A064397 -id:A064397 - cp's OEIS frontend

A230327 Index of smallest prime such that the sum of n consecutive primes starting with this specific prime is a square.

7, 6, 3, 42, 107, 6, 38, 1, 1631, 170, 38, 119, 5, 546, 78, 309, 85, 604, 199, 57, 270, 2, 3, 333, 45, 2, 178, 1708, 291, 2, 35, 72, 322, 19, 84, 5, 155, 346, 122, 2175, 1395, 24, 886, 2, 3108, 168, 14, 499, 340, 294, 156, 578, 325, 240, 115, 61, 283, 1035

Offset: 2

Michel Marcus, Oct 16 2013

nonn

			a(2)=7 because 17+19 (2 terms) = 36 is a square, 17 being the 7th prime.
a(3)=6 because 13+17+19 (3 terms) =49 is a square, 13 being the 6th prime.

Cf. A064397 (2 primes), A076305 (3 primes), A072849 (4 primes), A166255 (70 primes), A166261 (120 primes).

Cf. A132955 (primes themselves), A132956 (squares=sums), A132957 (square roots of sums).

a(n, lim=10^5) = {n --; pr = primes(lim); for (i = 1, lim-n, s = sum(k=i, i+n, pr[k]); if (issquare(s), return (i));); return (0);} \\ Michel Marcus, Oct 16 2013

A358156 a(n) is the smallest number k such that the sum of k consecutive prime numbers starting with the n-th prime is a square.

Original entry on oeis.org

9, 23, 4, 1862, 14, 3, 2, 211, 331, 163, 366, 3, 124, 48, 2, 449, 8403, 121, 35, 2, 4, 105, 77, 43, 190769, 1726, 234, 248, 200, 295, 293, 73, 4, 873, 32, 64, 2456139382, 8, 4519, 14, 123, 5, 9395, 296, 26, 5, 3479, 810, 9, 7091, 1669, 157, 1189, 12559, 269, 4930, 21, 376, 3

Offset: 1

Todor Szimeonov, Nov 01 2022

nonn

a(60) > 10^10 and a(68) > 10^13. - Martin Ehrenstein, Nov 09 2022

			For n=7, prime(7) = 17 and starting there 2 primes 17 + 19 = 36 which is square, so that a(7)=2.

Todor Szimeonov, Square of prime numbers

Cf. A000040, A000290, A105720, A230327 (exchanges the roles of n, k), A287027 (squares reached).

Indices of terms: A064397 (2's), A076305 (3's), A072849 (4's), A166255 (70's), A166261 (120's).

f:= proc(n) local p,s,k;
  p:= ithprime(n); s:= p;
  for k from 2 do
    p:= nextprime(p);
    s:= s+p;
    if issqr(s) then return k fi
  od
end proc:
map(f, [$1..36]); # Robert Israel, Nov 08 2022

a[n_] := Module[{p = s = Prime[n], k = 1}, While[! IntegerQ[Sqrt[s]], p = NextPrime[p]; s += p; k++]; k]; Array[a, 36] (* Amiram Eldar, Nov 08 2022 *)

a(25)-a(36) from Robert Israel, Nov 08 2022

a(37)-a(59) from Martin Ehrenstein, Nov 09 2022

A090346 Number of divisors of prime(n) + prime(n+1).

Original entry on oeis.org

2, 4, 6, 6, 8, 8, 9, 8, 6, 12, 6, 8, 12, 12, 9, 10, 16, 8, 8, 15, 8, 10, 6, 8, 12, 12, 16, 16, 8, 20, 8, 6, 12, 18, 18, 12, 14, 16, 12, 12, 24, 12, 16, 16, 18, 8, 8, 18, 16, 16, 8, 24, 12, 6, 16, 12, 24, 6, 12, 12, 21, 24, 8, 20, 24, 20, 6, 18, 16, 16, 8, 12, 12, 10, 8, 6, 8, 16, 20, 18

Offset: 1

Joseph L. Pe, Jan 28 2004

nonn

Positions of odd terms (A064397): 7,15,20,61,152,190,293,377,492,558,789,919,942,1768,2343,2429,... [From Zak Seidov, Oct 12 2010]

d[i_] := DivisorSigma[0, Prime[i] + Prime[i + 1]]; Table[d[i], {i, 1, 100}]
DivisorSigma[0,#]&/@(Total/@Partition[Prime[Range[90]],2,1]) (* Harvey P. Dale, Mar 03 2023 *)

A132746 Numbers k such that prime(k) + prime(k+1) is a perfect power.

Original entry on oeis.org

2, 7, 15, 18, 20, 28, 61, 152, 190, 293, 377, 492, 558, 564, 789, 919, 942, 1332, 1768, 2343, 2429, 2693, 2952, 3136, 3720, 3928, 4837, 5421, 5722, 6870, 7347, 8126, 8193, 9465, 9857, 9927, 10410, 10483, 10653, 12685, 13005, 13763, 13955, 16033, 16342

Offset: 1

Zak Seidov, Nov 17 2007

nonn

First terms absent in A064397: 2, 18, 28, 564, 1332, 3928, 12415, 13005, 16886.

			2 is a term because prime(2) + prime(3) = 3 + 5 = 8 = 2^3 (perfect power);
7 is a term because prime(7) + prime(8) = 17 + 19 = 36 = 6^2 (perfect power);
39867 is a term because prime(39867) + prime(39868) = 478241 + 478243 = 956484 = 978^2 (perfect power).

Cf. A064397 (numbers k such that prime(k) + prime(k+1) is a square).

Cf. A001043, A001597.

Select[Range[16342],ResourceFunction["PerfectPowerQ"][Prime[#]+Prime[#+1]]&] (* James C. McMahon, Mar 08 2025 *)

s=[];for(n=1,41530,a=prime(n)+prime(n+1);if(ispower(a),s=concat(s,n)));s

a(n) >> n^2/log^2 n. - Charles R Greathouse IV, Mar 08 2025

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A230327 Index of smallest prime such that the sum of n consecutive primes starting with this specific prime is a square.

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A358156 a(n) is the smallest number k such that the sum of k consecutive prime numbers starting with the n-th prime is a square.

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A090346 Number of divisors of prime(n) + prime(n+1).

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Mathematica

A132746 Numbers k such that prime(k) + prime(k+1) is a perfect power.

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