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 1-9 of 9 results.

A235696 Semiprimes which have one or more occurrences of exactly eight different digits.

Original entry on oeis.org

10234569, 10234657, 10234685, 10234687, 10234769, 10234795, 10234859, 10234865, 10234879, 10234957, 10234967, 10235469, 10235479, 10235489, 10235497, 10235679, 10235689, 10235769, 10235789, 10235798, 10235846, 10235847, 10235879, 10235894, 10235947, 10235986
Offset: 1

Views

Author

Colin Barker, Jan 14 2014

Keywords

Links

Crossrefs

Cf. A235690-A235695.
Cf. A235154-A235161.

Programs

  • Mathematica
    Select[Range[10234000,10236000],PrimeOmega[#]==Count[DigitCount[#],0]==2&] (* Harvey P. Dale, Sep 01 2014 *)
  • PARI
    list(lim)=my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, listput(v, t*q))); vecsort(Vec(v)) \\ From A001358
b=list(1030000); s=[]; for(n=1, #b, if(#vecsort(eval(Vec(Str(b[n]))),,8)==8, s=concat(s, b[n]))); s

A335843 a(n) is the number of n-digit positive integers with exactly two distinct base 10 digits.

Original entry on oeis.org

0, 81, 243, 567, 1215, 2511, 5103, 10287, 20655, 41391, 82863, 165807, 331695, 663471, 1327023, 2654127, 5308335, 10616751, 21233583, 42467247, 84934575, 169869231, 339738543, 679477167, 1358954415, 2717908911, 5435817903, 10871635887, 21743271855, 43486543791
Offset: 1

Views

Author

Stefano Spezia, Jul 18 2020

Keywords

Comments

a(n) is the number of n-digit numbers in A031955.

Examples

			a(1) = 0 since the positive integers must have at least two digits;
a(2) = 81 since #[99] - #[9] - #(11*[9]) = 99 - 9 - 9 = 81;
a(3) = 243 since #[999] - #[99] - #(111*[9]) - #{xyz in N | x,y,z are three different digits with x != 0} = 999 - 99 - 9 - 9*9*8 = 243;
...

Links

Crossrefs

Cf. A000225, A031955, A337127, A337313.
Cf. A055642, A235154, A235690, A235717.

Programs

  • Mathematica
    LinearRecurrence[{3,-2},{0,81},31]
  • PARI
    concat([0],Vec(81*x^2/(1-3*x+2*x^2)+O(x^31)))

Formula

O.g.f.: 81*x^2/(1 - 3*x + 2*x^2).
E.g.f.: 81*(exp(x) - 1)^2/2.
a(n) = 3*a(n-1) - 2*a(n-2) for n > 2.
a(n) = 81*(2^(n-1) - 1).
a(n) = 81*A000225(n-1).

Extensions

a(0) removed by Stefano Spezia, Sep 23 2020

A235691 Semiprimes which have one or more occurrences of exactly three different digits.

Original entry on oeis.org

106, 123, 129, 134, 142, 143, 145, 146, 158, 159, 169, 178, 183, 185, 187, 194, 201, 203, 205, 206, 209, 213, 214, 215, 217, 218, 219, 235, 237, 247, 249, 253, 254, 259, 265, 267, 274, 278, 287, 289, 291, 295, 298, 301, 302, 305, 309, 314, 319, 321, 326, 327
Offset: 1

Views

Author

Colin Barker, Jan 14 2014

Keywords

Comments

The first term having a repeated digit is 1003.

Examples

			91119111691966691969 is a term, because it is made of the 3 digits {1, 6, 9} and is the product of two primes 9397848521 and 9695741689. - _Giovanni Resta_, Jan 14 2014

Crossrefs

Cf. A235690, A235692-A235696.
Cf. A235154-A235161.

Programs

  • Mathematica
    Select[Range@999, Length@ Union@ IntegerDigits[#] == 3 && Total[Last /@ FactorInteger[#]] == 2 &] (* Giovanni Resta, Jan 14 2014 *)
  • PARI
    list(lim)=my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, listput(v, t*q))); vecsort(Vec(v)) \\ From A001358
b=list(10000); s=[]; for(n=1, #b, if(#vecsort(eval(Vec(Str(b[n]))),,8)==3, s=concat(s, b[n]))); s

A235693 Semiprimes which have one or more occurrences of exactly five different digits.

Original entry on oeis.org

10237, 10238, 10239, 10249, 10265, 10279, 10294, 10297, 10327, 10342, 10345, 10347, 10349, 10358, 10367, 10378, 10379, 10389, 10394, 10397, 10423, 10435, 10462, 10473, 10483, 10489, 10493, 10495, 10497, 10523, 10537, 10543, 10546, 10547, 10562, 10573, 10579
Offset: 1

Views

Author

Colin Barker, Jan 14 2014

Keywords

Comments

The first term having a repeated digit is 100235.
The first term that is a square is 12769. - Robert Israel, Jul 06 2018

Links

Crossrefs

Cf. A235690-A235692, A235694-A235696.
Cf. A235154-A235161.

Programs

  • Maple
    # to get all terms with 5 digits S:= combinat:-choose([$0..9],5):
f:= proc(x) local s,L;
      L:= convert(x,base,5);      if nops(L) < 5 then L:= [op(L),0$(5-nops(L))] fi;      if nops(convert(L,set))<5 then return NULL fi;
      op(select(t -> t > 10^4 and numtheory:-bigomega(t)=2, map(s -> add(s[L[i]+1]*10^(i-1),i=1..5),S)))
end proc:
sort(map(f, [$1..5^5-1])); # Robert Israel, Jul 06 2018
  • Mathematica
    Select[Range[10000,11000],PrimeOmega[#]==2&&Count[DigitCount[#],0]==5&] (* Harvey P. Dale, Apr 08 2015 *)
  • PARI
    list(lim)=my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, listput(v, t*q))); vecsort(Vec(v)) \\ From A001358
b=list(15000); s=[]; for(n=1, #b, if(#vecsort(eval(Vec(Str(b[n]))),,8)==5, s=concat(s, b[n]))); s

A247947 Four-digit odd semiprimes with all digits distinct.

Original entry on oeis.org

1027, 1037, 1043, 1047, 1057, 1059, 1067, 1073, 1079, 1203, 1205, 1207, 1243, 1247, 1253, 1257, 1263, 1267, 1273, 1285, 1293, 1329, 1345, 1347, 1349, 1357, 1369, 1379, 1385, 1387, 1389, 1397, 1403, 1405, 1437, 1457, 1465, 1469, 1473, 1497, 1507, 1509, 1527, 1529
Offset: 1

Views

Author

K. D. Bajpai, Sep 27 2014

Keywords

Comments

There are exactly 863 four-digit odd semiprimes with all distinct digits. The last few terms of the sequence are: 9563, 9571, 9573, 9607, 9617, 9627, 9637, 9641, 9647, 9651, 9671, 9673, 9683, 9687, 9701, 9703, 9713, 9731, 9745, 9753, 9761, 9763, 9813, 9827, 9841, 9847, 9853, 9863, 9865.
See the link with the b-file for all 863 entries.

Examples

			a(1) = 1027 = 13 * 79 is the smallest four-digit odd semiprime with all digits distinct.
a(863) = 9865 = 5 * 1973 is the largest four-digit odd semiprime with all digits distinct.

Links

Crossrefs

Cf. A001358, A046315, A074673, A235690.

Programs

  • Mathematica
    c = 0; Do[If[Length[Union[IntegerDigits[n]]] == 4 && PrimeOmega[n] == 2, c++; Print[c, "  ", n]], {n, 1001, 9999, 2}]

A235692 Semiprimes which have one or more occurrences of exactly four different digits.

Original entry on oeis.org

1027, 1037, 1042, 1043, 1046, 1047, 1057, 1059, 1067, 1073, 1079, 1082, 1094, 1203, 1205, 1207, 1234, 1238, 1243, 1247, 1253, 1257, 1263, 1267, 1273, 1285, 1286, 1293, 1294, 1306, 1329, 1345, 1346, 1347, 1349, 1354, 1357, 1369, 1379, 1382, 1385, 1387, 1389
Offset: 1

Views

Author

Colin Barker, Jan 14 2014

Keywords

Comments

The first term having a repeated digit is 10027.

Crossrefs

Cf. A235690, A235691, A235693-A235696.
Cf. A235154-A235161.

Programs

  • PARI
    list(lim)=my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, listput(v, t*q))); vecsort(Vec(v)) \\ From A001358
b=list(15000); s=[]; for(n=1, #b, if(#vecsort(eval(Vec(Str(b[n]))),,8)==4, s=concat(s, b[n]))); s

A235694 Semiprimes which have one or more occurrences of exactly six different digits.

Original entry on oeis.org

102347, 102349, 102369, 102379, 102385, 102386, 102387, 102389, 102394, 102395, 102398, 102439, 102457, 102458, 102463, 102467, 102469, 102473, 102478, 102479, 102493, 102549, 102569, 102574, 102589, 102637, 102639, 102649, 102658, 102659, 102683, 102689
Offset: 1

Views

Author

Colin Barker, Jan 14 2014

Keywords

Crossrefs

Cf. A235690-A235693, A235695, A235696.
Cf. A235154-A235161.

Programs

  • PARI
    list(lim)=my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, listput(v, t*q))); vecsort(Vec(v)) \\ From A001358
b=list(103000); s=[]; for(n=1, #b, if(#vecsort(eval(Vec(Str(b[n]))),,8)==6, s=concat(s, b[n]))); s

A235695 Semiprimes which have one or more occurrences of exactly seven different digits.

Original entry on oeis.org

1023469, 1023479, 1023547, 1023574, 1023586, 1023647, 1023649, 1023657, 1023689, 1023745, 1023746, 1023749, 1023794, 1023847, 1023879, 1023965, 1023985, 1024367, 1024369, 1024537, 1024538, 1024563, 1024567, 1024583, 1024637, 1024679, 1024687, 1024735
Offset: 1

Views

Author

Colin Barker, Jan 14 2014

Keywords

Crossrefs

Cf. A235690-A235694, A235696.
Cf. A235154-A235161.

Programs

  • PARI
    list(lim)=my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, listput(v, t*q))); vecsort(Vec(v)) \\ From A001358
b=list(1030000); s=[]; for(n=1, #b, if(#vecsort(eval(Vec(Str(b[n]))),,8)==7, s=concat(s, b[n]))); s

A247948 Five-digit odd semiprimes with all digits distinct.

Original entry on oeis.org

10237, 10239, 10249, 10265, 10279, 10297, 10327, 10345, 10347, 10349, 10367, 10379, 10389, 10397, 10423, 10435, 10473, 10483, 10489, 10493, 10495, 10497, 10523, 10537, 10543, 10547, 10573, 10579, 10583, 10587, 10623, 10637, 10643, 10645, 10649
Offset: 1

Views

Author

K. D. Bajpai, Sep 27 2014

Keywords

Comments

There are exactly 4858 five-digit odd semiprimes with all digits distinct. The last few terms of the sequence are: 98501, 98503, 98517, 98521, 98531, 98537, 98567, 98603, 98607, 98617, 98635, 98647, 98653, 98657, 98671, 98701, 98723, 98741, 98743, 98751, 98765.
See the link with the b-file for all 4858 entries.

Examples

			a(1) = 10237 = 29 * 353 is the smallest five-digit odd semiprime with all digits distinct.
a(4858) = 98765 = 5 * 19753 is the largest five-digit odd semiprime with all digits distinct.

Links

Crossrefs

Cf. A001358, A046315, A074671, A074673, A235690, A247947.

Programs

  • Mathematica
    c = 0; Do[If[Length[Union[IntegerDigits[n]]] == 5 && PrimeOmega[n] == 2, c++; Print[c, "  ", n]], {n, 10001, 99999, 2}]
Showing 1-9 of 9 results.

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