A073659 -id:A073659 - cp's OEIS frontend

A073660 Terms of A073659 which retain their relative positions.

1, 2, 4, 6, 12, 22, 26, 42, 58, 64, 96, 110, 128, 156, 216, 234, 242, 268, 298, 310, 330, 356, 380, 384, 408, 422, 424, 454, 462, 468, 490, 516, 580, 634, 648, 664, 708, 806, 824, 826, 864, 894, 896, 986, 1038, 1114, 1122, 1160, 1244, 1246, 1254, 1264, 1316, 1420

Offset: 1

Amarnath Murthy, Aug 10 2002

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A073659.

More terms from Sascha Kurz, Jan 28 2003

A291546 a(n) = Sum_{k=1..n} A073659(k).

Original entry on oeis.org

1, 3, 7, 13, 23, 31, 43, 59, 73, 97, 127, 149, 167, 193, 227, 263, 283, 311, 349, 389, 421, 463, 509, 557, 601, 653, 709, 769, 823, 881, 947, 997, 1061, 1123, 1193, 1277, 1367, 1439, 1531, 1607, 1693, 1787, 1861, 1949, 2017, 2099, 2179, 2281, 2377, 2477

Offset: 1

Seiichi Manyama, Aug 26 2017

nonn

For n > 1, a(n) is a prime.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000040, A008578, A073659. Apart from the first term the same as A291544.

a = {1}; Do[k = 2; While[Or[! PrimeQ[Total@ a + k], MemberQ[a, k]], k += 2]; AppendTo[a, k], {49}]; Accumulate@ a (* Michael De Vlieger, Aug 26 2017, after Jayanta Basu at A073659 *)

v=[1]; n=1; vp=[1]; while(n<200, if(isprime(p=n+vecsum(v)) && !vecsearch(vecsort(v), n), v=concat(v, n); vp = concat(vp, p); n=0); n++); vp \\ Michel Marcus, Aug 26 2017

A055265 a(n) is the smallest positive integer not already in the sequence such that a(n)+a(n-1) is prime, starting with a(1)=1.

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 5, 8, 9, 10, 13, 16, 15, 14, 17, 12, 11, 18, 19, 22, 21, 20, 23, 24, 29, 30, 31, 28, 25, 34, 27, 26, 33, 38, 35, 32, 39, 40, 43, 36, 37, 42, 41, 48, 49, 52, 45, 44, 53, 50, 47, 54, 55, 46, 51, 56, 57, 70, 61, 66, 65, 62, 69, 58, 73, 64, 63, 68, 59, 72, 67, 60

Offset: 1

Henry Bottomley, May 09 2000

The sequence is well-defined (the terms must alternate in parity, and by Dirichlet's theorem a(n+1) always exists). - N. J. A. Sloane, Mar 07 2017
Does every positive integer eventually occur? - Dmitry Kamenetsky, May 27 2009. Reply from Robert G. Wilson v, May 27 2009: The answer is almost certainly yes, on probabilistic grounds.
It appears that this is the limit of the rows of A051237. That those rows do approach a limit seems certain, and given that that limit exists, that this sequence is the limit seems even more likely, but no proof is known for either conjecture. - Robert G. Wilson v, Mar 11 2011, edited by Franklin T. Adams-Watters, Mar 17 2011
The sequence is also a particular case of "among the pairwise sums of any M consecutive terms, N are prime", with M = 2, N = 1. For other M, N see A055266 & A253074 (M = 2, N = 0), A329333, A329405 - A329416, A329449 - A329456, A329563 - A329581, and the OEIS Wiki page. - M. F. Hasler, Feb 11 2020

			a(5) = 7 because 1, 2, 3 and 4 have already been used and neither 4 + 5 = 9 nor 4 + 6 = 10 are prime while 4 + 7 = 11 is prime.

Zak Seidov, Table of n, a(n) for n = 1..10000 (First 1000 terms from T. D. Noe)
N. J. A. Sloane, Table of n, a(n) for n = 1..100000 (computed using Orlovsky's Mma program)
M. F. Hasler, Prime sums from neighboring terms, OEIS Wiki, Nov. 23, 2019
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A117922; fixed points: A117925; A117923=a(a(n)). - Reinhard Zumkeller, Apr 03 2006
Cf. A036440, A051237, A051239, A055266, A088643. A010051.
Cf. A086527 (the primes a(n)+a(n-1)).
Cf. A070942 (n's such that a(1..n) is a permutation of (1..n)). - Zak Seidov, Oct 19 2011
See also A076990, A243625.
See A282695 for deviation from identity sequence.
A073659 is a version where the partial sums must be primes.

import Data.List (delete)
a055265 n = a055265_list !! (n-1)
a055265_list = 1 : f 1 [2..] where
   f x vs = g vs where
     g (w:ws) = if a010051 (x + w) == 1
                   then w : f w (delete w vs) else g ws
-- Reinhard Zumkeller, Feb 14 2013

A055265 := proc(n)
    local a,i,known ;
    option remember;
    if n =1 then
        1;
    else
        for a from 1 do
            known := false;
            for i from 1 to n-1 do
                if procname(i) = a then
                    known := true;
                    break;
                end if;
            end do:
            if not known and isprime(procname(n-1)+a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A055265(n),n=1..100) ; # R. J. Mathar, Feb 25 2017

f[s_List] := Block[{k = 1, a = s[[ -1]]}, While[ MemberQ[s, k] || ! PrimeQ[a + k], k++ ]; Append[s, k]]; Nest[f, {1}, 71] (* Robert G. Wilson v, May 27 2009 *)
q=2000; a={1}; z=Range[2,2*q]; While[Length[z]>q-1, k=1; While[!PrimeQ[z[[k]]+Last[a]], k++]; AppendTo[a,z[[k]]]; z=Delete[z,k]]; Print[a] (*200 times faster*) (* Vladimir Joseph Stephan Orlovsky, May 03 2011 *)

v=[1];n=1;while(n<50,if(isprime(v[#v]+n)&&!vecsearch(vecsort(v),n), v=concat(v,n);n=0);n++);v \\ Derek Orr, Jun 01 2015

U=-a=1; vector(100,k, k=valuation(1+U+=1<M. F. Hasler, Feb 11 2020

a(2n-1) = A128280(2n-1) - 1, a(2n) = A128280(2n) + 1, for all n >= 1. - M. F. Hasler, Feb 11 2020

Corrected by Hans Havermann, Sep 24 2002

A054408 a(n) = smallest positive integer not already in sequence such that the partial sum a(1)+...+a(n) is prime.

Original entry on oeis.org

2, 1, 4, 6, 10, 8, 12, 16, 14, 24, 30, 22, 18, 26, 34, 36, 20, 28, 38, 40, 32, 42, 46, 48, 44, 52, 56, 60, 54, 58, 66, 50, 64, 62, 70, 84, 90, 72, 92, 76, 86, 94, 74, 88, 68, 82, 80, 102, 96, 100, 114, 98, 78, 112, 120, 110, 108, 106, 126, 122, 130, 132, 134, 124, 128, 118

Offset: 1

Henry Bottomley, May 09 2000

1 is the only odd number in this sequence. - Derek Orr, Feb 07 2015

Conjecture: Every even numbers appears. - N. J. A. Sloane, May 29 2017

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms n = 1..4100 from N. J. A. Sloane).

Cf. A254337.
See A073659 for another version.
In A055265 only pairs of adjacent terms add to primes.

t = {2}; Do[i = 1; While[! PrimeQ[Total[t] + i] || MemberQ[t, i], i++]; AppendTo[t, i], {65}]; t (* Jayanta Basu, Jul 04 2013 *)

v=[2];n=1;while(n<100,if(isprime(vecsum(v)+n)&&!vecsearch(vecsort(v),n),v=concat(v,n);n=0);n++);v \\ Derek Orr, Feb 07 2015

from sympy import isprime
def aupton(terms):
    alst, aset, asum = [], set(), 0
    while len(alst) < terms:
        an = 1
        while True:
            while an in aset: an += 1
            if isprime(asum + an):
                alst, aset, asum = alst + [an], aset | {an}, asum + an
                break
            an += 1
    return alst
print(aupton(66)) # Michael S. Branicky, Jun 05 2021

A075574 a(1) = 1, then the smallest number (obviously even) greater than the previous term such that every partial sum is prime.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 16, 18, 26, 30, 36, 48, 52, 54, 56, 58, 60, 66, 74, 78, 88, 90, 96, 104, 106, 108, 122, 126, 144, 154, 156, 158, 172, 188, 190, 192, 206, 210, 214, 228, 240, 242, 250, 258, 260, 262, 284, 286, 288, 290, 298, 300, 302, 318, 324, 328, 332, 340

Offset: 1

Amarnath Murthy, Sep 25 2002

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A073659, A073660.

A075574:=proc(n) local i,j,k,t,s; j:=1; s:=1; t:=1; for i to n do k:=s; s:=nextprime(s+j); j:=s-k; t:=t,j; od; t; end; # Floor van Lamoen, Oct 21 2005

nxt[{ps_,a_}]:=Module[{c=a+2},While[!PrimeQ[ps+c],c+=2];{ps+c,c}]; Join[ {1},NestList[nxt,{3,2},60][[All,2]]] (* Harvey P. Dale, Sep 19 2021 *)

More terms from David Wasserman, Jan 20 2005

A256270 a(1) = 1; for n > 1, a(n) is the smallest positive integer not already in the sequence such that a(1)^2 + a(2)^2 + ... + a(n)^2 is prime.

Original entry on oeis.org

1, 2, 6, 24, 12, 54, 18, 36, 30, 48, 60, 84, 42, 90, 66, 72, 96, 78, 120, 126, 102, 150, 144, 108, 156, 114, 138, 132, 180, 204, 210, 168, 186, 192, 234, 174, 162, 216, 198, 222, 246, 228, 264, 348, 240, 270, 300, 330, 336, 402, 390, 414, 288, 306, 258, 294, 318, 324, 276, 252, 372, 426, 366, 282, 432, 354, 378

Offset: 1

Derek Orr, Jun 01 2015

1 is the only odd number in this sequence.

Excluding 1 and 2, this is a permutation of the positive multiples of 6.

Cf. A073659, A073852.

v=[1];n=1;while(n<10^3,if(isprime(n^2+sum(i=1,#v,v[i]^2))&&!vecsearch(vecsort(v),n),v=concat(v,n);n=1);n++);v

a(n)^2 = A073852(n).

A287662 a(n) is the smallest positive integer not already in sequence such that a(1) + ... + a(n) is a prime power, with a(1) = 1.

Original entry on oeis.org

1, 2, 4, 6, 3, 7, 8, 10, 12, 11, 9, 16, 14, 18, 28, 20, 22, 32, 33, 13, 24, 38, 30, 36, 34, 26, 42, 48, 40, 44, 46, 50, 60, 52, 68, 54, 58, 5, 15, 64, 78, 56, 66, 70, 74, 76, 84, 62, 72, 82, 90, 80, 55, 21, 92, 106, 104, 88, 98, 100, 96, 108, 102, 86, 114, 94, 116, 118, 122, 120, 130, 126, 107, 31, 132, 138

Offset: 1

Ilya Gutkovskiy, May 29 2017

nonn

It appears that the sequence contains all even numbers.

			a(8) = 10 because 1, 2, 3, 4, 6, 7 and 8 have already been used in the sequence, 1 + 2 + 4 + 6 + 3 + 7 + 8 + 5 = 36 is not prime power, 1 + 2 + 4 + 6 + 3 + 7 + 8 + 9 = 40 is not prime power while 1 + 2 + 4 + 6 + 3 + 7 + 8 + 10 = 41 is a prime power.

Cf. A000961, A054408, A073659, A073879, A075562, A084693, A284049.

t = {1}; Do[i = 1; While[! PrimePowerQ[Total[t] + i] || MemberQ[t, i], i++]; AppendTo[t, i], {75}]; t

A356188 a(1)=1; for n > 1, if a(n-1) is prime then a(n) = the smallest number not yet in the sequence. Otherwise a(n) = a(n-1) + n - 1.

Original entry on oeis.org

1, 2, 3, 4, 8, 13, 5, 6, 14, 23, 7, 9, 21, 34, 48, 63, 79, 10, 28, 47, 11, 12, 34, 57, 81, 106, 132, 159, 187, 216, 246, 277, 15, 48, 82, 117, 153, 190, 228, 267, 307, 16, 58, 101, 17, 18, 64, 111, 159, 208, 258, 309, 361, 414, 468, 523, 19, 20, 78, 137, 22, 83, 24, 87, 151, 25, 91

Offset: 1

John Tyler Rascoe, Jul 28 2022

			a(8) = 6 because a(7) is prime and 6 is the smallest number that has not appeared in the sequence thus far.
a(9) = 6 + 9 - 1 = 14 because a(8) is not prime.

Cf. A000040, A060735, A073659, A331603.

f[s_] := Module[{k=1, t}, t = If[!PrimeQ[s[[-1]]], s[[-1]] + Length[s], While[!FreeQ[s, k], k++]; k]; Join[s, {t}]]; Nest[f, {1}, 66] (* Amiram Eldar, Sep 28 2022 *)

from sympy import isprime
from itertools import count, filterfalse
A356188 = A = [1]
for n in range(1,100):
      if isprime(A[-1]):
            y = next(filterfalse(set(A)._contains_, count(1)))
      else:
            y = A[-1] + n
      A.append(y)

cp's OEIS Frontend

A073660 Terms of A073659 which retain their relative positions.

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A291546 a(n) = Sum_{k=1..n} A073659(k).

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A055265 a(n) is the smallest positive integer not already in the sequence such that a(n)+a(n-1) is prime, starting with a(1)=1.

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A054408 a(n) = smallest positive integer not already in sequence such that the partial sum a(1)+...+a(n) is prime.

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A075574 a(1) = 1, then the smallest number (obviously even) greater than the previous term such that every partial sum is prime.

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Maple

Mathematica

Extensions

A256270 a(1) = 1; for n > 1, a(n) is the smallest positive integer not already in the sequence such that a(1)^2 + a(2)^2 + ... + a(n)^2 is prime.

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A287662 a(n) is the smallest positive integer not already in sequence such that a(1) + ... + a(n) is a prime power, with a(1) = 1.

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A356188 a(1)=1; for n > 1, if a(n-1) is prime then a(n) = the smallest number not yet in the sequence. Otherwise a(n) = a(n-1) + n - 1.

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