A265109 -id:A265109

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A035346

Let F(n) = Q(n) - P(n) be the Fortunate numbers (A005235); sequence gives n such that F(n) = prime(n+1).

+10
6

1, 2, 3, 6, 7, 8, 14, 16, 17, 21, 73, 801, 1971, 3332, 3469, 3509, 4318, 7986, 41292

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Positive n such that A002110(n) + A000040(n+1) is prime. - Robert Israel, Dec 02 2015

Subsequence of A265109. - Altug Alkan, Dec 02 2015

LINKS

Table of n, a(n) for n=1..19.

Antonín Čejchan, Michal Křížek, and Lawrence Somer, On Remarkable Properties of Primes Near Factorials and Primorials, Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.4.

S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209-210.

EXAMPLE

a(10) = 21 because A002110(21) + prime(22) = 40729680599249024150621323549 = 2*3*5*...*67*71*73 + 79 is prime.

MAPLE

p:= 3:

A[1]:= 1:

count:= 1:

Primorial:= 2:

for n from 2 to 1000 do

Primorial:= Primorial*p;

p:= nextprime(p);

if isprime(Primorial + p) then

count:= count+1;

A[count]:= n;

od:

seq(A[i], i=1..count); # Robert Israel, Dec 02 2015

MATHEMATICA

Select[Range@ 801, PrimeQ[Product[Prime@ k, {k, #}] + Prime[# + 1]] &] (* Michael De Vlieger, Dec 02 2015 *)

PROG

(PARI) lista(nn) = {s = 1; for(k=1, nn, s *= prime(k); if(ispseudoprime(s + prime(k+1)), print1(k, ", ")); ); } \\ Altug Alkan, Dec 02 2015

CROSSREFS

Cf. A000040, A002110, A005235, A006862, A035345, A265109.

KEYWORD

nonn,more

AUTHOR

N. J. A. Sloane

EXTENSIONS

a(10)-a(11) were found by Labos Elemer, May 02 2000

a(12) from Ralf Stephan, Oct 20 2002

Offset changed by Altug Alkan, Dec 02 2015

a(13) from Michael De Vlieger, Dec 02 2015

a(14)-a(18) from Altug Alkan, Dec 02 2015

a(19) from Henri Lifchitz, Nov 08 2024

STATUS

approved

A100380

a(n) = least k such that prime(n) + A002110(k) is prime.

+10
3

0, 1, 1, 2, 1, 2, 1, 4, 2, 1, 2, 2, 1, 3, 2, 2, 1, 2, 2, 1, 2, 3, 2, 5, 2, 1, 2, 1, 3, 5, 3, 2, 1, 4, 1, 2, 2, 3, 2, 2, 1, 3, 1, 2, 1, 3, 3, 2, 1, 4, 2, 1, 3, 2, 2, 2, 1, 2, 2, 1, 3, 4, 2, 1, 4, 3, 2, 3, 1, 3, 2, 3, 2, 2, 3, 2, 3, 4, 3, 3, 1, 4, 1, 2, 5, 2, 3, 2, 1, 4, 4, 3, 5, 3, 4, 2, 4, 1, 4, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

Conjecture: every prime number can be written as +- p(n) -+ p(k)# where p(i)=i-th prime, p(i)#=i-th primorial.

The sequence grows remarkably slowly. The largest number occurring within the first 50000 elements is 90. - Stefan Steinerberger, Apr 10 2006

a(1) = 0 is the minimum value of a(n). It is also unrepeated in this sequence. - Altug Alkan, Dec 02 2015

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000 (corrected by Ray Chandler, Jan 19 2019)

EXAMPLE

p(8)=19;

19 + 2 = 21 = 3*7,

19 + 6 = 25 = 5*5, and

19 + 30 = 49 = 7*7, but

19 + 210 = 229, which is prime; 210=p(4)#, so a(8)=4.

MAPLE

primorial:= proc(n) option remember: ithprime(n)*procname(n-1) end proc:

primorial(0):= 1:

f:= proc(n) local k, p;

p:= ithprime(n);

for k from 0 do if isprime(p+primorial(k)) then return k fi od:

end proc:

map(f, [$1..100]); # Robert Israel, Aug 27 2015

MATHEMATICA

Table[k := 0; While[Not[PrimeQ[Prime[n]+Product[Prime[i], {i, 1, k}]]], k++ ]; k, {n, 1, 100}] (* Stefan Steinerberger, Apr 10 2006 *)

PROG

(PARI) primo(n) = prod(i=1, n, prime(i));

a(n) = {k=0; while(!isprime(prime(n)+primo(k)), k++); k; } \\ Michel Marcus, Aug 27 2015

CROSSREFS

Cf. A002110, A265109.

KEYWORD

easy,nonn

AUTHOR

Pierre CAMI, Dec 30 2004

EXTENSIONS

More terms from Stefan Steinerberger, Apr 10 2006

a(1) = 0 added and name edited by Altug Alkan, Dec 02 2015

STATUS

approved

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