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.

A050694 Composite numbers k such that all prime factors of k are a substring of k.

Original entry on oeis.org

25, 32, 125, 128, 135, 175, 243, 250, 256, 324, 375, 432, 512, 625, 735, 875, 1024, 1250, 1352, 1372, 1593, 1675, 1715, 1792, 2048, 2176, 2304, 2500, 2510, 2560, 2570, 2744, 3072, 3087, 3125, 3375, 3645, 3675, 3792, 4232, 4375, 5120, 5210, 5230, 5832
Offset: 1

Views

Author

Patrick De Geest, Aug 15 1999

Keywords

Examples

			1675 = 5*5*67 -> 167{5} and 1{67}5.

Links

Crossrefs

Cf. A050695, A050696, A035140, A050697.

Programs

  • Mathematica
    d[n_]:=IntegerDigits[n]; t={}; Do[le1=Max@@Length/@(t1=d[First/@FactorInteger[n]]); t2=Flatten[Table[Partition[d[n],i,1],{i,le1}],1]; If[!PrimeQ[n]&&Complement[t1,t2]=={},AppendTo[t,n]],{n,20,5850}]; t (* Jayanta Basu, May 31 2013 *)
  • PARI
    substr(m,n)=my(a=#Str(m),b=#Str(n)); for(i=0,a-b,if(valuation(m-n,10)>=b, return(1)); m\=10); 0
is(n)=if(isprime(n)||n<9, return(0)); my(f=factor(n)[,1]); for(i=1,#f,if(!substr(n,f[i]), return(0))); 1 \\ Charles R Greathouse IV, Jul 09 2015

Formula

a(n) << n log n. - Charles R Greathouse IV, Jul 09 2015

A050695 Composite numbers k such that none of the prime factors of k is a substring of k.

Original entry on oeis.org

4, 6, 8, 9, 10, 14, 16, 18, 21, 27, 34, 38, 40, 44, 46, 48, 49, 51, 54, 56, 57, 58, 60, 64, 66, 68, 69, 74, 76, 78, 80, 81, 84, 86, 87, 88, 90, 91, 94, 96, 98, 99, 100, 104, 106, 108, 111, 114, 116, 117, 118, 119, 121, 129, 133, 134, 136, 140, 141, 143, 144, 146
Offset: 1

Views

Author

Patrick De Geest, Aug 15 1999

Keywords

Comments

A131929(a(n)) = 0; together with 1, complement of A131930. - Reinhard Zumkeller, Jul 30 2007

Examples

			114 = 2*3*19.

Links

Crossrefs

Cf. A050694, A050696, A035139, A050698.

Programs

  • Mathematica
    d[n_]:=IntegerDigits[n]; t={}; Do[le1=Max@@Length/@(t1=d[First/@FactorInteger[n]]); t2=Flatten[Table[Partition[d[n],i,1],{i,le1}],1]; If[!PrimeQ[n]&&Intersection[t1,t2]=={},AppendTo[t,n]],{n,2,146}]; t (* Jayanta Basu, May 31 2013 *)
npfsQ[n_]:=Module[{idn=IntegerDigits[n],f=FactorInteger[n][[All,1]]}, And@@ Table[SequenceCount[idn,IntegerDigits[f[[i]]]]==0,{i, Length[ f]}]]; Select[Range[200],CompositeQ[#] && npfsQ[#]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 28 2016 *)

A161990 Composites which have the same largest prime factor as their index.

Original entry on oeis.org

10, 12, 14, 25, 36, 39, 42, 45, 77, 124, 132, 140, 147, 224, 234, 266, 345, 365, 370, 375, 380, 385, 390, 494, 621, 638, 660, 671, 682, 782, 899, 945, 1001, 1086, 1140, 1377, 1558, 1577, 1628, 1696, 1728, 1760, 1798, 1885, 2046, 2145, 2484, 2550, 2970, 3101, 3122, 3477
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jun 24 2009

Keywords

Comments

If A052369(k) = A006530(k), we add the associated composite A002808(k) to the sequence.

Examples

			The 6th composite is 12=2^2*3 with largest prime factor 3, and the largest prime factor of the index 6=2*3 is also 3, which adds 12 to the sequence.
The 7th composite is 14=2*7 with largest prime factor 7, and the largest prime factor of the index 7 is also 7, which adds 14 to the sequence.

Links

Crossrefs

Cf. A002808, A050696.

Programs

  • Maple
    A006530 := proc(n) sort(convert(numtheory[factorset](n),list)); op(-1,%) ; end:
A002808 := proc(n) if n = 1 then 4; else for a from procname(n-1)+1 do if not isprime(a) then RETURN(a) ; fi; od: fi; end:
A052369 := proc(n) A006530(A002808(n)) ; end:
for n from 1 to 10000 do if A052369(n) = A006530(n) then printf("%d,",A002808(n)) ; fi; od: # R. J. Mathar, Aug 14 2009
# More efficient alternative:
N:= 10000: # to get terms <= N
Lpf:= [seq(max(numtheory:-factorset(n)),n=1..N)]:
comps:= select(n -> Lpf[n]Robert Israel, Mar 05 2018
  • Mathematica
    lpf[n_] := FactorInteger[n ][[-1, 1]];
cc = Select[Range[10000], CompositeQ];
Select[{Range[Length[cc]], cc} // Transpose, lpf[#[[1]]] == lpf[#[[2]]]&][[All, 2]] (* Jean-François Alcover, Aug 19 2020 *)

Extensions

Corrected and extended by R. J. Mathar, Aug 14 2009
Showing 1-3 of 3 results.

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