A068638 -id:A068638 - cp's OEIS frontend

A052349 Lexicographically earliest sequence of distinct positive integers such that no subsequence sums to a prime.

1, 8, 24, 25, 86, 1260, 1890, 14136, 197400, 10467660, 1231572090, 682616834970

Offset: 1

Carlos Rivera, Mar 07 2000

This set was defined by T. W. A. Baumann in The Prime Puzzles and Problems pages. He and C. Rivera obtained the first 10 members. Chris Nash proved that this sequence is infinite.

			a(4) = 25 as 25+1, 25+8, 25+24, 25+1+8, 25+1+24, 25+8+24 and finally 25+1+8+24 all are composite numbers.

Chris Nash, Proof that A052349, A128687, and A128688 are infinite [Cached copy of proof, from The Prime Puzzles and Problems website]
Carlos Rivera, Puzzle 84. Non-primes adding up to non-primes, The Prime Puzzles and Problems Connection.

Cf. A068638.

Cf. A128687 (restricted to odd numbers), A128688 (restricted to even numbers).

a[1]=1; a[n_]:=a[n]=(s=Subsets[Array[a,n-1],n-1]; c=a[n-1]; While[d=1;While[!PrimeQ[Total[s[[d]]]+c]&&dGiorgos Kalogeropoulos, Nov 19 2021 *)

from sympy import isprime
from itertools import islice
def agen(start=1): # generator of terms
    alst, k, sums = [start], 1, {0} | {start}
    while True:
        yield alst[-1]
        while any(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(), 9))) # Michael S. Branicky, Dec 12 2022

One more term from T. D. Noe, Mar 20 2007
a(12) from Donovan Johnson, Jun 26 2010
New name from Charles R Greathouse IV, Jan 13 2014
Name clarified by Peter Kagey, Jan 07 2017

A025044 a(n) not of form prime - a(k), k < n.

Original entry on oeis.org

0, 1, 8, 14, 20, 24, 25, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 90, 92, 94, 98, 104, 110, 116, 118, 120, 122, 128, 134, 140, 144, 146, 152, 158, 160, 164, 170, 176, 182, 184, 188, 194, 200, 206, 212, 218, 220, 224, 230, 234, 236, 242, 248, 254, 260, 264, 266, 272, 274

Offset: 1

David W. Wilson

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A025043, A068638, A072545, A254337.

a1 = {0}; nmax = 275; Do[ If[Select[n + a1, PrimeQ] == {}, AppendTo[a1, n]] , {n, nmax}]; a1 (* Ray Chandler, Jan 15 2017 *)

A072545 a(0) = 1, a(n) for n > 0 is the smallest number > a(n-1) such that a(n)-a(k) is nonprime for 0 <= k < n.

Original entry on oeis.org

1, 2, 10, 11, 26, 35, 36, 50, 56, 86, 92, 101, 116, 122, 126, 134, 146, 156, 170, 176, 188, 196, 206, 218, 236, 248, 254, 260, 266, 290, 296, 302, 310, 311, 320, 326, 336, 344, 356, 362, 376, 386, 392, 396, 404, 416, 426, 446, 452, 470, 476, 482, 486, 494

Offset: 0

Amarnath Murthy, Aug 04 2002

nonn

a(0) = 1, a(3) = 11, a(5) = 35, a(11) = 101 and a(33) = 311 are the only odd elements <= 10^6 and probably the only ones. If so, then for n >= 34, a(n) is the smallest even k >= a(n-1)+4 for which none of k-1, k-11, k-35, k-101 or k-311 is prime. - David W. Wilson, Dec 14 2006

			26 is the smallest number > 11 which differs from 1, 2, 10, 11 by a nonprime (25, 24, 16, 15), so 26 is the next term after 11.

D. W. Wilson, Table of n, a(n) for n = 0..10000

Cf. A025043, A025044, A068638, A084834, A254337.

print1(a=1,","); v=[1]; n=1; while(n<55,a++; k=1; while(k<=n&&!isprime(a-v[k]), k++); if(k>n,n++; v=concat(v,a); print1(a,",")))

Edited and extended by Klaus Brockhaus, Aug 09 2002

A068640 Define f(n) = 2n+1, a(n) = largest prime of the form f(f(f(...(n))). If no such prime exists then a(n) = 0.

Original entry on oeis.org

7, 47, 7, 0, 47, 13, 0, 17, 19, 0, 47, 0, 0, 59, 31, 0, 0, 37, 0, 167, 43, 0, 47, 0, 0, 107, 0, 0, 59, 61, 0, 0, 67, 0, 71, 73, 0, 0, 79, 0, 167, 0, 0, 2879, 0, 0, 0, 97, 0, 101, 103, 0, 107, 109, 0, 227, 0, 0, 0, 0, 0, 0, 127, 0, 263, 0, 0, 137, 139, 0, 0, 0, 0, 149, 151, 0, 0, 157, 0

Offset: 1

Amarnath Murthy, Feb 27 2002

nonn

			a(2) = 47 as f(2) = 5, f(5) = 11,f(11) = 23, f(23) = 47 is the largest such prime . f(47) = 95 is not a prime. a(4) = 0 as f(4) = 9 is composite.

Cf. A068638.

for k from 1 to 500 do a := 2*k+1; while(isprime(a)) do a := 2*a+1; end do; c[k] := (a-1)/2; if(not isprime(c[k])) then c[k] := 0; end if; if(c[k]<2*k+1) then c[k] := 0; end if; end do:q := seq(c[i],i=1..500);

More terms from Sascha Kurz, Mar 17 2002

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

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A052349 Lexicographically earliest sequence of distinct positive integers such that no subsequence sums to a prime.

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A025044 a(n) not of form prime - a(k), k < n.

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A072545 a(0) = 1, a(n) for n > 0 is the smallest number > a(n-1) such that a(n)-a(k) is nonprime for 0 <= k < n.

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A068640 Define f(n) = 2n+1, a(n) = largest prime of the form f(f(f(...(n))). If no such prime exists then a(n) = 0.

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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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