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 1-5 of 5 results.

A050697 Numbers that appear in A035140 but not in A050694.

Original entry on oeis.org

121, 132, 143, 187, 295, 312, 341, 351, 451, 671, 679, 781, 928, 932, 1023, 1053, 1057, 1207, 1243, 1255, 1324, 1325, 1328, 1331, 1359, 1375, 1377, 1379, 1392, 1395, 1539, 1573, 1592, 1703, 1775, 1972, 1975, 1982, 2139, 2189, 2317, 2321, 2349, 2367
Offset: 1

Views

Author

Patrick De Geest, Aug 15 1999

Keywords

Comments

Digits of prime factors of a(n) all appear in a(n) but not all prime factors of a(n) are a substring of a(n).

Examples

			187 = 11*17 -> digits 1 and 7 appear in {1}8{7} and 11 and 17 aren't substrings of "187".

Crossrefs

Cf. A050694, A035140.

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]!={}&&Complement[Flatten[t1],d[n]]=={},AppendTo[t,n]],{n,20,2380}]; t (* Jayanta Basu, May 31 2013 *)

A035139 Digits of prime factors of k do not appear in k.

Original entry on oeis.org

1, 4, 6, 8, 9, 10, 14, 16, 18, 21, 27, 34, 38, 40, 44, 46, 48, 49, 54, 56, 57, 58, 60, 64, 66, 68, 69, 76, 78, 80, 81, 84, 86, 87, 88, 90, 96, 98, 99, 100, 106, 108, 111, 116, 118, 129, 134, 140, 144, 146, 148, 158, 160, 161, 166, 168, 174, 177, 180, 184, 188, 189, 196
Offset: 1

Views

Author

Patrick De Geest, Nov 15 1998

Keywords

Examples

			161 = 7 * 23 since {2,3,7} and {1,6} are separate digit sets.

Links

Crossrefs

Cf. A035140, A035141.

Programs

  • Magma
    [k:k in [1..200]| forall{a: a in PrimeDivisors(k)|Set(Intseq(a)) meet Set(Intseq(k)) eq {}}]; // Marius A. Burtea, Oct 08 2019
  • Maple
    q:= n-> (f-> is(map(f, numtheory[factorset](n)) intersect
        {f(n)}={}))(d-> convert(d, base, 10)[]):
select(q, [$1..200])[];  # Alois P. Heinz, Oct 11 2021
  • Mathematica
    Fac[n_] := Flatten[IntegerDigits[Take[FactorInteger[n],All,1]]]; t={1}; Do[ If[!PrimeQ[n] && Intersection[Fac[n], IntegerDigits[n]] == {}, AppendTo[t,n]], {n,2,196}]; t (* Jayanta Basu, May 02 2013 *)
  • PARI
    digsf(n) = my(f=factor(n), list=List()); for (k=1, #f~, my(dk=digits(f[k,1])); for (i=1, f[k,2], for (j=1, #dk, listput(list, dk[j])))); Vec(list);
isok(m) = my(df=digsf(m), d=digits(m)); (#setintersect(Set(df), Set(d)) == 0); \\ Michel Marcus, Oct 11 2021
  • Python
    from sympy import factorint
def ok(n):
    return set(str(n)) & set("".join(str(p) for p in factorint(n))) == set()
print(list(filter(ok, range(1601))))  # Michael S. Branicky, Oct 11 2021

Extensions

Offset corrected and a(1) added by Giovanni Resta, May 02 2013

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

A035141 Composite numbers k such that digits in k and in juxtaposition of prime factors of k are the same (apart from multiplicity).

Original entry on oeis.org

132, 312, 735, 1255, 1377, 1775, 1972, 3792, 4371, 4773, 5192, 6769, 7112, 7236, 7371, 7539, 9321, 11009, 11099, 11132, 11163, 11232, 11255, 11375, 11913, 12176, 12326, 12595, 12955, 13092, 13175, 13312, 13377, 13491, 13755, 14842, 15033
Offset: 1

Views

Author

Patrick De Geest, Nov 15 1998

Keywords

Examples

			1972 = {1,2,7,9} -> 2 * 2 * 17 * 29, so 1972 is a term.

Links

Crossrefs

Cf. A035139, A035140.

Programs

  • Mathematica
    Fac[n_]:=Sort[DeleteDuplicates[Flatten[IntegerDigits[Take[FactorInteger[n], All,1]]]]];Fn[n_]:=Sort[DeleteDuplicates[IntegerDigits[n]]];t={};Do[If[! PrimeQ[n]&&Fac[n]===Fn[n],AppendTo[t, n]],{n,2,15100}];t (* Jayanta Basu, May 02 2013 *)
  • PARI
    is(n)=if(isprime(n)||n<9,return(0));my(f=factor(n)[,1],v=[]);for(i=1,#f,v=concat(v,digits(f[i])));vecsort(digits(n),,8)==vecsort(v,,8) \\ Charles R Greathouse IV, May 02 2013

Formula

a(n) ~ n. Proof: the density of numbers without a given decimal digit in their prime factors is 0, which can be seen by looking at the first (or second, in the case of 0) digit and removing all primes with that digit. Taken with the 0 density of numbers missing any decimal digit the result is obtained. - Charles R Greathouse IV, May 02 2013

Extensions

Definition corrected by Charles R Greathouse IV, May 02 2013

A372384 The smallest composite number k such that the digits of k and its prime factors, both written in base n, contain the same set of distinct digits.

Original entry on oeis.org

4, 8, 30, 25, 57, 16, 27, 192, 132, 121, 185, 169, 465, 32, 306, 289, 489, 361, 451, 2250, 552, 529, 125, 1586, 81, 1652, 985, 841, 1057, 64, 1285, 86166, 2555, 1332, 1387, 1369, 4752, 3240, 2005, 1681, 2649, 1849, 2047, 5456, 2256, 2209, 343, 5050, 2761, 5876, 2862, 2809, 3097, 15512
Offset: 2

Views

Author

Scott R. Shannon, Apr 29 2024

Keywords

Examples

			a(4) = 30 as 30 = 2 * 3 * 5 = 132_4 = 2_4 * 3_4 * 11_4, and both 132_4 and its prime factors contain the same distinct digits 1, 2, and 3.
a(10) = 132 as 132 = 2 * 3 * 11, and both 132 and its prime factors contain the same distinct digits 1, 2, and 3. See also A035141.
a(14) = 465 as 465 = 3 * 5 * 31 = 253_14 = 3_14 * 5_14 * 23_14, and both 253_14 and its prime factors contain the same distinct digits 2, 3, and 5.

Links

Crossrefs

Cf. A027746, A035141, A035140, A372309, A371695, A000668.

Formula

a(n) = 2*n + 2 if n = 2^k - 1 with k >= 2, otherwise a(n) = n^2 if n is prime.
Showing 1-5 of 5 results.

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