A052349 -id:A052349 - cp's OEIS frontend

A333522 Lexicographically earliest sequence of distinct positive integers such that for any nonempty set of k positive integers, say {m_1, ..., m_k}, a(m_1) XOR ... XOR a(m_k) is neither null nor prime (where XOR denotes the bitwise XOR operator).

1, 8, 48, 68, 1158, 4752, 81926, 1059600, 713949458, 299601649920

Offset: 1

Rémy Sigrist, Mar 26 2020

This sequence is infinite (the proof is similar to that of the infinity of A333403).
This sequence has similarities with A052349; here we combine terms with the XOR operator, there with the classical addition.
All terms, except a(1) = 1, are even.

			For n = 1:
- we can choose a(1) = 1.
For n = 2:
- 2 is prime,
- 3 is prime,
- 4 XOR 1 = 5 is prime,
- 5 is prime,
- 6 XOR 1 = 7 is prime,
- 7 is prime,
- neither 8 nor 8 XOR 1 = 9 is prime,
- so a(2) = 8.

Rémy Sigrist, PARI program for A333522

Cf. A052349, A333403.

PARI
```
See Links section.
```

a(n) = A333403(2^(n-1)).

a(10) from Giovanni Resta, Mar 30 2020

A133266 a(1) = 30; for n >= 2, choose smallest a(n) so that no sum of 2 or more terms equals a prime.

Original entry on oeis.org

30, 32, 33, 52, 60, 63, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420, 450, 480, 510, 540, 570, 600, 630, 660, 690, 720, 750, 780

Offset: 1

Robert G. Wilson v, Jan 01 2008

Cf. A052349, A133660, A133661.

(* first do *) Needs [ "Combinatorica`" ] (* then *) lst = {30}; g [ k_ ] := Block [ {j = 1, l = 2^Length@ lst}, While [ j < l && !PrimeQ [ Plus @@ NthSubset [ j, lst ] + k ], j++ ]; If [ j == l, False, True ] ]; f [ n_ ] := Block [ {k = lst [ [ -1 ] ] + 1}, While [ g@k == True, k++ ]; AppendTo [ lst, k ]; k ]; Do [ Print@ f@n, {n, 30} ]

a(n+1) = a(n) + 30 for n >= 7 (conjectured). - Chai Wah Wu, Feb 15 2020

A305579 Square array read by antidiagonals upwards in which row k has k as its first term and each subsequent term is the least possible value such that the sum of any 2 or more terms does not equal a prime.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 4, 5, 6, 87, 5, 5, 7, 8, 87, 6, 7, 11, 83, 10, 1151, 7, 8, 9, 29, 235, 12, 5371, 8, 8, 10, 79, 215, 395, 14, 199276, 9, 10, 13, 12, 131, 511, 5275, 16, 32281747, 10, 11, 12, 37, 14, 196, 8729, 76128, 18, 16946784207, 11, 11, 13, 14, 67, 16, 3983, 20526, 9782734, 20

Offset: 1

Randy L. Ekl and Robert G. Wilson v, Jun 05 2018

Rows which appear to have consecutive even numbers are for k = 2, 6, 8, 14, 18, 20, 26, 36, 44, 48, 50, 54,56, 68, 74, 78, 86, 96, 114, ..., .

Conjecture: these row terms are a proper subset of A005843.

			Row 1 is A133660 and is a good illustration of the definition.
Array begins:
============================================================================
k\n|  1   2   3   4    5     6      7       8         9           10
---|------------------------------------------------------------------------
1  |  1,  3,  5, 87, 113, 1151,  5371, 199276, 32281747, 16946784207, ..., ;
2  |  2,  4,  6,  8,  10,   12,    14,     16,       18,          20, ..., ;
3  |  3,  5,  7, 83, 235,  395,  5275,  76128,  9782734, ..., ;
4  |  4,  5, 11, 29, 215,  511,  8729,  20526,  9745499, ..., ;
5  |  5,  7,  9, 79, 131,  196,  3983,  16380,   270270, ..., ;
6  |  6,  8, 10, 12,  14,   16,    18,     20,       22,          24, ..., ;
7  |  7,  8, 13, 37,  67, 1087,  5128, 137886,  6353767, ..., ;
8  |  8, 10, 12, 14,  16,   18,    20,     22,       24,          26, ..., ;
9  |  9, 11, 13, 71, 112,  281,  1952, 147630,  1729159, ..., ;
10 | 10, 11, 14, 25,  94,  756,  2394,  28480,  1466566, ..., ;
11 | 11, 13, 14, 25, 109,  559,  2719,  57985,  2589731, ..., ;
12 | 12, 13, 14, 37,  79,  673,  2929, 113256,  9708060, ..., ;
13 | 13, 14, 19, 31,  97,  882,  2028, 161340,  3635970, ..., ;
14 | 14, 16, 18, 20,  22,   24,    26,     28,       30,          32, ..., ;
15 | 15, 17, 18, 31, 137,  502,  7983, 599346, 27105801, ..., ;
16 | 16, 17, 18, 47, 107,  395,  6480,  91140,   467730, ..., ;
17 | 17, 18, 21, 31,  77,  637,  3609,  77910,   652680, ..., ;
18 | 18, 20, 22, 24,  26,   28,    30,     32,       34,          36, ..., ;
19 | 19, 20, 25, 30,  61,  235,  2965,   4415,   394170,     5769540, ..., ;
20 | 20, 22, 24, 26,  28,   30,    32,     34,       36,          38, ..., ;
21 | 21, 23, 25, 47,  73,  797, 20419, 235665,      ..., ;
22 | 22, 23, 27, 42,  69,  462,   672,    783,    71652,      935298, ..., ;
23 | 23, 25, 26, 37,  73, 1555,  4219, 196260,  3698520, ..., ;
24 | 24, 25, 26, 31, 193,  504,  3756,  91831,  7703843, ..., ;
25 | 25, 26, 29, 31,  39,  750,  4350,  85830,   661350, ..., ;
26 | 26, 28, 30, 32,  34,   36,    38,     40,       42,          44, ..., ;
27 | 27, 28, 29, 35, 232,  888,  5670, 134400,  4058376, ..., ;
28 | 28, 29, 34, 53,  59, 1045,  3696, 249240,  9475589, ..., ;
29 | 29, 31, 33, 55,  57,  674,  6581, 126272,  2549747, ..., ;
30 | 30, 32, 33, 52,  60,   63,    90,    120,      150,         180, ..., ;
31 | 31, 32, 33, 54,  90,  714,  9450, 188850,  2598573, ..., ;
32 | 32, 33, 37, 45, 138,  597,  2703, 101055,  2754885, ..., ;
33 | 33, 35, 37, 47, 133,  555,  4155, 332885,  3090195, ..., ;
34 | 34, 35, 41, 43,  77,  594,  2940,  35700,  2323246, ..., ;
35 | 35, 37, 39, 43, 210, 1061, 10125, 372955, 30373014, ..., ;
36 | 36, 38, 40, 42,  44,   46,    48,     50,       52,          54, ..., ;
37 | 37, 38, 39, 47,  48,  631,  8862, 124851,  4972506, ..., ;
..., etc.

Cf. A005843, A052349, A133660, A133661, first column: A000027.

(* first do *) Needs["Combinatorica`"] (* then *) lst = {k}; g[k_] := Block[{j = 1, l = 2^Length@lst}, While[j < l && !PrimeQ[Plus @@ NthSubset[j, lst] + k], j++ ]; If[j == l, False, True]]; f[n_] := Block[{k = lst[[-1]] + 1}, While[PrimeQ@k || g[k] == True, k++; k++ ]; AppendTo[lst, k]; k]; Do[ Print@ f@ n, {n, 10}] (* Robert G. Wilson v, Jun 05 2018 *)

A359065 Lexicographically earliest sequence of distinct positive composite integers such that no subsequence sums to a prime and in which all terms are coprime.

Original entry on oeis.org

4, 21, 65, 209, 391, 3149, 9991, 368131, 57556589, 14865154981

Offset: 1

Conor Houghton, Dec 15 2022

The sequences A052349 and A068638 are composed of integers starting from one with the rule that no subsequence has a prime sum; these sequences start with one. Starting with a different number seems to result in straightforward geometric sequences, for example the sequence with no prime subsequence sums starting with four is 4, 6, 8, and so on. One way to avoid this is to enforce a coprime rule, requiring that the entries to the sequence are coprime. It is not clear whether the sequence is infinite.

Cf. A052349, A068638.

k = 4; K = {k}; f = {2}; q = Subsets[K]; While[Length@K < 10, k++; If[! PrimeQ[k] && ! IntersectingQ[FactorInteger[k][[All, 1]], f], s = k; z = 0; For[p = 1, p <= Length@q, p++, If[PrimeQ[Total[q[[p]]] + k], z = 1; Break[]]]; If[z == 0, AppendTo[K, k]; q = Subsets[K]; AppendTo[f, FactorInteger[k][[All, 1]]]; f = Flatten[f]]]]; Print[K] (* Samuel Harkness, Apr 11 2023 *)

import sys
import math
from sympy.ntheory import primefactors
from sympy.ntheory import primerange
def intersection(lst1, lst2):
    lst3 = [value for value in lst1 if value in lst2]
    return len(lst3)
n_primes=1000000
factors=[primefactors(n) for n in range(0,n_primes)]
primes=list(primerange(0, n_primes))
sequence=[4]
sums=[sequence[0]]
prime_factors=[f for f in factors[sequence[0]]]
big_n=8
while len(sequence)

from math import gcd
from sympy import isprime
from itertools import islice
def agen(start=4): # generator of terms
    alst, k, sums = [start], start+1, {0} | {start}
    while True:
        yield alst[-1]
        while any(gcd(k, an) != 1 for an in alst) or \
              any(k+s not in sums and isprime(k+s) for s in sums):
            k += 1
        alst.append(k)
        sums.update([k + s for s in sums])
        k += 1
print(list(islice(agen(), 8))) # Michael S. Branicky, Dec 16 2022

a(8)-a(9) from Michael S. Branicky, Dec 15 2022

a(10) from Rémy Sigrist, Dec 16 2022

cp's OEIS Frontend

A333522 Lexicographically earliest sequence of distinct positive integers such that for any nonempty set of k positive integers, say {m_1, ..., m_k}, a(m_1) XOR ... XOR a(m_k) is neither null nor prime (where XOR denotes the bitwise XOR operator).

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A133266 a(1) = 30; for n >= 2, choose smallest a(n) so that no sum of 2 or more terms equals a prime.

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Mathematica

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A305579 Square array read by antidiagonals upwards in which row k has k as its first term and each subsequent term is the least possible value such that the sum of any 2 or more terms does not equal a prime.

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Examples

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Mathematica

A359065 Lexicographically earliest sequence of distinct positive composite integers such that no subsequence sums to a prime and in which all terms are coprime.

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Mathematica

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Python

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