

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.


1



2, 5, 17, 29, 41, 101, 107, 137, 149, 179, 197, 269, 281, 457, 461, 499, 521, 569, 593, 617, 641, 673, 727, 809, 821, 827, 857, 881, 1049, 1061, 1229, 1277, 1289, 1301, 1321, 1451, 1453, 1481, 1483, 1619, 1697, 1721, 1753, 1777, 1861, 1873, 1877, 1949, 1997, 2027
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OFFSET

1,1


COMMENTS

In most cases a power of 2 has a larger exponent than any odd prime power.


LINKS



EXAMPLE

5 is a member as 6 is divisible by 3^1 as well as by 2^1.
17 is a member as 18 is divisible by 3^2 but not by 2^2.


MATHEMATICA

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 *)


PROG

(PARI) f(p) = prod(k=p+1, nextprime(p+1)1, k); \\ A061214
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


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS

Wrong term removed and more terms added by Amiram Eldar, Apr 01 2021


STATUS

approved



