A077380 -id:A077380 - cp's OEIS frontend

A077377 Squarefree numbers whose external digits form a squarefree number.

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, 103, 105, 107, 109, 110, 111, 113

Offset: 1

Amarnath Murthy, Nov 06 2002

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A005117, A077376, A077378, A077379, A077380.

A077377Q[k_] := SquareFreeQ[k] && (k < 10 || SquareFreeQ[FromDigits[IntegerDigits[k][[{1, -1}]]]]);
Select[Range[200], A077377Q] (* Paolo Xausa, Feb 19 2025 *)

list(lim)=my(v=List(),n,t); for(d=0,logint(lim\=1,10), t=10^d; forsquarefree(k=t,min(10*t-1,lim), if(issquarefree(k[1]\t*10 + k[1]%10), listput(v,k[1])))); Vec(v) \\ Charles R Greathouse IV, Feb 14 2018

a(1) inserted by Charles R Greathouse IV, Feb 14 2018

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

Amarnath Murthy, Nov 06 2002

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

Paolo Xausa, Table of n, a(n) for n = 1..10000

Intersection of A077376 and A077377.

Cf. A005117, A077379, A077380.

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 *)

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

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

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

Amarnath Murthy, Nov 06 2002

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

Cf. A077377, A077378, A077379, A077380.

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 *)

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

Amarnath Murthy, Nov 06 2002

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

			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.

Cf. A077376, A077377, A077378, A077380.

a(5)-a(12) from Jonathan Vos Post, Mar 22 2006

More terms from Franklin T. Adams-Watters, Jun 13 2006

cp's OEIS Frontend

A077377 Squarefree numbers whose external digits form a squarefree number.

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

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PARI

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A077376 Squarefree numbers whose internal digits form a squarefree number.

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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.

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