, 240, 360, 720 and the least difference is 2, between 16 and 18.

References

  • L. Tutescu, "On a Conjecture Concerning the Smarandache Function." Abstracts of Papers Presented to the Amer. Math. Soc. 17, 583, 1996.

Links

Crossrefs

Cf. A099118 (number of times S(k+n) = S(k)), A099119 (greatest k such that S(k) = S(k-n)).

Programs

  • Mathematica
    (*See A002034 for the Kempner function*) a=Table[Kempner[n], {n, 10!}]; Table[lst=Flatten[Position[a, n]]; mn=Infinity; Do[mn=Min[mn, lst[[i+1]]-lst[[i]]], {i, Length[lst]-1}]; mn, {n, 10}]

A099143 Least k such that S(k) = S(k+n), or 0 if there is no k, where S is the Kempner function A002034.

Original entry on oeis.org

0, 16, 3, 4, 5, 0, 7, 4, 9, 5, 11, 12, 13, 7, 5, 8, 17, 18, 19, 4, 7, 11, 23, 48, 5, 13, 9, 7, 16, 10, 31, 16, 11, 17, 5, 9, 37, 19, 9, 20, 41, 14, 43, 11, 15, 23, 47, 10192, 7, 10, 17, 13, 53, 18, 5, 7, 19, 29, 59, 60, 61, 18, 7, 16, 13, 11, 67, 17, 23, 14, 9, 18, 73, 16, 25, 19, 7, 13
Offset: 1

Views

Author

T. D. Noe, Sep 30 2004

Keywords

Comments

The case of n=1 corresponds to the Tutescu conjecture, which states that the equation S(k) = S(k+1) has no solutions. It is conjectured that S(k) = S(k+6) also has no solutions. For odd prime p, a(p) = p. It appears that a(n) <= n, except for n = 1, 2, 6, 24, 48, 120, 240, 720 (A099144).

References

  • L. Tutescu, "On a Conjecture Concerning the Smarandache Function." Abstracts of Papers Presented to the Amer. Math. Soc. 17, 583, 1996.

Links

Crossrefs

Cf. A099118 (number of times S(k+n) = S(k)), A099119 (greatest k such that S(k) = S(k-n)).

Programs

  • Mathematica
    (*See A002034 for the Kempner function*) Table[i=1; While[i<10^5&&Kempner[i] != Kempner[i+n], i++ ]; If[i<10^5, i, 0], {n, 100}]
Showing 1-3 of 3 results.

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