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A077217 Prime(k) such that the prime power with largest exponent that divides the product P(k) of composite numbers between prime(k) and prime(k+1) is an odd number, i.e., if p^r and 2^s divide P(k) then r >= s, p is an odd prime.

%I A077217 #16 Apr 01 2021 08:28:03
%S A077217 2,5,17,29,41,101,107,137,149,179,197,269,281,457,461,499,521,569,593,
%T A077217 617,641,673,727,809,821,827,857,881,1049,1061,1229,1277,1289,1301,
%U A077217 1321,1451,1453,1481,1483,1619,1697,1721,1753,1777,1861,1873,1877,1949,1997,2027
%N A077217 Prime(k) such that the prime power with largest exponent that divides the product P(k) of composite numbers between prime(k) and prime(k+1) is an odd number, i.e., if p^r and 2^s divide P(k) then r >= s, p is an odd prime.
%C A077217 In most cases a power of 2 has a larger exponent than any odd prime power.
%C A077217 Primes p = prime(k) such that A051903(A000265(A061214(k))) >= A007814(A061214(k)). - _Amiram Eldar_, Apr 01 2021
%H A077217 Amiram Eldar, <a href="/A077217/b077217.txt">Table of n, a(n) for n = 1..10000</a>
%e A077217 5 is a member as 6 is divisible by 3^1 as well as by 2^1.
%e A077217 17 is a member as 18 is divisible by 3^2 but not by 2^2.
%t A077217 q[p_] := Module[{prod = Product[k, {k, p + 1, NextPrime[p] - 1}], e2}, e2 = IntegerExponent[prod, 2]; Max[FactorInteger[prod/2^e2][[;; , 2]]] >= e2]; Select[Range[2000], PrimeQ[#] && q[#] &] (* _Amiram Eldar_, Apr 01 2021 *)
%o A077217 (PARI) f(p) = prod(k=p+1, nextprime(p+1)-1, k);\\ A061214
%o A077217 isok(p) = {my(prd = f(p), e = valuation(prd, 2), ofprd = prd/2^e); if (prd > 1, (ofprd == 1) || (e <= vecmax(factor(ofprd)[,2])));} \\ _Michel Marcus_, Apr 01 2021
%Y A077217 Cf. A000265, A007814, A051903, A061214.
%K A077217 nonn
%O A077217 1,1
%A A077217 _Amarnath Murthy_, Nov 02 2002
%E A077217 Wrong term removed and more terms added by _Amiram Eldar_, Apr 01 2021