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A277225 Initial values of runs of 5 consecutive numbers all of which are squares, primes, or products of one prime and one square. 2
1, 16, 97, 241, 3302449, 123402049, 2163898897, 4168758049, 10062413521, 18006326449, 21376916449, 55487798449, 77248867921, 108396336049, 141466251697, 183357512449, 198504183697, 200642120449, 278584843921, 282955076449, 341261911249, 723039200449, 770057874097, 847823354449 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The existence of runs of 6 or 7 numbers with the same property is an open question; a run of length 8 is impossible. - Andrey Zabolotskiy, Nov 07 2016
Numbers n such that each of the five consecutive integers k in n..n+4 has A162642(k)<2. - Jason Kimberley, Aug 27 2017
There is no n in this sequence below 10^10000 such that n+1 is also in this sequence. In other words, no n below 10^10000 such that each of the six consecutive integers n..n+5 has squarefree rank less than two. - Roger Eggleton and Jason Kimberley, Sep 02 2017
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).
EXAMPLE
For n=16, the five numbers in the group are 16 (square) 17 (prime) 18 (prime * square) 19 (prime) 20 (prime * square).
PROG
(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, ", ")))
(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
CROSSREFS
Sequence in context: A044267 A044648 A041488 * A265841 A248883 A223902
KEYWORD
nonn
AUTHOR
Charles Bowyer, Oct 14 2016
EXTENSIONS
a(7)-a(24) from Charles R Greathouse IV, Oct 14 2016
STATUS
approved

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Last modified June 24 23:28 EDT 2024. Contains 373691 sequences. (Running on oeis4.)