A083236 -id:A083236 - cp's OEIS frontend

A128280 a(n) is the least number not occurring earlier such that a(n)+a(n-1) is prime, a(0) = 0.

0, 2, 1, 4, 3, 8, 5, 6, 7, 10, 9, 14, 15, 16, 13, 18, 11, 12, 17, 20, 21, 22, 19, 24, 23, 30, 29, 32, 27, 26, 33, 28, 25, 34, 37, 36, 31, 40, 39, 44, 35, 38, 41, 42, 47, 50, 51, 46, 43, 54, 49, 48, 53, 56, 45, 52, 55, 58, 69, 62, 65, 66, 61, 70, 57, 74, 63, 64, 67, 60, 71, 68, 59

Offset: 0

Zak Seidov, May 03 2007

nonn

Original definition: start with a(1) = 2. See A055265 for start with a(1) = 1.
The sequence may well be a rearrangement of natural numbers. Interestingly, subsets of first n terms are permutations of 1..n for n = {2, 4, 8, 10, 18, 22, 24, 56, ...}. E.g., first 56 terms: {2, 1, 4, 3, 8, 5, 6, 7, 10, 9, 14, 15, 16, 13, 18, 11, 12, 17, 20, 21, 22, 19, 24, 23, 30, 29, 32, 27, 26, 33, 28, 25, 34, 37, 36, 31, 40, 39, 44, 35, 38, 41, 42, 47, 50, 51, 46, 43, 54, 49, 48, 53, 56, 45, 52, 55} are a permutation of 1..56.
Without altering the definition nor the existing values, one can as well start with a(0) = 0 and get (conjecturally) a permutation of the nonnegative integers. This sequence is in some sense the "arithmetic" analog of the "digital" variant A231433: Here we add subsequent terms, there the digits are concatenated. - M. F. Hasler, Nov 09 2013
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 A329333, A329405 ff, A329449 ff and the OEIS Wiki page. - M. F. Hasler, Nov 24 2019

Lars Blomberg, Table of n, a(n) for n = 0..9999
M. F. Hasler, Prime sums from neighboring terms, OEIS wiki, Nov. 23, 2019.

Cf. A083236.

Cf. A055265 for the variant starting with a(1) = 1, and A329333, A329405, ..., A329425 and A329449, ..., A329581 for other variants. - M. F. Hasler, Nov 24 2019

PARI
```
{a=0;u=0; for(n=1, 99, u+=1<
				
```

a(2n-1) = A055265(2n-1) + 1, a(2n) = A055265(2n) - 1, for all n >= 1. - M. F. Hasler, Feb 11 2020

Initial a(0) = 0 prefixed by M. F. Hasler, Nov 09 2013

A083237 First order recursion: a(0)=5; a(n)=prime(n)-a(n-1).

Original entry on oeis.org

5, -3, 6, -1, 8, 3, 10, 7, 12, 11, 18, 13, 24, 17, 26, 21, 32, 27, 34, 33, 38, 35, 44, 39, 50, 47, 54, 49, 58, 51, 62, 65, 66, 71, 68, 81, 70, 87, 76, 91, 82, 97, 84, 107, 86, 111, 88, 123, 100, 127, 102, 131, 108, 133, 118, 139, 124, 145, 126, 151, 130, 153, 140, 167, 144, 169, 148, 183, 154, 193, 156, 197, 162, 205, 168, 211, 172

Offset: 0

Labos Elemer, Apr 23 2003

sign

Same function as in A083236 but initial value = 5.

Cf. A000040, A001223.

A083237 := proc(n)
    option remember ;
    if n = 0 then
        5 ;
    else
        ithprime(n)-procname(n-1) ;
    end if;
end proc:
seq(A083237(n),n=0..100) ; # R. J. Mathar, Jun 20 2021

RecursionLimit$=10000; f[x_] := Prime[x]-f[x-1]; f[0]=5; Table[f[w], {w, 1, 100}]

Same implicit relationship: a(n-1)+a(n)=prime(n), the n-th prime;

It follows also that A001223(n)=a(n+1)-a(n-1).

a(0) preprended. - R. J. Mathar, Jun 20 2021

A083238 First order recursion: a(0)=1; a(n) = sigma(1,n) - a(n-1).

Original entry on oeis.org

1, 0, 3, 1, 6, 0, 12, -4, 19, -6, 24, -12, 40, -26, 50, -26, 57, -39, 78, -58, 100, -68, 104, -80, 140, -109, 151, -111, 167, -137, 209, -177, 240, -192, 246, -198, 289, -251, 311, -255, 345, -303, 399, -355, 439, -361, 433, -385, 509, -452, 545, -473, 571, -517, 637, -565, 685, -605, 695, -635, 803, -741, 837, -733, 860

Offset: 0

Labos Elemer, Apr 23 2003

sign

Provide interesting decomposition: sigma(n)=u+w, where u and w consecutive terms of this sequence; this depends also on initial value.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000203, A083236, A083237.

f[x_] := DivisorSigma[1, x]-f[x-1] f[0]=1; Table[f[w], {w, 1, 100}]
nxt[{n_,a_}]:={n+1,DivisorSigma[1,n+1]-a}; NestList[nxt,{0,1},70][[;;,2]] (* Harvey P. Dale, May 10 2024 *)

lista(nn) = {my(last = 1, v=vector(nn)); for (n=1, nn, v[n] = sigma(n) - last; last = v[n]; ); concat(1, v); } \\ Michel Marcus, Mar 28 2020

It follows that a(n)+a(n-1) = A000203(n).

A083239 First order recursion: a(0) = 1; a(n) = phi(n) - a(n-1) = A000010(n) - a(n-1).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, -1, 7, -3, 9, -5, 15, -11, 23, -17, 25, -17, 33, -27, 45, -37, 49, -39, 61, -53, 73, -61, 79, -67, 95, -87, 117, -101, 121, -105, 129, -117, 153, -135, 159, -143, 183, -171, 213, -193, 217, -195, 241, -225, 267, -247, 279, -255, 307, -289, 329, -305, 341, -313, 371, -355, 415, -385, 421, -389, 437, -417

Offset: 0

Labos Elemer, Apr 23 2003

Provides interesting decomposition: phi(n) = u+w, where u and w consecutive terms of this sequence. Depends also on initial value.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000010, A068773, A083236, A083237, A083238.

A083239 := proc(n)
    option remember ;
    if n = 0 then
        1 ;
    else
        numtheory[phi](n)-procname(n-1) ;
    end if;
end proc:
seq(A083239(n),n=0..100) ; # R. J. Mathar, Jun 20 2021

a[n_] := a[n] = EulerPhi[n] -a[n-1]; a[0] = 1; Table[a[n], {n, 0, 100}]

# uses programs from A002088 and A049690
def A083239(n): return A002088(n)-(A049690(n>>1)<<1)-1 if n&1 else 1+(A049690(n>>1)<<1)-A002088(n) # Chai Wah Wu, Aug 04 2024

a(n) + a(n-1) = A000010(n).

a(n) = (-1)^n * (1 - A068773(n)) for n >= 1. - Amiram Eldar, Mar 05 2024

a(0)=1 prepended by R. J. Mathar, Jun 20 2021

A273960 a(n) = (-1)^n*prime(n).

Original entry on oeis.org

-2, 3, -5, 7, -11, 13, -17, 19, -23, 29, -31, 37, -41, 43, -47, 53, -59, 61, -67, 71, -73, 79, -83, 89, -97, 101, -103, 107, -109, 113, -127, 131, -137, 139, -149, 151, -157, 163, -167, 173, -179, 181, -191, 193, -197, 199, -211, 223, -227, 229, -233, 239, -241, 251, -257, 263, -269, 271

Offset: 1

Terry D. Grant, Jun 12 2016

sign

			a(1) = 2*(-1)^1 = -2.
a(2) = 3*(-1)^2 = 3.
a(3) = 5*(-1)^3 = -5.

Cf. A000040, A083236.

Table[Prime[n]*(-1)^n,{n,1,70}]
Table[If[OddQ[n],-Prime[n],Prime[n]],{n,70}] (* Harvey P. Dale, Jul 01 2016 *)

a(n) = prime(n)*(-1)^n \\ Felix Fröhlich, Jun 15 2016

a(n) = A000040(n)*A033999(n). - Felix Fröhlich, Jun 15 2016

cp's OEIS Frontend

A128280 a(n) is the least number not occurring earlier such that a(n)+a(n-1) is prime, a(0) = 0.

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A083237 First order recursion: a(0)=5; a(n)=prime(n)-a(n-1).

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A083238 First order recursion: a(0)=1; a(n) = sigma(1,n) - a(n-1).

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A083239 First order recursion: a(0) = 1; a(n) = phi(n) - a(n-1) = A000010(n) - a(n-1).

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A273960 a(n) = (-1)^n*prime(n).

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