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.

A077378 Squarefree numbers whose external as well as internal digits form a squarefree number.

Original entry on oeis.org

123, 127, 129, 130, 131, 133, 134, 137, 139, 151, 154, 155, 157, 159, 161, 163, 165, 167, 170, 173, 174, 177, 179, 221, 222, 223, 226, 229, 231, 233, 239, 251, 253, 259, 262, 263, 266, 269, 271, 273, 321, 323, 327, 329, 330, 331, 334, 335, 337, 339, 353, 354, 355, 357, 358
Offset: 1

Views

Author

Amarnath Murthy, Nov 06 2002

Keywords

Comments

Here 1 is treated as not squarefree. - Andrew Howroyd, Sep 19 2024

Links

Crossrefs

Intersection of A077376 and A077377.
Cf. A005117, A077379, A077380.

Programs

  • Mathematica
    A077378Q[k_] := k >= 100 && SquareFreeQ[k] && SquareFreeQ[FromDigits[#[[{1, -1}]]]] && FromDigits[#[[2;; -2]]] > 1 && SquareFreeQ[FromDigits[#[[2;; -2]]]] & [IntegerDigits[k]];
Select[Range[500], A077378Q] (* Paolo Xausa, Feb 19 2025 *)
  • PARI
    isok(k)={if(issquarefree(k) && k>=100, my(b=10^logint(k,10), m=k%b\10); m!=1 && issquarefree(m) && issquarefree(k\b*10+k%10), 0)} \\ Andrew Howroyd, Sep 19 2024

Extensions

Offset changed and a(29) onwards from Andrew Howroyd, Sep 19 2024

A077380 Largest n-digit squarefree number whose internal as well as external digits form a squarefree number, or 0 if no such number exists.

Original entry on oeis.org

0, 0, 977, 9977, 99987, 999987, 9999987, 99999977, 999999985, 9999999987, 99999999987, 999999999987, 9999999999987, 99999999999987, 999999999999987, 9999999999999987, 99999999999999987, 999999999999999987
Offset: 1

Views

Author

Amarnath Murthy, Nov 06 2002

Keywords

Crossrefs

Cf. A077376, A077377, A077378, A077379.

Programs

  • Maple
    with(numtheory) ; A077380 := proc(n) local anmax,leadd,edit,idit ; anmax := 10^n-1 ; while anmax >= 10^(n-1) do leadd := floor(anmax/10^(n-1)) ; edit := 10*leadd + ( anmax mod 10 ); idit := floor(anmax/10) -leadd*10^(n-2) ; if issqrfree(anmax) and issqrfree(edit) and issqrfree(idit) then RETURN(anmax) ; fi ; anmax := anmax-1 ; od ; RETURN(0) ; end: printf("0,") ; for n from 2 to 30 do printf("%d,",A077380(n)) ; od ; # R. J. Mathar, Sep 26 2006

Extensions

More terms from R. J. Mathar, Sep 26 2006

A077376 Squarefree numbers whose internal digits form a squarefree number.

Original entry on oeis.org

122, 123, 127, 129, 130, 131, 133, 134, 137, 138, 139, 151, 154, 155, 157, 158, 159, 161, 163, 165, 166, 167, 170, 173, 174, 177, 178, 179, 221, 222, 223, 226, 227, 229, 230, 231, 233, 235, 237, 238, 239, 251, 253, 254, 255, 257, 258, 259, 262, 263, 265, 266
Offset: 1

Views

Author

Amarnath Murthy, Nov 06 2002

Keywords

Comments

Here 1 is treated as not squarefree. - Harvey P. Dale, Jul 22 2012

Crossrefs

Cf. A077377, A077378, A077379, A077380.

Programs

  • Mathematica
    sf2Q[n_]:=Module[{c=FromDigits[Most[Rest[IntegerDigits[n]]]]},c!=1&&SquareFreeQ[ n]&&SquareFreeQ[c]]; Select[Range[100,300],sf2Q] (* Harvey P. Dale, Jul 22 2012 *)

Extensions

More terms from Harvey P. Dale, Jul 22 2012
Offset changed by Andrew Howroyd, Sep 19 2024

A077379 Smallest n-digit squarefree number whose internal as well as external digits form a squarefree number greater than 1; or 0 if no such number exists.

Original entry on oeis.org

0, 0, 123, 1021, 10021, 100021, 1000021, 10000021, 100000021, 1000000021, 10000000021, 100000000021, 1000000000021, 10000000000021, 100000000000021, 1000000000000021, 10000000000000021, 100000000000000021, 1000000000000000021, 10000000000000000021, 100000000000000000021, 1000000000000000000021
Offset: 1

Views

Author

Amarnath Murthy, Nov 06 2002

Keywords

Comments

Note that the pattern 10^(n-1)+21 is broken for any n == 3 (mod 22), where 10^(n-1)+21 is divisible by 11^2. - Franklin T. Adams-Watters, Jun 13 2006

Examples

			a(5) = 10021 = 11 * 911 is squarefree, internal digits 002 squarefree, external digits 11 squarefree.
a(6) = 100021 = 29 * 3449 has exactly those same properties.
a(7) = 1000021 = 11 * 90911 has exactly those same properties.
a(8) = 10000021 = 97 * 103093 has exactly those same properties.
a(9) = 100000021 = 11 * 19 * 23 * 71 * 293 is similar, though not semiprime.

Crossrefs

Cf. A077376, A077377, A077378, A077380.

Extensions

a(5)-a(12) from Jonathan Vos Post, Mar 22 2006
More terms from Franklin T. Adams-Watters, Jun 13 2006

A367801 Numbers that are both exponentially odd (A268335) and exponentially odious (A270428).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106
Offset: 1

Views

Author

Amiram Eldar, Dec 01 2023

Keywords

Comments

First differs from its subsequence A005117 at n = 79: a(79) = 128 is not a squarefree number.
First differs from A077377 at n = 63, and from A348506 at n = 68.
Numbers whose prime factorization contains only exponents that are odd odious numbers (A092246).
The asymptotic density of this sequence is Product_{p prime} f(1/p) = 0.61156148494581943994..., where f(x) = (1-x) * (1 + x/(2*(1-x^2)) + (Product_{k>=0} (1-(-x)^(2^k)) - Product_{k>=0} (1-x^(2^k))))/2.

Links

Crossrefs

Intersection of A268335 and A270428.
Cf. A367802, A367803, A367804.
Subsequences: A005117, A092759.
Cf. A092246.
Cf. A077377, A348506.

Programs

  • Mathematica
    odQ[n_] := OddQ[n] && OddQ[DigitCount[n, 2, 1]]; Select[Range[150], AllTrue[FactorInteger[#][[;;, 2]], odQ] &]
  • PARI
    is(n) = {my(f = factor(n)); for (i = 1, #f~, if(!(f[i, 2]%2 && hammingweight(f[i, 2])%2), return (0))); 1;}
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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