A131930 -id:A131930 - 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 *)

A131929 Number of prime factors of n that are contained in decimal representation of n.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1

Offset: 1

Reinhard Zumkeller, Jul 30 2007

a(A050695(n)) = 0; a(A131930(n)) > 0;

a(A131931(n)) = n and a(m) <> n for m < A131931(n).

			a(230) = a(23*5*2) = #{2,23} = 2.

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

Cf. A131930, A131931.

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]

A319022 a(n) is the smallest number with n distinct prime factors whose decimal representation contains all its prime factors, taking multiplicity into account.

Original entry on oeis.org

2, 119911, 2510, 21372, 1943795, 73171842, 113678565, 13121675970, 297115923720, 73605381139290, 255360234137190, 43759729761726090

Offset: 1

Altug Alkan, Sep 08 2018

A version of A318965.
If a term t is divisible by a prime power p^k, then p must appear at least k times in the decimal representation of t. The copies of p are allowed to overlap; however, in the first 12 terms of the sequence, this does not occur, and only a(2), a(4), and a(9) are nonsquarefree.
a(n) >= A318965(n) by definition. If A318965(n) is squarefree, then a(n) = A318965(n). But a(n) = A318965(n) can also hold for a(n) that is nonsquarefree; e.g., a(4) = A318965(4).

			a(2) = 119911 = 11 * 11 * 991.
a(3) = 2510 = 2 * 5 * 251.
a(4) = 21372 = 2 * 2 * 3 * 13 * 137.
a(5) = 1943795 = 5 * 7 * 19 * 37 * 79.
a(6) = 73171842 = 2 * 3 * 17 * 31 * 73 * 317.
a(7) = 113678565 = 3 * 5 * 7 * 11 * 13 * 67 * 113.

Cf. A083359, A131930, A318965.

Terms computed by Giovanni Resta, Sep 08 2018

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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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A131929 Number of prime factors of n that are contained in decimal representation of n.

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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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A319022 a(n) is the smallest number with n distinct prime factors whose decimal representation contains all its prime factors, taking multiplicity into account.

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