Search: a096915 -id:a096915
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1, 3, 14, 71, 76, 131, 188, 196, 232, 314, 535, 695, 1451, 2474, 3868, 7717, 41284, 52462, 90760, 119008, 264433, 487534, 618691, 935477, 959456, 3232220, 5149055, 6734713, 24668330, 92436217, 144794399, 603533275, 927592756, 969468212, 1182908572
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OFFSET
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1,2
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LINKS
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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3, 7, 23, 29, 43, 47, 59, 61, 79, 89, 97, 131, 179, 239, 283, 313, 367, 373, 379, 541, 577, 601, 607, 617, 857, 911, 953, 1063, 1301, 1321, 1499, 1783, 1867, 1913, 1933
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OFFSET
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1,1
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LINKS
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A088606
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Smallest number k such that concatenation of k and prime(n) is a prime, or 0 if no other number exists. a(1) = a(3) = 0.
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+10
3
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0, 1, 0, 1, 2, 1, 3, 4, 2, 2, 1, 1, 2, 4, 3, 3, 3, 4, 1, 2, 1, 1, 2, 3, 1, 5, 1, 5, 1, 2, 4, 2, 2, 4, 11, 1, 4, 1, 3, 6, 2, 1, 3, 1, 5, 6, 4, 1, 5, 1, 5, 2, 4, 2, 3, 6, 2, 3, 1, 2, 1, 2, 1, 2, 3, 6, 3, 4, 2, 4, 6, 3, 1, 1, 6, 2, 2, 4, 12, 1, 5, 4, 5, 1, 1, 5, 3, 3, 3, 3, 2, 5, 1, 3, 1, 2, 17, 2, 1, 3, 3, 2, 5, 5
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OFFSET
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1,5
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COMMENTS
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Subsidiary sequences: (set(1)) Index of the start of the first occurrence of a string of n consecutive 1's or 2's or 3's etc. (set (2)): a(n) = smallest prime such that concatenation of 1 with n successive primes starting from a(n) gives primes in each case. (n primes are obtained.) Similarly for 2, 3, etc. Conjecture: The subsidiary sequences are infinite.
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LINKS
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PROG
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(PARI) a(n) = if ((n==1) || (n==3), 0, my(k=1); while (!isprime(eval(Str(k, prime(n)))), k++); k); \\ Michel Marcus, Jul 11 2021
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A089777
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a(n) = smallest prime of the form n followed by a prime.
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+10
2
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13, 23, 37, 43, 53, 67, 73, 83, 97, 103, 113, 127, 137, 1423, 157, 163, 173, 1811, 193, 2011, 2111, 223, 233, 2411, 257, 263, 277, 283, 293, 307, 313, 3217, 337, 347, 353, 367, 373, 383, 397, 4013, 4111, 4211, 433, 443, 457, 463, 4723, 487, 4919, 503, 5113
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OFFSET
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1,1
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COMMENTS
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Open problem(?): show that a(n) always exists.
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LINKS
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MAPLE
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cat2 := proc(a, b) local dgs ; dgs := max(1, ilog10(b)+1) ; a*10^dgs+b ; end: A089777 := proc(k) local i, p, q ; for i from 1 do p := ithprime(i) ; q := cat2(k, p) ; if isprime(q) then RETURN(q) ; fi; od: end: for k from 1 to 80 do printf("%d, ", A089777(k)) ; od: # R. J. Mathar, Jan 05 2009
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MATHEMATICA
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Table[k=2; While[p=FromDigits[Join[IntegerDigits[n], IntegerDigits[Prime[k]]]]; !PrimeQ[p], k++ ]; p, {n, 100}] (* T. D. Noe, Jan 06 2009 *)
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CROSSREFS
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KEYWORD
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base,easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A103835
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Smallest prime p, larger than previous term, such that concatenation of n and p is a prime.
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+10
1
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3, 11, 13, 19, 23, 31, 43, 53, 67, 97, 113, 149, 151, 173, 193, 223, 239, 251, 373, 389, 397, 409, 431, 439, 457, 479, 487, 499, 569, 577, 601, 647, 739, 757, 797, 809, 811, 821, 827, 829, 863, 929, 991, 1109, 1181, 1297, 1301, 1303, 1327, 1367, 1409, 1429
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OFFSET
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1,1
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COMMENTS
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LINKS
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EXAMPLE
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a(10)=97 because 1097 is prime, while 1071,1073,1079,1083,1089 are all composite.
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PROG
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(Python)
from sympy import isprime, nextprime
def ispal(n): s = str(n); return s == s[::-1]
def aupto(lim):
n, p, alst = 1, 2, []
while p <= lim:
if isprime(int(str(n)+str(p))): n, alst = n + 1, alst + [p]
p = nextprime(p)
return alst
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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A103836
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Smallest palindromic prime p, larger than previous term, such that concatenation of n and p is a prime.
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+10
1
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3, 11, 181, 373, 12821, 14741, 32323, 72227, 74747, 77977, 78887, 79997, 90709, 94049, 94849, 98689, 1055501, 1065601, 1114111, 1129211, 1134311, 1177711, 1180811, 1186811, 1190911, 1262621, 1333331, 1338331, 1407041, 1409041, 1411141, 1461641, 1463641
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OFFSET
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1,1
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COMMENTS
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LINKS
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EXAMPLE
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a(4) = 373 because 4373 is prime, while 4191, 4313, 4353 are all composite.
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PROG
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(Python)
from sympy import isprime, nextprime
def ispal(n): s = str(n); return s == s[::-1]
def aupto(lim):
n, p, alst = 1, 2, []
while p <= lim:
if ispal(p) and isprime(int(str(n)+str(p))): n, alst = n + 1, alst + [p]
p = nextprime(p)
return alst
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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A131757
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Period 10: repeat 3, 3, 3, -7, 3, 3, -7, 3, 3, -7.
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+10
0
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3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3, 3, 3, -7, 3, 3, -7, 3, 3, -7, 3
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OFFSET
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0,1
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LINKS
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MATHEMATICA
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PadRight[{}, 120, {3, 3, 3, -7, 3, 3, -7, 3, 3, -7}] (* Harvey P. Dale, Aug 07 2018 *)
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PROG
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(PARI) a(n)=[3, 3, 3, -7, 3, 3, -7, 3, 3, -7][n%10+1] \\ Charles R Greathouse IV, Jul 13 2016
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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