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A225873 Squares that become prime when their most significant (or leftmost) digit is removed. 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

See A249589 for the square roots. - M. F. Hasler, Nov 02 2014

LINKS

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]

EXAMPLE

2401 = 49^2 becomes the prime number 401 when 2 is removed. 5041 = 71^2 becomes the prime number 41 when 5 is removed.

MATHEMATICA

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 *)

PROG

(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)

CROSSREFS

Cf. A225885.

Sequence in context: A125413 A160084 A265967 * A029987 A017582 A210430

Adjacent sequences:  A225870 A225871 A225872 * A225874 A225875 A225876

KEYWORD

nonn,base

AUTHOR

Kevin L. Schwartz and Christian N. K. Anderson, May 19 2013

EXTENSIONS

Extended by Davin Park, Dec 30 2016

STATUS

approved

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Last modified May 20 03:10 EDT 2019. Contains 323412 sequences. (Running on oeis4.)