A233555
Prime(m), where m is such that (Sum_{i=1..m} prime(i)^17) / m is an integer.
Original entry on oeis.org
2, 5724469, 10534369, 16784723, 33330911, 189781037, 8418091991, 58605633953, 109388266843, 448366797199, 1056238372873, 24603683667221, 86982253895059, 100316149840769, 164029709175817, 542295448805641, 685217940914237, 1701962315686097, 23064173255594491
Offset: 1
a(1) = 2, because 2 is the 1st prime and the sum of the first 1 primes^17 = 131072 when divided by 1 equals 131072 which is an integer.
Cf.
A085450 (smallest m > 1 that divide Sum_{k=1..m} prime(k)^n).
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t = {}; sm = 0; Do[sm = sm + Prime[n]^17; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
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is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^17); s==0 \\ Charles R Greathouse IV, Nov 30 2013
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S=n=0;forprime(p=1,,(S+=p^17)%n++||print1(p",")) \\ M. F. Hasler, Dec 01 2013
A233575
Prime(m), where m is such that (Sum_{i=1..m} prime(i)^18) / m is an integer.
Original entry on oeis.org
2, 157, 1697, 190573, 167719729, 22092660553, 57613776809, 4386989244577, 91982826261331, 13432259712845291
Offset: 1
a(2) = 157, because 157 is the 37th prime and the sum of the first 37 primes^18 = 7222759943091280921446062146835136523956 when divided by 37 equals 195209728191656241120163841806355041188 which is an integer.
Cf.
A085450 = smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.
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t = {}; sm = 0; Do[sm = sm + Prime[n]^18; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
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is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^18); s==0 \\ Charles R Greathouse IV, Nov 30 2013
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S=n=0;forprime(p=1,,(S+=p^18)%n++||print1(p",")) \\ - M. F. Hasler, Dec 01 2013
A233767
Prime(n), where n is such that (Sum_{i=1..n} prime(i)^19) / n is an integer.
Original entry on oeis.org
2, 97, 3203, 5059, 6469, 8081, 35051, 39719, 42209, 109049, 154591, 523297, 6621827, 20059771, 258196441, 731584957, 1427109029, 1899496631, 8428550519, 50790885203, 7475902096387, 22626378502139, 38855796912367, 162082298018497, 589085299527401, 4271778258271487
Offset: 1
97 is a term, because 97 is the 25th prime and the sum of the first 25 primes^19 = 71486619210134792705255313675343157050 when divided by 25 equals 2859464768405391708210212547013726282 which is an integer.
Cf.
A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
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t = {}; sm = 0; Do[sm = sm + Prime[n]^19; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
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is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^19); s==0 \\ Charles R Greathouse IV, Nov 30 2013
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my(S=n=0);forprime(p=1,,(S+=p^19)%n++||print1(p", ")) \\ M. F. Hasler, Dec 01 2013
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