A277225 Initial values of runs of 5 consecutive numbers all of which are squares, primes, or products of one prime and one square.
1, 16, 97, 241, 3302449, 123402049, 2163898897, 4168758049, 10062413521, 18006326449, 21376916449, 55487798449, 77248867921, 108396336049, 141466251697, 183357512449, 198504183697, 200642120449, 278584843921, 282955076449, 341261911249, 723039200449, 770057874097, 847823354449
Offset: 1
Keywords
Examples
For n=16, the five numbers in the group are 16 (square) 17 (prime) 18 (prime * square) 19 (prime) 20 (prime * square).
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 1..400
- Moscow Mathematical Olympiad, LXXVIII Moscow Mathematical Olympiad: Problems and Solutions, 2015 (in Russian), Question 4 for Grade 8 by D. M. Krekov (pages 5 and 15-16).
Programs
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PARI
sp(x)=local(c);c=core(x);isprime(c)||(c==1) for(n=1, 100000000000, if(vector(5, i, 1)==vector(5, i, sp(n+i-1)), print1(n, ", ")))
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PARI
has(n)=if(isprime(n), return(1)); forprime(p=2,97, if(n%p==0, my(e=valuation(n,p)); if(e%2, return(issquare(n/p)), n/=p^e))); issquare(n) || isprime(core(n)) is(n)=for(k=n,n+4,if(!has(k), return(0))); 1 print1(1); for(s=2,1e7, t=2*s^2; if(!has(t-1) || !has(t+1), next); for(n=t-3,t-1, if(is(n), print1(", "n)))) \\ Charles R Greathouse IV, Oct 14 2016
Extensions
a(7)-a(24) from Charles R Greathouse IV, Oct 14 2016
Comments