A260181 -id:A260181 - cp's OEIS frontend

A254751 Numbers such that, in base 10, all their proper prefixes and suffixes represent primes.

22, 23, 25, 27, 32, 33, 35, 37, 52, 53, 55, 57, 72, 73, 75, 77, 237, 297, 313, 317, 373, 537, 597, 713, 717, 737, 797, 2337, 2397, 2937, 3113, 3137, 3173, 3797, 5937, 5997, 7197, 7337, 7397, 29397, 31373, 37937, 59397, 73313, 739397

Offset: 1

Stanislav Sykora, Feb 15 2015

A proper prefix (or suffix) of a number m is one which is neither void, nor identical to m.
Alternative definition: Slicing the decimal expansion of a(n) in any way into two nonempty parts, each part represents a prime number.
Every proper prefix of each member a(n) is a member of A024770, and every proper suffix is a member of A024785. Since these are finite sequences, a(n) is also finite. It has 45 members, the largest of which is 739397 and happens to be a prime.
The sequence is a union of A254753 and A020994.
A subsequence of A260181. - M. F. Hasler, Sep 16 2016

			6 is not a member because its expansion cannot be sliced in two.
597 is a member because (5,97,59, and 7) are all primes.
2331 is excluded because 233 is prime, but 1 is not. - _Gordon Hamilton_, Feb 20 2015

Cf. A020994, A024770, A024785, A254750, A254752, A254753, A254754, A254756.

Cf. A260181.

fQ[n_] := (p = {2, 3, 5, 7}; If[ Union@ Join[p, {Mod[n, 10]}] != p, {False}, Block[{idn = IntegerDigits@ n, lng = Floor@ Log10@ n}, Union@ PrimeQ@ Flatten@ Table[{FromDigits[ Take[idn, i]], FromDigits[ Take[idn, -lng + i - 1]]}, {i, lng}] == {True}]]); Select[ Range@1000000, fQ] (* Robert G. Wilson v, Feb 21 2015 *)
Select[Range[10,750000],AllTrue[Flatten[Table[FromDigits/@TakeDrop[IntegerDigits[#],n],{n,IntegerLength[#]-1}]],PrimeQ]&] (* Harvey P. Dale, Feb 13 2024 *)

slicesIntoPrimes(n,b=10) = {my(k=b);if(n0,if(!isprime(n\k)||!isprime(n%k),return(0););k*=b;);return(1);}

def breakIntoPrimes(n):
    D=n.digits()
    for i in [1..len(D)-1]:
        if not(is_prime(sum(D[i:][j]*10^j for j in range(len(D[i:])))) and is_prime(sum(D[:i][j]*10^j for j in range(len(D[:i]))))):
            return False
        else:
            continue
    return True
[n for n in [10..1000] if breakIntoPrimes(n)] # Tom Edgar, Feb 20 2015

A197652 Numbers that are congruent to 0 or 1 mod 10.

Original entry on oeis.org

0, 1, 10, 11, 20, 21, 30, 31, 40, 41, 50, 51, 60, 61, 70, 71, 80, 81, 90, 91, 100, 101, 110, 111, 120, 121, 130, 131, 140, 141, 150, 151, 160, 161, 170, 171, 180, 181, 190, 191, 200, 201, 210, 211, 220, 221, 230, 231, 240, 241, 250, 251, 260, 261, 270, 271

Offset: 1

Philippe Deléham, Oct 16 2011

From Wesley Ivan Hurt, Sep 26 2015: (Start)
Numbers with last digit 0 or 1.
Complement of (A260181 Union A262389). (End)
Numbers k such that floor(k/2) = 5*floor(k/10). - Bruno Berselli, Oct 05 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A020714, A030308, A260181, A262389.

[5*n-7-2*(-1)^n: n in [1..60]]; // Vincenzo Librandi, Jul 11 2012

A197652:=n->5*n-7-2*(-1)^n: seq(A197652(n), n=1..100); # Wesley Ivan Hurt, Sep 26 2015

CoefficientList[Series[x*(1+9*x)/((1+x)*(1-x)^2),{x,0,50}],x] (* Vincenzo Librandi, Jul 11 2012 *)

a(n)=n\2*10+n%2*9-9 \\ Charles R Greathouse IV, Oct 25 2011

def A197652(n): return 5*n-(5 if n&1 else 9) # Chai Wah Wu, Oct 29 2024

a(n) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=1 and b(k) = 5*2^k = A020714(k) for k>0.
From Zak Seidov, Oct 20 2011: (Start)
a(n) = a(n-2) + 10.
a(n) = 5*n - 7 - 2*(-1)^n. (End)
From Vincenzo Librandi, Jul 11 2012: (Start)
G.f.: x^2*(1+9*x)/((1+x)*(1-x)^2).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3. (End)
E.g.f.: 9 + (5*x - 7)*exp(x) - 2*exp(-x). - David Lovler, Sep 03 2022

A226099 Positive integers that yield a prime when their most significant (i.e., leftmost) decimal digit is removed.

Original entry on oeis.org

12, 13, 15, 17, 22, 23, 25, 27, 32, 33, 35, 37, 42, 43, 45, 47, 52, 53, 55, 57, 62, 63, 65, 67, 72, 73, 75, 77, 82, 83, 85, 87, 92, 93, 95, 97, 102, 103, 105, 107, 111, 113, 117, 119, 123, 129, 131, 137, 141, 143, 147, 153, 159, 161, 167, 171, 173, 179, 183, 189, 197, 202, 203, 205, 207, 211, 213, 217

Offset: 1

Jonathan Vos Post, May 26 2013

Terms < 110 are the same as in A260181, numbers whose last digit is prime. - M. F. Hasler, Dec 20 2019

These are numbers with decimal expansion of the form k = xp where p is a prime and x is a single digit. Whether or not the number k itself is a prime is irrelevant. - N. J. A. Sloane, Dec 21 2019

			a(1) = 12 because when its most significant (or leftmost) digit (1) is removed, the remaining number 2 is prime, and it is the least such number.
102, 103, 105 and 107 are in the sequence because if the first digit is dropped, what is left is a 1-digit prime with a leading digit '0'.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040, A217657 (n without initial digit), A000030 (initial digit of n), A260181 (last digit is prime), A202262 (substrings are composite).

[k:k in [1..220]| IsPrime( k-Reverse(Intseq(k))[1]*10^(#Intseq(k)-1 ))]; // Marius A. Burtea, Dec 21 2019

Select[Range@ 300, PrimeQ@ FromDigits@ Rest@ IntegerDigits@ # &] (* Giovanni Resta, Dec 20 2019 *)

select( is(n)=isprime(n%10^logint(n+!n,10)), [0..222]) \\ M. F. Hasler, Dec 20 2019

From M. F. Hasler, Dec 21 2019: (Start)
n in A226099 (this sequence) <=> A217657(n) in A000040 (prime).
a(n) = a(n-4) + 10 for 4 < n < 41, i.e., 20 < a(n) < 110; a(n) = a(n-25) for 61 < n < 287, i.e., 200 < a(n) < 1100, etc. (End)

A262389 Numbers whose last digit is composite.

Original entry on oeis.org

4, 6, 8, 9, 14, 16, 18, 19, 24, 26, 28, 29, 34, 36, 38, 39, 44, 46, 48, 49, 54, 56, 58, 59, 64, 66, 68, 69, 74, 76, 78, 79, 84, 86, 88, 89, 94, 96, 98, 99, 104, 106, 108, 109, 114, 116, 118, 119, 124, 126, 128, 129, 134, 136, 138, 139, 144, 146, 148, 149

Offset: 1

Wesley Ivan Hurt, Sep 21 2015

Numbers ending in 4, 6, 8 or 9.
Union of A017317, A017341, A017365 and A017377.
Subsequence of A118951 (numbers containing at least one composite digit).
Complement of (A197652 Union A260181).

Gerald Hillier and Didier Lachieze, Last Digit Composite.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A017317, A017341, A017365, A017377.

Cf. A118951, A197652, A260181 (last digit is prime).

[(5*n+1-(-1)^n+(3+(-1)^n)*(-1)^((2*n-3-(-1)^n) div 4) div 2) div 2: n in [1..70]]; // Vincenzo Librandi, Sep 21 2015

A262389:=n->(5*n+1-(-1)^n+(3+(-1)^n)*(-1)^((2*n-3-(-1)^n)/4)/2)/2: seq(A262389(n), n=1..100);

Table[(5n+1-(-1)^n+(3+(-1)^n)*(-1)^((2n-3-(-1)^n)/4)/2)/2, {n, 100}]
LinearRecurrence[{1, 0, 0, 1, -1}, {4, 6, 8, 9, 14}, 80] (* Vincenzo Librandi, Sep 21 2015 *)
CoefficientList[Series[(4 + 2*x + 2*x^2 + x^3 + x^4)/((x - 1)^2*(1 + x + x^2 + x^3)), {x, 0, 80}], x] (* Wesley Ivan Hurt, Sep 21 2015 *)
Select[Range[200],CompositeQ[Mod[#,10]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 21 2019 *)

G.f.: x*(4+2*x+2*x^2+x^3+x^4)/((x-1)^2*(1+x+x^2+x^3)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (5*n+1-(-1)^n+(3+(-1)^n)*(-1)^((2*n-3-(-1)^n)/4)/2)/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(10-2*sqrt(5))*Pi - sqrt(5)*arccoth(3/sqrt(5)) - 4*log(2))/20. - Amiram Eldar, Jul 30 2024

Name edited by Jon E. Schoenfield, Feb 15 2018

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A254751 Numbers such that, in base 10, all their proper prefixes and suffixes represent primes.

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A197652 Numbers that are congruent to 0 or 1 mod 10.

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A226099 Positive integers that yield a prime when their most significant (i.e., leftmost) decimal digit is removed.

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A262389 Numbers whose last digit is composite.

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