A244283 -id:A244283 - cp's OEIS frontend

A244282 Consider a prime number p with m decimal digits. The sequence lists the primes p such that the prefix of length m-1 and the suffix of length m-1 are both prime numbers.

23, 37, 53, 73, 113, 131, 137, 173, 179, 197, 311, 313, 317, 373, 379, 419, 431, 479, 613, 617, 619, 673, 719, 797, 971, 1013, 1019, 1031, 1097, 1277, 1373, 1499, 1571, 1733, 1811, 1997, 2113, 2239, 2293, 2719, 3079, 3137, 3313, 3373, 3491, 3499, 3593, 3673, 3677, 3733

Offset: 1

Michel Lagneau, Jun 25 2014

Let x(0)x(1)... x(q-1)x(q) denote the decimal expansion of a prime p. The sequence lists the primes p such that the prefix x(0)x(1)... x(q-1) and the suffix x(1)... x(q-1)x(q) are primes.

Superset of A051362; a(n) first differs from A051362 when n=12.

			The prime number 179 is in the sequence because 17 and 79 are primes.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A000040, A051362, A244283.

with(numtheory):
for m from 1 to 200 do:
      n:=ithprime(m):x:=convert(n, base, 10):n1:=nops(x):
      s1:=sum('x[i]*10^(i-1) ', 'i'=1..n1-1):
      s2:=(n-irem(n,10))/10:
      if type(s1,prime)=true and type(s2,prime)=true
      then
        printf(`%d, `, n):
        else
      fi:
od:

Select[Prime[Range[1000]], (id = IntegerDigits[#]; PrimeQ[FromDigits[Take[id, {1, -2}]]] && PrimeQ[FromDigits[Take[id, {2, -1}]]]) &] (* César Eliud Lozada, Mar 31 2024 *)
Select[Prime[Range[600]],With[{idp=IntegerDigits[#]},AllTrue[FromDigits/@{Rest[idp],Most[idp]},PrimeQ]]&] (* Harvey P. Dale, Feb 06 2025 *)

Inserted missing term a(49) and corrected a(50) by Paolo P. Lava, Dec 04 2017

A343855 Numbers whose second digit is not zero and such that removing either the first or last digit leaves a square number.

Original entry on oeis.org

11, 14, 19, 41, 44, 49, 91, 94, 99, 164, 364, 649, 816, 1441, 1961, 2256, 4841, 6256, 7841, 31369, 46241, 51849, 54761, 73969, 79216, 94096, 116641, 141616, 148841, 219044, 292416, 361009, 368644, 466564, 961009, 973441, 2580644, 3249001, 4651249, 6561001

Offset: 1

Andrew Howroyd, May 26 2021

The requirement that the second digit is not zero is so that both of the two squares have the same number of digits.
For k > 2, the number of k-digit terms is given by A344570(k-1).
All terms have last digit either 1, 4, 6, or 9. A term cannot have last digit 0 since that would mean one of the squares ends in an odd number of zeros and all squares end in an even number of zeros. A term cannot have last digit 5 since squares ending in 5 have 25 as last 2 digits and there are no squares having last digit 2. The last 2 digits of terms must be one of 01, 04, 09, 16, 41, 44, 49, 56, 61, 64, 69, 96. - Chai Wah Wu, May 27 2021

			14 is a term because both 1 and 4 are square numbers.
164 is a term because both 16 = 4^2 and 64 = 8^2 are square numbers.
1441 is a term because both 144 = 12^2 and 441 = 21^2 are square numbers.

Chai Wah Wu, Table of n, a(n) for n = 1..2348

Subsequence of A244283.

Cf. A344570.

sQ[n_] := IntegerQ@Sqrt[n];
selQ[n_] := With[{dd = IntegerDigits[n]}, If[dd[[2]] == 0 || FreeQ[dd[[-1]], 1|4|6|9], False, sQ[FromDigits[Rest[dd]]] && sQ[FromDigits[Most[dd]]]]];
Select[Range[11, 10^6], selQ] (* Jean-François Alcover, May 29 2021 *)

A244394 Consider a number n with m decimal digits, m>1. The sequence lists the numbers n such that the prefix of length m-1 and the suffix of length m-1 both have the same sum of divisors.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 335, 444, 534, 551, 555, 579, 589, 666, 682, 777, 888, 954, 956, 999, 1102, 1111, 2138, 2204, 2222, 2246, 2649, 3190, 3333, 4354, 4408, 4428, 4444, 5332, 5376, 5555, 5644, 5925, 6294, 6666, 6933, 7480, 7528

Offset: 1

Michel Lagneau, Jun 27 2014

Let x(0)x(1)... x(q-1)x(q) denote the decimal expansion of a number n. The sequence lists the numbers n such that sigma(p) = sigma(s) where p is the prefix x(0)x(1)... x(q-1) and s is the suffix x(1)... x(q-1)x(q).

			5376 is in the sequence because sigma(537) = sigma(376) = 720.

Cf. A244282, A244283.

with(numtheory):
for n from 10 to 10000 do:
      x:=convert(n, base, 10):n1:=nops(x):
      s1:=sum('x[i]*10^(i-1) ', 'i'=1..n1-1):
      s2:=(n-irem(n,10))/10:
      x1:=sigma(s1):x2:=sigma(s2):
      if x1 = x2
        then
        printf(`%d, `, n):
        else
      fi:
od:

cp's OEIS Frontend

A244282 Consider a prime number p with m decimal digits. The sequence lists the primes p such that the prefix of length m-1 and the suffix of length m-1 are both prime numbers.

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A343855 Numbers whose second digit is not zero and such that removing either the first or last digit leaves a square number.

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Author

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Comments

Examples

Links

Crossrefs

Programs

Mathematica

A244394 Consider a number n with m decimal digits, m>1. The sequence lists the numbers n such that the prefix of length m-1 and the suffix of length m-1 both have the same sum of divisors.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple