A077389 -id:A077389 - cp's OEIS frontend

A082596 a(n) is the number of values of k such that A077389(k) = A060863(n).

1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Naohiro Nomoto, May 14 2003

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A060863, A077389, A082653.

A082592 A077389 sorted and duplicates removed.

Original entry on oeis.org

2, 4, 5, 9, 12, 17, 23, 29, 30, 31, 37, 38, 39, 49, 51, 52, 56, 70, 71, 79, 89, 95, 100, 105, 110, 111, 113, 124, 125, 133, 137, 139, 149, 151, 153, 155, 157, 169, 177, 179, 185, 187, 199, 200, 201, 214, 227, 230, 242, 251, 252, 261, 272, 273, 274, 275, 280, 281

Offset: 1

Naohiro Nomoto, May 13 2003

Cf. A060863.

A054892 Smallest prime a(n) such that the sum of n consecutive primes starting with a(n) is divisible by n.

Original entry on oeis.org

2, 3, 3, 5, 71, 5, 7, 17, 239, 13, 29, 5, 43, 23, 5, 5, 7, 7, 79, 17, 47, 11, 2, 73, 97, 53, 271, 13, 263, 23, 41, 61, 97, 101, 181, 41, 47, 13, 233, 13, 53, 13, 359, 151, 71, 61, 239, 73, 443, 859, 29, 131, 2, 61, 313, 101, 19, 151, 521, 3, 571, 31, 7, 79, 109, 97, 53

Offset: 1

Labos Elemer, May 23 2000

nonn

See A132809 for another version.

In some cases (n=1,2,25,..), like a(25)=97, the sum of 25 consecutive primes starts with the 25th prime and is divided by 25: Sum=97+...+227=3925=25*157

			a(8) = 17 since the sum of the 8 consecutive primes starting with 17 is 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 = 240, which is divisible by 8.  No prime less than 17 has this property: for example, 7 + 11 + ... + 31 = 150 which is not divisible by 8.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A054643, A024672, A034961, A077388, A077389, A122820, A132809.

f[n_] := Block[{k = 1, t}, While[t = Table[Prime[i], {i, k, k + n - 1}]; Mod[Plus @@ t, n] > 0, k++ ]; t]; First /@ Table[f[n], {n, 67}] (* Ray Chandler, Oct 09 2006 *)
Module[{prs=Prime[Range[250]]},Table[SelectFirst[Partition[prs,n,1],Mod[Total[#],n]==0&],{n,70}]][[;;,1]] (* Harvey P. Dale, Jul 11 2023 *)

a(n) = min{q_1 | Sum_{i=1..n} q_i = n*X}, q_i is a prime (rarely only a(n) = prime(n)).

A077388 Row sums of the triangle in A122820.

Original entry on oeis.org

2, 8, 15, 36, 395, 72, 119, 240, 2349, 300, 539, 276, 923, 714, 435, 496, 629, 702, 2375, 1120, 1995, 1144, 874, 3192, 3925, 2938, 9531, 1960, 10063, 2670, 3441, 4448, 5907, 6358, 9835, 4464, 5069, 3610, 13533, 4000, 6273, 4410, 21113, 12012, 8325

Offset: 1

Amarnath Murthy, Nov 06 2002

nonn

19318176 = A077388(1296) = A077388(2088). Are there any other pairs?. - Naohiro Nomoto, May 17 2003

Cf. A054892, A077389, A082638, A122820.

f[n_] := Block[{k = 1, t},While[t = Table[Prime[i], {i, k, k + n - 1}]; Mod[Plus @@ t, n] > 0, k++ ];t];Total /@ Table[f[n], {n, 45}] (* Ray Chandler, Oct 09 2006 *)

More terms from Sascha Kurz, Jan 30 2003

Name corrected by Sean A. Irvine, May 18 2025

A122820 Array read by rows in which n-th row contains n successive primes with least sum divisible by n.

Original entry on oeis.org

2, 3, 5, 3, 5, 7, 5, 7, 11, 13, 71, 73, 79, 83, 89, 5, 7, 11, 13, 17, 19, 7, 11, 13, 17, 19, 23, 29, 17, 19, 23, 29, 31, 37, 41, 43, 239, 241, 251, 257, 263, 269, 271, 277, 281, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 5, 7, 11, 13, 17

Offset: 1

Ray Chandler, Sep 27 2006

			Triangle begins:
2
3 5
3 5 7
5 7 11 13
71 73 79 83 89
5 7 11 13 17 19
7 11 13 17 19 23 29
17 19 23 29 31 37 41 43
239 241 251 257 263 269 271 277 281
13 17 19 23 29 31 37 41 43 47
29 31 37 41 43 47 53 59 61 67 71
5 7 11 13 17 19 23 29 31 37 41 43

Cf. A054892 (first term of each row), A077388 (row sum), A077389 (row average).

Cf. A082592, A082596, A082638, A082653, A082661.

f[n_] := Block[{k = 1, t},While[t = Table[Prime[i], {i, k, k + n - 1}]; Mod[Plus @@ t, n] > 0, k++ ];t];Flatten[Table[f[n], {n, 12}]]

A082653 For smallest numbers k such that A082596(k) = n, sequence gives A060863(k).

Original entry on oeis.org

2, 30, 910, 15203

Offset: 1

Naohiro Nomoto, May 17 2003

			For n=2; smallest number k = 20, A082596(20) = 2. so a(2) = A060863(20) = 30.

Cf. A060863, A077389, A082592.

cp's OEIS Frontend

A082596 a(n) is the number of values of k such that A077389(k) = A060863(n).

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A082592 A077389 sorted and duplicates removed.

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A054892 Smallest prime a(n) such that the sum of n consecutive primes starting with a(n) is divisible by n.

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A077388 Row sums of the triangle in A122820.

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A122820 Array read by rows in which n-th row contains n successive primes with least sum divisible by n.

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Mathematica

A082653 For smallest numbers k such that A082596(k) = n, sequence gives A060863(k).

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