A131929 -id:A131929 - cp's OEIS frontend

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

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

Patrick De Geest, Aug 15 1999

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

			114 = 2*3*19.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A050694, A050696, A035139, A050698.

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 *)

A131930 Numbers k such that the decimal representation of k contains at least one prime factor of k.

Original entry on oeis.org

2, 3, 5, 7, 11, 12, 13, 15, 17, 19, 20, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 35, 36, 37, 39, 41, 42, 43, 45, 47, 50, 52, 53, 55, 59, 61, 62, 63, 65, 67, 70, 71, 72, 73, 75, 77, 79, 82, 83, 85, 89, 92, 93, 95, 97, 101, 102, 103, 105, 107, 109, 110, 112, 113, 115, 120

Offset: 1

Reinhard Zumkeller, Jul 30 2007

Numbers k such that A131929(k) > 0; complement of A050695 including 1.

All primes are terms.

			55 is a term because 5 | 55.

James C. McMahon, Table of n, a(n) for n = 1..10000

Cf. A131929, A050695.

Select[Range[2,120],ContainsAny[Rest[FromDigits/@Subsets[IntegerDigits[#]]],First/@FactorInteger[#]]&] (* James C. McMahon, Mar 02 2025 *)

A131931 Smallest number containing exactly n prime factors in its decimal representation.

Original entry on oeis.org

1, 2, 132, 735, 21372, 271362, 4773132, 113678565, 11317129824

Offset: 0

Reinhard Zumkeller, Jul 30 2007

A131929(a(n)) = n and A131929(m) <> n for m

			a(2) = 2^2 * 3 * 11 = 132 containing 2 and 3;
a(3) = 3 * 5 * 7^2 = 735 containing 3, 5 and 7;
a(4) = 2^2 * 3 * 13 * 137 = 21372 containing 2, 3, 13 and 137;
a(5) = 2 * 3 * 7^2 * 13 * 71 = 271362 containing 2, 3, 7, 13 and 71.

Cf. A131930, A131929, A318965.

Offset changed by and a(6)-a(8) from Giovanni Resta, Sep 06 2018

A318965 a(n) is the smallest number containing all its n prime factors in its decimal representation.

Original entry on oeis.org

2, 135, 735, 21372, 271362, 4773132, 113678565, 11317129824, 131175822960, 7113719552940, 255360234137190, 12411792985131540

Offset: 1

Giovanni Resta, Sep 06 2018

			a(2) = 135 = 3^3 * 5,
a(3) = 735 = 3 * 5 * 7^2,
a(4) = 21372 = 2^2 * 3 * 13 * 137,
a(5) = 271362 = 2 * 3 * 7^2 * 13 * 71,
a(6) = 4773132 = 2^2 * 3^2 * 7 * 13 * 31 * 47.
a(7) = 113678565 = 3 * 5 * 7 * 11 * 13 * 67 * 113.

Cf. A131929, A131930, A131931, A239060, A115738.

a[n_] := Block[{k=1, s, f}, While[True, k++; If[Length[f = FactorInteger[k]] == n, s = ToString@k; If[AllTrue[First /@ f, StringPosition[s, ToString@ #] != {} &], Break[]]]]; k]; Array[a, 5]

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

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A131930 Numbers k such that the decimal representation of k contains at least one prime factor of k.

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A131931 Smallest number containing exactly n prime factors in its decimal representation.

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A318965 a(n) is the smallest number containing all its n prime factors in its decimal representation.

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