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A225873
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Squares that become prime when their most significant (or leftmost) digit is removed.
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12
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25, 289, 361, 441, 529, 729, 841, 961, 1089, 1521, 2401, 2601, 2809, 4761, 5041, 5929, 6241, 7569, 8281, 9409, 20449, 21609, 22801, 24649, 25281, 26569, 29241, 29929, 34969, 36481, 39601, 40401, 52441, 53361, 54289, 57121, 58081, 59049, 61009, 63001, 71289
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OFFSET
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1,1
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COMMENTS
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a(1)=25 is the only term in the sequence that ends in 5. Proof: Any number ending in 5 is divisible by 5, and no square ending in 5 can have all 0 internal digits. Let N=A+B where A=N-5 and B=5. Then N^2 = A^2 + 2AB + B^2. B^2 is 25, and because A ends in a zero, A^2 and 2AB ends in two zeros; therefore the sum ends in 25.
All other terms end in 1 or 9, because no square ends in 3 or 7.
Observation: The sequence often experiences large gaps when the most-significant digit is square. For example, there is a gap of over 10^8 between a(764)=99420841 and a(765)=200307409, and over 10^9 between a(9156)=39980402401 and a(9157)=50000984881.
These gaps occur because if n^2 = (10^k*d+r)^2 = 10^(2k)d^2+r*(2*10^k+r) with d=1, 2, or 3 and r small enough so that the first digit of n^2 is d^2, then removing that digit d^2 we are left with r*(2*10^k+r) which is divisible by r and thus cannot be prime if r>1. - Giovanni Resta, May 23 2013
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LINKS
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Christian N. K. Anderson and Davin Park, Table of n, a(n) for n = 1..20000 [Terms 1 through 10000 were computed by Christian N. K. Anderson and terms 10001 through 20000 were computed by Davin Park]
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EXAMPLE
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2401 = 49^2 becomes the prime number 401 when 2 is removed. 5041 = 71^2 becomes the prime number 41 when 5 is removed.
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MATHEMATICA
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b^2 /. Flatten[Outer[Solve[a + #2*10^#1 == b^2 && 0 <= a < 10^#1 && Sqrt[#2*10^#1] <= b < Sqrt[10^(#1 + 1)] && a \[Element] Primes, {a, b}, Integers] &, Range[0, 10], Range[9]], 2] (* Davin Park, Dec 30 2016 *)
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PROG
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(R)no0<-function(s){ while(substr(s, 1, 1)=="0" & nchar(s)>1) s=substr(s, 2, nchar(s)); s};
issquare<-function(x) ifelse(as.bigz(x)<2, T, all(table(as.numeric(gmp::factorize(x)))%%2==0));
which(sapply(1:200, function(x) isprime(no0(substr(x^2, 2, ndig(x^2)))))>0)^2
(PARI) is_A225873(n)=isprime(n%10^(#Str(n)-1))&&issquare(n)
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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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EXTENSIONS
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STATUS
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approved
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