A144103 -id:A144103 - cp's OEIS frontend

A337767 Array T(n,k) (n >= 1, k >= 1) read by upward antidiagonals and defined as follows. Let N(p,i) denote the result of applying "nextprime" i times to p; T(n,k) = smallest prime p such that N(p,n) - p = 2*k, or 0 if no such prime exists.

3, 0, 7, 0, 3, 23, 0, 0, 5, 89, 0, 0, 0, 23, 139, 0, 0, 0, 3, 19, 199, 0, 0, 0, 0, 7, 47, 113, 0, 0, 0, 0, 3, 17, 83, 1831, 0, 0, 0, 0, 0, 5, 23, 211, 523, 0, 0, 0, 0, 0, 0, 17, 43, 109, 887, 0, 0, 0, 0, 0, 0, 3, 13, 79, 317, 1129, 0, 0, 0, 0, 0, 0, 0, 7, 19, 107, 619, 1669

Offset: 1

Robert G. Wilson v, Sep 19 2020

The positive entries in each row and column are distinct.
Number of zeros right of 3 are 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 3, 3, 3, 6, 5, 5, 4, 6, ..., .
Number of zeros in the n-th row are 0, 1, 3, 4, 6, 7, 10, 13, 14, 17, 18, 20, 22, 25, 28, 30, 32, 36, 37, 40, 45, 47, 51, 52, 55, ..., .
The usual convention in the OEIS is to use -1 in the "escape clause" - that is, when "no such terms exists". It is probably too late to change this sequence, but it should not be cited as a role model for other sequences. - N. J. A. Sloane, Jan 19 2021
a(1416), a(1637), and a(1753) were provided by Brian Kehrig. - Martin Raab, Jun 28 2024

			The initial rows of the array are:
  3, 7, 23, 89, 139, 199, 113, 1831, 523, 887, 1129, 1669, 2477, 2971, 4297, ...
  0, 3, 5, 23, 19, 47, 83, 211, 109, 317, 619,  199, 1373, 1123, 1627, 4751, ...
  0, 0, 0,  3,  7, 17, 23,  43,  79, 107, 109,  113,  197,  199,  317,  509, ...
  0, 0, 0,  0,  3,  5, 17,  13,  19,  47,  79,   73,  113,  109,  193,  317, ...
  0, 0, 0,  0,  0,  0,  3,   7,  11,  17,  19,   43,   71,   73,  107,  191, ...
  0, 0, 0,  0,  0,  0,  0,   3,   5,  11,   7,   13,   41,   31,   67,  107, ...
  0, 0, 0,  0,  0,  0,  0,   0,   0,   3,   0,    5,   11,   13,   23,   47, ...
  0, 0, 0,  0,  0,  0,  0,   0,   0,   0,   0,    0,    3,    0,    7,   29, ...
  0, 0, 0,  0,  0,  0,  0,   0,   0,   0,   0,    0,    0,    3,    0,    5, ...
The initial antidiagonals are:
  [3]
  [0, 7]
  [0, 3, 23]
  [0, 0, 5, 89]
  [0, 0, 0, 23, 139]
  [0, 0, 0, 3, 19, 199]
  [0, 0, 0, 0, 7, 47, 113]
  [0, 0, 0, 0, 3, 17, 83, 1831]
  [0, 0, 0, 0, 0, 5, 23, 211, 523]
  [0, 0, 0, 0, 0, 0, 17, 43, 109, 887]
  [0, 0, 0, 0, 0, 0, 3, 13, 79, 317, 1129]
  ...

Martin Raab, Table of n, a(n) for n = 1..1830 (antidiagonals 1..60)

Cf. A000230, A144103, A339943, A339944 (rows 1 to 4), A086153.

t[r_, c_] := If[ 2c <= Prime[r + 2] - 5, 0, Block[{p = 3}, While[ NextPrime[p, r] != 2c + p && p < 52000000, p = NextPrime@ p]; If[p > 52000000, 0, p]]]; Table[ t[r -c +1, c], {r, 11}, {c, r}] // Flatten

T(n,k) = 0 if prime(n+2)-5 <= 2k. A089038.

T(n,k) = 3 if prime(n+2) = 2k+6. A067076.

Entry revised by N. J. A. Sloane, Nov 07 2020

Deleted a-file and b-file because entries were unreliable. - N. J. A. Sloane, Nov 01 2021

A339943 Let N(p,i) denote the result of applying "nextprime" i times to p; a(n) = smallest prime p such that N(p,3) - p = 2*n, or -1 if no such prime exists.

Original entry on oeis.org

-1, -1, -1, 3, 7, 17, 23, 43, 79, 107, 109, 113, 197, 199, 317, 509, 523, 773, 1823, 1237, 1319, 3119, 1321, 2473, 2153, 4159, 2477, 6491, 5581, 7351, 9551, 9973, 18803, 18593, 24247, 30559, 31883, 33211, 19603, 66191, 37699, 31393, 83117, 43801, 107351, 107357, 69499, 38461, 130859

Offset: 1

Robert G. Wilson v, Dec 23 2020

sign

This sequence is the third row of A337767.a(n) > 0 and that there are multiple instances for some k where (p_(k+3) - p_k)/2 - 3 = n.
This sequence only cites the first such occurrence.
n:
4:    3,   5,  11, 101, 191, 821, 1481, 1871, 2081, 3251, 3461, 5651, ...,
5:    7,  13,  37,  97, 103, 223,  307,  457,  853,  877, 1087, 1297, ...,
6:   17,  19,  29,  31,  41,  59,   61,   67,   71,  127,  227,  229, ...,
7:   23,  47,  53,  89, 137, 149,  167,  179,  257,  263,  419,  449, ...,
8:   43,  73, 151, 157, 163, 181,  277,  337,  367,  373,  433,  487, ...,
9:   79,  83, 131, 139, 173, 193,  211,  233,  239,  251,  331,  349, ...,
10: 107, 293, 311, 353, 359, 389,  401,  479,  503,  653,  719,  839, ...,
etc.

			a(4) = 3. This represents the beginning of the run of primes {3, 5, 7, 11}. (11 - 3)/2 = 4 and it is the first prime to do so. Others are 5, 11, 101, 191, etc.;
a(5) = 7. This represents the beginning of the run of primes {7, 11, 13, 17}. (17 - 7)/2 = 5 and it is the first prime to do so. Others are 13, 37, 97, 103, etc.;
a(6) = 17. This represents the beginning of the run of primes {17, 19, 23 & 29}. (29 - 17)/2 = 6 and it is the first prime to do so. Others are 19, 29, 31, 41, etc.

Martin Raab, Table of n, a(n) for n = 1..504 (Terms 1..345 from Robert G. Wilson v)

Cf. A000230, A001223, A144103, A337767, A339944.

p = 3; q = 5; r = 7; s = 11; tt[_] := 0; While[p < 250000, d = (s - p)/2; If[ tt[d] == 0, tt[d] = p]; {p, q, r, s} = {q, r, s, NextPrime@ s}]; tt@# & /@ Range@ 75

A339944 Let N(p,i) denote the result of applying "nextprime" i times to p; a(n) = smallest prime p such that N(p,4) - p = 2*n, or -1 if no such prime exists.

Original entry on oeis.org

-1, -1, -1, -1, 3, 5, 17, 13, 19, 47, 79, 73, 113, 109, 193, 317, 313, 521, 503, 523, 887, 1499, 1231, 1319, 1373, 1321, 1307, 3947, 2473, 2143, 2477, 7369, 5573, 5939, 9967, 16111, 18587, 20773, 18593, 31883, 17209, 19597, 24251, 19609, 25471, 31397, 44389, 18803, 38459, 38461, 66191, 69557, 103183

Offset: 1

Robert G. Wilson v, Dec 23 2020

sign

This sequence is the fourth row of A337767.
From Robert G. Wilson v, Dec 30 2020: (Start)
a(n) > -1 for all n >= 5.
It seems likely that for almost all values of n there is more than one integer k such that prime(k+4) - prime(k) = 2*n; a(n) = prime(k) for the smallest such k.
.
   n | list of numbers k such that prime(k+4) - prime(k) = 2*n
  ---+-----------------------------------------------------------------
   5 |  3.
   6 |  5,  7,  11,  97, 101, 1481, 1867, 3457, 5647, 15727, 16057, ...
   7 | 17, 29,  59, 227, 269, 1277, 1289, 1607, 2129,  2789,  3527, ...
   8 | 13, 31,  37,  67, 223, 1087, 1291, 1423, 1483,  1597,  1861, ...
   9 | 19, 23,  41,  43,  53,   61,   71,   89,  149,   163,   179, ...
  10 | 47, 83, 131, 137, 173,  191,  251,  257,  347,   419,   443, ...
etc.
(End)

			a(1) = 3. This represents the beginning of the run of primes {3, 5, 7, 11, 13}. (13 - 3)/2 = 5 and it is the only prime to do so.
a(2) = 5. This represents the beginning of the run of primes {5, 7, 11, 13, 17}. (17 - 5)/2 = 6 and it is the first prime to do so. Others are 7, 11, 97, 101, etc.
a(3) = 17. This represents the beginning of the run of primes {17, 19, 23, 29, 31}. (31 - 17)/2 = 7 and it is the first prime to do so. Others are 29, 59, 227, 269, etc.

Martin Raab, Table of n, a(n) for n = 1..546 (Terms 1..342 from Robert G. Wilson)

Cf. A337767, A000230, A144103, A339943.

p = 3; q = 5; r = 7; s = 11; t = 13; tt[] := 0; While[p < 450000, d = (t - p)/2; If[ tt[d] == 0, tt[d] = p]; {p, q, r, s, t} = {q, r, s, t, NextPrime@ t}]; tt@# & /@ Range@ 75 (* _Robert G. Wilson v, Dec 30 2020 *)

A359199 Least prime p such that 2n can be written as a signed sum of p and the next 3 primes, or -1 if no such prime exists.

Original entry on oeis.org

5, 3, 3, 3, 7, 3, 3, 5, 3, 19, 3, 5, 79, 3, 113, 17, 467, 7, 5, 11, 19, 17, 19, 13, 7, 17, 1123, 17, 19, 23, 11, 23, 19, 31, 2153, 31, 13, 23, 29, 31, 29, 37, 43, 37, 17, 31, 19081, 37, 43, 41, 19319, 19, 37897, 53, 43, 54193, 35671, 47, 43, 53, 23, 53, 59, 47, 35603, 61

Offset: 0

Karl-Heinz Hofmann and Peter Munn, Jun 04 2023

nonn

We require each of the 4 primes to appear in the sum exactly once.
Inspired by the study of problems about the signed sum of consecutive primes, for example, by Rivera in 2000 (see link).
The equivalent sequence with 2 rather than 4 primes is A363544, which is closely related to A000230, which concerns prime gaps.
Conjecture: a satisfactory prime p exists for all n.
The magnitude of the terms exhibits a notable variation that depends on the number of negations in the sum. See the visualization in the links.
All odd primes appear in the sequence. When 2n = A034963(k) we see the last occurrence of the k-th prime. Obviously, these last occurrences correspond to the sums where all the signs are positive. Do any odd primes occur only once?

			The signed sums of 2, 3, 5 and 7 are all odd, so cannot be 2n for any n. So all terms are >= 3, the 2nd prime.
The 16 possible signed sums of 3, 5, 7 and 11 give 8 nonnegative totals: 2, 4, 6, 10, 12, 16, 20, 26. So a(1) = a(2) = a(3) = a(5) = a(6) = a(8) = a(10) = a(13) = 3.
0 was not one of the 8 totals, and 0 = 5 - 7 - 11 + 13. So a(0) = 5.

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..532
Carlos Rivera, Conjecture 21. Rivera's conjecture, The Prime Puzzles and Problems Connection.
Karl-Heinz Hofmann, Solutions for n = 0 to 100.
Karl-Heinz Hofmann, Visualization of possible sums.
Karl-Heinz Hofmann, Visualization of possible sums and results.

Cf. A000040, A000230, A034963, A144103, A327467, A362465, A363544.

from sympy import nextprime
import numpy as np
aupto = 100
A359199 = np.zeros(aupto+1, dtype=object)
signset = np.array([[ 1,  1,  1,  1] , # green line in visualizations (see links)
                    [ 1,  1,  1, -1] , # red ribbon
                    [ 1,  1, -1,  1] , # red ribbon
                    [ 1, -1,  1,  1] , # red ribbon
                    [ 1,  1, -1, -1] , # magenta ribbon
                    [ 1, -1,  1, -1] , # magenta ribbon
                    [ 1, -1, -1,  1] , # magenta ribbon
                    [ 1, -1, -1, -1]], # red ribbon
                    dtype="i4")
primeset = np.array([3, 5, 7, 11], dtype=object)
while all(A359199) == 0:
    for signs in signset:
        asum = abs(sum(signs * primeset)) // 2
        if asum <= aupto and A359199[asum] == 0: A359199[asum] = primeset[0]
    primeset = np.append(primeset, nextprime(primeset[-1]))[1:]
print(list(A359199))

For k >= 2, a(A034963(k)/2) = A000040(k).

cp's OEIS Frontend

A337767 Array T(n,k) (n >= 1, k >= 1) read by upward antidiagonals and defined as follows. Let N(p,i) denote the result of applying "nextprime" i times to p; T(n,k) = smallest prime p such that N(p,n) - p = 2*k, or 0 if no such prime exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A339943 Let N(p,i) denote the result of applying "nextprime" i times to p; a(n) = smallest prime p such that N(p,3) - p = 2*n, or -1 if no such prime exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A339944 Let N(p,i) denote the result of applying "nextprime" i times to p; a(n) = smallest prime p such that N(p,4) - p = 2*n, or -1 if no such prime exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A359199 Least prime p such that 2n can be written as a signed sum of p and the next 3 primes, or -1 if no such prime exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula