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 1-7 of 7 results.

A062332 Primes starting and ending with 1.

Original entry on oeis.org

11, 101, 131, 151, 181, 191, 1021, 1031, 1051, 1061, 1091, 1151, 1171, 1181, 1201, 1231, 1291, 1301, 1321, 1361, 1381, 1451, 1471, 1481, 1511, 1531, 1571, 1601, 1621, 1721, 1741, 1801, 1811, 1831, 1861, 1871, 1901, 1931, 1951, 10061, 10091, 10111, 10141
Offset: 1

Views

Author

Amarnath Murthy, Jun 21 2001

Keywords

Comments

Complement of A208261 (nonprime numbers with all divisors starting and ending with digit 1) with respect to A208262 (numbers with all divisors starting and ending with digit 1). - Jaroslav Krizek, Mar 04 2012
Intersection of A030430 and A045707. - Michel Marcus, Jun 08 2013

Examples

			102701 is a member as it is a prime and the first and the last digits are both 1.

Links

Crossrefs

Cf. A208259 (Numbers starting and ending with digit 1).
Cf. A010051, A017281, A131835, A000030.

Programs

  • Haskell
    a062332 n = a062332_list !! (n-1)
a062332_list = filter ((== 1) . a010051') a208259_list
-- Reinhard Zumkeller, Jul 16 2014
  • Mathematica
    fl1Q[n_]:=Module[{idn=IntegerDigits[n]},First[idn]==Last[idn]==1]; Select[ Prime[Range[1300]],fl1Q] (* Harvey P. Dale, Apr 30 2012 *)
  • PARI
    { n=-1; t=log(10); forprime (p=2, 5*10^5, if ((p-10*(p\10)) == 1 && (p\10^(log(p)\t)) == 1, write("b062332.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 05 2009

Formula

A010051(a(n)) * A000030(a(n)) * (a(n) mod 10) = 1. - Reinhard Zumkeller, Jul 16 2014

Extensions

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jun 29 2001
Missing term a(36)=1901 added by Harry J. Smith, Aug 05 2009

A227858 Numbers which start and end with the same digit in decimal.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494, 505, 515, 525, 535, 545, 555, 565, 575, 585, 595, 606, 616, 626
Offset: 1

Views

Author

Brian E. Foose and Todd R. Haldeman, Nov 01 2013

Keywords

Comments

Supersequence of A002113 and A208259.
The first term that is not a palindromic number is a(109) = 1011. - Alonso del Arte, Nov 01 2013
Primes in this sequence are A077652. - Jonathan Vos Post, Nov 02 2013
Differs from A110751 which contains for example 1089. - R. J. Mathar, Jun 29 2025

Links

Crossrefs

Cf. A002113, A077652, A208259.

Programs

  • Mathematica
    Select[Range[700],IntegerDigits[#][[1]]==IntegerDigits[#][[-1]]&] (* Harvey P. Dale, Nov 28 2019 *)
  • PARI
    is(n)=n=digits(n);n[1]==n[#n] \\ Charles R Greathouse IV, Nov 01 2013

Formula

a(n) = 10n + O(1). - Charles R Greathouse IV, Nov 01 2013

A208260 Nonprime numbers starting and ending with digit 1.

Original entry on oeis.org

1, 111, 121, 141, 161, 171, 1001, 1011, 1041, 1071, 1081, 1101, 1111, 1121, 1131, 1141, 1161, 1191, 1211, 1221, 1241, 1251, 1261, 1271, 1281, 1311, 1331, 1341, 1351, 1371, 1391, 1401, 1411, 1421, 1431, 1441, 1461, 1491, 1501, 1521, 1541, 1551, 1561, 1581, 1591
Offset: 1

Views

Author

Jaroslav Krizek, Feb 24 2012

Keywords

Comments

Complement of A062332 with respect to A208259. Supersequence of A208261 (nonprime numbers with all divisors starting and ending with digit 1).

Links

Crossrefs

Cf. A208259 (number starting and ending with a number 1), A062332 (primes starting and ending with a number 1).
Cf. A010051, A017281, A131835, A000030.

Programs

  • Haskell
    a208260 n = a208260_list !! (n-1)
a208260_list = filter ((== 0) . a010051') a208259_list
-- Reinhard Zumkeller, Jul 16 2014
  • Mathematica
    Select[Range[2000], ! PrimeQ[#] && First[IntegerDigits[#]] == 1 && Last[IntegerDigits[#]] == 1 &] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2012 *)
Join[{1},Select[Range[2000],CompositeQ[#]&&NumberDigit[#,0] == NumberDigit[ #,IntegerLength[ #]-1]==1&]] (* Harvey P. Dale, Aug 01 2021 *)

Formula

(1 - A010051(a(n))) * A000030(a(n)) * (a(n) mod 10) = 1. - Reinhard Zumkeller, Jul 16 2014

A208262 Numbers with all divisors starting and ending with digit 1.

Original entry on oeis.org

1, 11, 101, 121, 131, 151, 181, 191, 1021, 1031, 1051, 1061, 1091, 1111, 1151, 1171, 1181, 1201, 1231, 1291, 1301, 1321, 1331, 1361, 1381, 1441, 1451, 1471, 1481, 1511, 1531, 1571, 1601, 1621, 1661, 1721, 1741, 1801, 1811, 1831, 1861, 1871, 1901, 1931, 1951, 1991
Offset: 1

Views

Author

Jaroslav Krizek, Feb 24 2012

Keywords

Comments

Union of A062332 and A208261. Subsequence of A208259 (numbers starting and ending with digit 1).

Examples

			Divisors of 1111: 1, 11, 101, 1111.

Crossrefs

Cf. A062332 (primes starting and ending with digit 1), A208261 (nonprime numbers with all divisors starting and ending with digit 1).

Programs

  • Mathematica
    Select[Table[n, {n, 2000}], Union[First /@ IntegerDigits[Divisors[#]]] == {1} && Union[Last /@ IntegerDigits[Divisors[#]]] == {1} &] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2012 *)

A178318 Primes which remain prime after reflection across a vertical line through the middle of the number (numbers are written as digital clock style numerals).

Original entry on oeis.org

2, 5, 11, 101, 181, 1051, 1181, 1201, 1811, 10151, 11251, 11551, 12101, 12211, 12511, 15121, 18181, 100151, 100501, 101501, 101581, 102001, 102101, 102181, 102551, 105211, 105251, 108881, 110051, 110581, 110881, 111521, 111581, 115021, 115201
Offset: 1

Views

Author

David Nacin, May 24 2010

Keywords

Comments

Apart from first two terms: subsequence of A208259. - Reinhard Zumkeller, Jul 16 2014

Examples

			For example 1051 becomes 1201 under this reflection and since these are both prime, these number are part of the sequence. Note that a number must be composed only of the digits 0,1,2,5,8 to qualify.

Links

Crossrefs

Cf. A125308, A178317.
Cf. A010051, A004086, A208259, A062332.

Programs

  • Haskell
    import Data.List (intersect, genericIndex)
a178318 n = a178318_list !! (n-1)
a178318_list = 2 : 5 : filter f a062332_list where
   f p = null (show p `intersect` "34679") && a010051' (r 0 p) == 1
   r y 0 = y
   r y x = r (10 * y + genericIndex [0,1,5,0,0,2,0,0,8,0] d) x'
             where (x', d) = divMod x 10
-- Reinhard Zumkeller, Jul 16 2014

A270237 Primes which are the decimal concatenation of 1, a prime, and 1.

Original entry on oeis.org

131, 151, 1171, 1231, 1291, 1471, 1531, 1831, 11071, 11131, 11311, 11491, 11731, 11971, 12391, 12511, 13171, 13591, 13831, 14011, 14431, 15031, 15091, 15991, 16411, 16831, 17011, 17191, 17431, 17971, 18211, 19471, 19531, 110311, 110491, 111031, 111091, 111871
Offset: 1

Views

Author

Emre APARI, Mar 13 2016

Keywords

Comments

No padding 0's are allowed: 1021 is not a member even though 02 is prime.
These are the primes that result from A069687.
Subsequence of A208259.

Examples

			1171 is a prime which is the concatenation of 1, 17 and 1.

Links

Crossrefs

Cf. A000040, A069687, A208259.

Programs

  • Mathematica
    Select[Map[FromDigits@ Flatten@ {1, IntegerDigits@ Prime@ #, 1} &, Range@ 200], PrimeQ] (* Michael De Vlieger, Mar 15 2016 *)
Select[FromDigits[Flatten[IntegerDigits/@Join[{1},#,{1}]]]&/@Prime[ Range[ 200]], PrimeQ] (* Harvey P. Dale, Jul 26 2016 *)
  • PARI
    lista(nn) = forprime(p=2, nn, if(isprime(P=eval(Str(1, p, 1))), print1(P, ", "))); \\ Altug Alkan, Mar 13 2016
  • PARI
    is(n)=my(d=digits(n)); isprime(n) && d[1]==1 && #d>2 && d[2] && d[#d]==1 && isprime(fromdigits(d[2..#d-1])) \\ Charles R Greathouse IV, Mar 15 2016

A337184 Numbers divisible by their first digit and their last digit.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99, 101, 102, 104, 105, 111, 112, 115, 121, 122, 123, 124, 125, 126, 128, 131, 132, 135, 141, 142, 144, 145, 147, 151, 152, 153, 155, 156, 161, 162, 164, 165, 168, 171, 172, 175, 181, 182
Offset: 1

Views

Author

Bernard Schott, Jan 29 2021

Keywords

Comments

The first 23 terms are the same first 23 terms of A034838 then a(24) = 101 while A034838(24) = 111.
Terms of A034709 beginning with 1 and terms of A034837 ending with 1 are terms.
All positive repdigits (A010785) are terms.
There are infinitely many terms m for any of the 53 pairs (first digit, last digit) of m described below: when m begins with {1, 3, 7, 9} then m ends with any digit from 1 to 9; when m begins with {2, 4, 6, 8}, then m must also end with {2, 4, 6, 8}; to finish, when m begins with 5, m must only end with 5. - Metin Sariyar, Jan 29 2021

Links

Crossrefs

Intersection of A034709 and A034837.
Subsequences: A010785\{0}, A034838, A043037, A043040, A208259, A066622.
Cf. A139138.

Programs

  • Mathematica
    Select[Range[175], Mod[#, 10] > 0 && And @@ Divisible[#, IntegerDigits[#][[{1, -1}]]] &] (* Amiram Eldar, Jan 29 2021 *)
  • PARI
    is(n) = n%10>0 && n%(n%10)==0 && n % (n\10^logint(n,10)) == 0 \\ David A. Corneth, Jan 29 2021
  • Python
    def ok(n): s = str(n); return s[-1] != '0' and n%int(s[0])+n%int(s[-1]) == 0
print([m for m in range(180) if ok(m)]) # Michael S. Branicky, Jan 29 2021

Formula

(10n-9)/9 <= a(n) < 45n. (I believe the liminf of a(n)/n is 3.18... and the limsup is 6.18....) - Charles R Greathouse IV, Nov 26 2024
Conjecture: 3n < a(n) < 7n for n > 75. - Charles R Greathouse IV, Dec 02 2024
Showing 1-7 of 7 results.

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

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