A050695 -id:A050695 - cp's OEIS frontend

A050698 Numbers that appear in A050695 but not in A035139.

51, 74, 91, 94, 104, 114, 117, 119, 121, 133, 136, 141, 143, 154, 156, 159, 164, 169, 171, 176, 178, 186, 187, 194, 198, 203, 207, 209, 221, 253, 261, 291, 299, 301, 314, 319, 338, 341, 344, 346, 361, 364, 368, 374, 376, 377, 386, 391, 394, 398, 403, 404

Offset: 1

Patrick De Geest, Aug 15 1999

None of the prime factors of a(n) are a substring of a(n) and at least one of the digits of the prime factors of a(n) appears in a(n).

			51 = 3*17 -> digit 1 appears in 5{1}.

Cf. A050695, A035139.

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

Patrick De Geest, Aug 15 1999

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

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 150 terms from Lava)
Gil Broussard, Integers containing prime factors as substrings.

Cf. A050695, A050696, A035140, A050697.

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 *)

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

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

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 *)

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.

A050696 At least one prime factor of composite a(n) is a substring of a(n).

Original entry on oeis.org

12, 15, 20, 22, 24, 25, 26, 28, 30, 32, 33, 35, 36, 39, 42, 45, 50, 52, 55, 62, 63, 65, 70, 72, 75, 77, 82, 85, 92, 93, 95, 102, 105, 110, 112, 115, 120, 122, 123, 124, 125, 126, 128, 130, 132, 135, 138, 142, 145, 147, 150, 152, 153, 155, 162, 165, 170, 172, 175

Offset: 1

Patrick De Geest, Aug 15 1999

			26 is in the sequence because 26 = 2 * 13 and the factor 2 appears in the decimal representation. Though 13 does not appear, the 2 is enough for 26 to be in the sequence.
27 is not in the sequence since 27 = 3 * 3 * 3, which does not appear in the decimal representation.

Vincenzo Librandi, Table of n, a(n) for n = 1..4720

Cf. A050694, A050695.

digs[n_] := IntegerDigits[n]; A050696 = {}; Do[le1 = Max@@Length/@(prFDigs = digs[First/@FactorInteger[n]]); dSubStrs = Flatten[Table[Partition[digs[n], i, 1], {i, le1}], 1]; If[!PrimeQ[n] && Intersection[prFDigs, dSubStrs] != {}, AppendTo[A050696, n]],{n, 2, 180}]; A050696 (* Jayanta Basu, May 31 2013 *)

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A050698 Numbers that appear in A050695 but not in A035139.

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A050694 Composite numbers k such that all prime factors of k are 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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A131929 Number of prime factors of n that are contained in decimal representation of n.

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A050696 At least one prime factor of composite a(n) is a substring of a(n).

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