A040976 -id:A040976 - cp's OEIS frontend

A244797 Number of moduli m such that (prime(n) mod m) = 2.

0, 0, 1, 1, 2, 1, 3, 1, 3, 3, 1, 3, 3, 1, 5, 3, 3, 1, 3, 3, 1, 3, 4, 3, 3, 5, 1, 7, 1, 3, 3, 3, 7, 1, 5, 1, 3, 3, 7, 5, 3, 1, 7, 1, 7, 1, 3, 3, 8, 1, 7, 3, 1, 3, 7, 5, 3, 1, 5, 5, 1, 3, 3, 3, 1, 11, 3, 3, 7, 1, 7, 7, 3, 3, 3, 3, 5, 3, 7, 3, 3, 1, 7, 1, 3, 8

Offset: 1

Clark Kimberling, Jul 06 2014

Except for the initial 0,0, this is column 2 of the array at A244740.

			prime(5) = 11 = (2 mod m) for m = 3, 9 so that a(5) = 2.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A244740, A244796, A244798.

Cf. A032741, A040976.

z = 300; f[n_, m_] := If[Mod[Prime[n], m] == 2, 1, 0];
t = Table[f[n, m], {n, 1, z}, {m, 1, Prime[n]}];
Table[Count[t[[k]], 1], {k, 1, z}] (* A244797 *)

a(n) = A032741(A040976(n)). - Ridouane Oudra, Mar 17 2024

A249426 Record values in A249150.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 17, 21, 27, 29, 35, 39, 41, 45, 51, 57, 59, 62, 65, 69, 71, 77, 79, 81, 83, 87, 89, 95, 99, 101, 104, 105, 107, 111, 118, 125, 129, 131, 135, 137, 143, 147, 149, 153, 155, 159, 161, 164, 165, 167, 171, 177, 179, 181, 189, 191, 194, 195, 197, 199, 207, 209, 219, 221, 225, 227, 231, 237, 239, 249, 255

Offset: 1

Antti Karttunen, Oct 28 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..999

One less than A249427.
Cf. A249150, A249425.
Differs from A040976 a(n) = prime(n) - 2 for the first time at n=19, where a(n) = 62, while A040976(19) = 65. And larger terms differs only a few times.

(define (A249426 n) (A249150 (A249425 n)))

a(n) = A249150(A249425(n)).

a(n) = A249427(n) - 1.

A140777 a(n) = 2*prime(n) - 4.

Original entry on oeis.org

0, 2, 6, 10, 18, 22, 30, 34, 42, 54, 58, 70, 78, 82, 90, 102, 114, 118, 130, 138, 142, 154, 162, 174, 190, 198, 202, 210, 214, 222, 250, 258, 270, 274, 294, 298, 310, 322, 330, 342, 354, 358, 378, 382, 390, 394, 418, 442, 450, 454, 462, 474, 478, 498, 510, 522

Offset: 1

Leroy Quet, May 29 2008, May 31 2008

A number n is included if (p + n/p) is prime, where p is the smallest prime that divides n. Since all terms of this sequence are even (or otherwise p + n/p would be even and not a prime), p is always 2. So this sequence is the set of all even numbers n where (2 + n/2) is prime.
The entries are also encountered via the bilinear transform approximation to the natural log (unit circle). Specifically, evaluating 2(x-1)/(x+1) at x = 2, 3, 4, ..., the terms of this sequence are seen ahead of each new prime encountered. Additionally, the position of those same primes will occur at the entry positions. For clarity, the evaluation output is 2, 3, 1, 1, 6, 5, 4, 3, 10, 7, 3, 2, 14, 9, 8, 5, 18, 11, ..., where the entries ahead of each new prime are 2, 6, 10, 18, ... . As an aside, the same mechanism links this sequence to A165355. - Bill McEachen, Jan 08 2015
As a follow-up to previous comment, it appears that the numerators and denominators of 2(x-1)/(x+1) are respectively given by A145979 and A060819, but with different offsets. - Michel Marcus, Jan 14 2015
Subset of the union of A017641 & A017593. - Michel Marcus, Sep 01 2020

			The smallest prime dividing 42 is 2. Since 2 + 42/2 = 23 is prime, 42 is included in this sequence.

Cf. A000040, A020639, A040976, A111234, A140775, A140776.
Cf. A060819, A145979, A165355.
Cf. A017641, A017593.

[2*NthPrime(n)-4: n in [1..80]]; // Vincenzo Librandi, Feb 19 2015

A020639 := proc(n) local dvs,p ; dvs := sort(convert(numtheory[divisors](n),list)) ; for p in dvs do if isprime(p) then RETURN(p) ; fi ; od: error("%d",n) ; end: A111234 := proc(n) local p ; p := A020639(n) ; p+n/p ; end: isA140777 := proc(n) RETURN(isprime(A111234(n))) ; end: for n from 2 to 1200 do if isA140777(n) then printf("%d,",n) ; fi ; od: # R. J. Mathar, May 31 2008
seq(2*ithprime(i)-4, i=1..1000); # Robert Israel, Jan 09 2015

fQ[n_] := Block[{p = First@ First@ Transpose@ FactorInteger@ n}, PrimeQ[p + n/p] == True]; Select[ Range[2, 533], fQ@# &] (* Robert G. Wilson v, May 30 2008 *)
Table[2 Prime[n] - 4, {n, 60}] (* Vincenzo Librandi, Feb 19 2015 *)

vector(100, n, 2*prime(n) - 4) \\ Michel Marcus, Jan 09 2015

a(n) = 2*A040976(n). - Michel Marcus, Jan 09 2015

More terms from Robert G. Wilson v and R. J. Mathar, May 30 2008

A215747 a(n) = (-2)^n mod n.

Original entry on oeis.org

0, 0, 1, 0, 3, 4, 5, 0, 1, 4, 9, 4, 11, 4, 7, 0, 15, 10, 17, 16, 13, 4, 21, 16, 18, 4, 1, 16, 27, 4, 29, 0, 25, 4, 17, 28, 35, 4, 31, 16, 39, 22, 41, 16, 28, 4, 45, 16, 19, 24, 43, 16, 51, 28, 12, 32, 49, 4, 57, 16, 59, 4, 55, 0, 33, 64, 65, 16, 61, 44, 69, 64, 71, 4, 7

Offset: 1

Alex Ratushnyak, Aug 23 2012

n^(n+2) mod (n+2) is essentially the same.
Indices of 0's: 2^k - 1, k>=0.
Indices of 1's: A006521 except the first term.
Indices of 3's: A015940.
Indices of 5's: 7, 133, 1517, 11761, ...
a(A000040(n)) = A000040(n)-2 = A040976(n).

			a(5) = (-2)^5 mod 5 = -32 mod 5 = 3.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A015910, A082493, A082494, A110146, A213381.

a:= n-> (-2)&^n mod n:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 08 2015

a[n_]:=Mod[(-2)^n ,n]; Array[a,75] (* Stefano Spezia, Aug 25 2025 *)

for n in range(1, 333):
    print((-2)**n % n, end=',')

A359262 a(n) is the largest number m such that prime(n)^m is in A359260.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 3, 1, 1, 5, 3, 1, 3, 1, 1, 1, 5, 3, 1, 3, 3, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 3, 1, 5, 3, 3, 1, 1, 1, 5, 1, 3, 1, 3, 9, 3, 1, 3, 1, 1, 5, 1, 1, 1, 1, 5, 3, 1, 3, 1, 3, 1, 3, 1, 5, 3, 1, 3, 1, 1, 3, 3, 3, 1, 1, 3, 1, 3, 1, 9, 1, 3, 3, 1, 1

Offset: 1

Amiram Eldar, Dec 23 2022

a(n) is the largest number m such that the arithmetic mean of {1, p, p^2, ..., p^k} is an integer for all k in 1..m.
Apparently, all the terms are of the form prime(k)-2 (A040976). Conjecture: The asymptotic density of the occurrences of prime(k)-2 is (1/s(k-1)-1/s(k)), where s(k) = A005867(k) = phi(prime(k)#), and prime(k)# is the k-th primorial number (A002110).
The sums of the first 10^k terms, for k = 1, 2, ..., are 15, 221, 2291, 23287, 233641, 2337007, 23379901, 233814475, 2338211029, 23382168187, ... . If the mentioned above conjecture is correct, then the asymptotic mean of this sequence is Sum_{k>=1} (prime(k)-2)*(1/s(k-1)-1/s(k)) = 2.33821872365981424748... .
Apparently, the indices of records after n = 1 occur at A000720(A073917(n)) (verified for the first 12 terms of A073917) with record values a(A000720(A073917(n))) = prime(n+1) - 2 (verified for the first 150 terms of A073917).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000720, A002110, A040976, A073917, A005867, A359260, A359261.

Cf. A002476, A073102, A132230.

a[n_] := Module[{p = Prime[n], k = 1, r = s = 1}, While[Divisible[s, k], k++; r *= p; s += r]; k - 2]; Array[a, 100]

a(n) = {my(p = prime(n), k = 1, r = s = 1); while(!(s%k), k++; r *= p; s += r); k - 2; }

a(n) >= 1 for n >= 2.
a(n) >= 3 iff prime(n) == 1 (mod 6) (prime(n) is in A002476).
Conjectures:
a(n) >= 5 iff prime(n) == 1 (mod 30) (prime(n) is in A132230).
a(n) >= 9 iff prime(n) == 1 (mod 210) (prime(n) is in A073102).
a(n) >= prime(k) - 2 iff prime(n) == 1 (mod A002110(k-1)).

A360735 Even integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has only two elements.

Original entry on oeis.org

16, 22, 26, 32, 44, 46, 52, 56, 58, 62, 70, 74, 76, 82, 86, 88, 92, 100, 106, 112, 116, 118, 122, 128, 130, 136, 140, 142, 146, 148, 152, 158, 160, 166, 170, 172, 176, 182, 184, 194, 196, 200, 202, 206, 212, 214, 218, 224, 226, 232, 236, 242, 244, 250, 254, 256, 262, 266, 268

Offset: 1

Bernard Schott, Feb 19 2023

nonn

Similar sequence with odd integers d is A040976 \ {0}.
Terms are even numbers that are not divisible by 3 and that are not also in A206037.
These longest corresponding APs are of the form (q, q+d) with q odd primes (see examples).
This subsequence of A359408 corresponds to the second case '2 is one less than prime 3' (see A173919); the first case is linked to A040976.
A342309(d) gives the first element of the smallest such AP with 2 elements whose common difference is a(n) = d.

			d = 16 is a term because the first longest APs of primes with common difference 16 are (3, 19), (7,23), (13, 29), ... and all have 2 elements because next elements should be respectively 35, 39 and 45 that are all composite; the first such AP that starts with A342309(16) = 3 is (3, 19).
d = 22 is a term because the first longest APs of primes with common difference 22 are (7, 29), (19, 41), (31, 53), ... and all have 2 elements because next elements should be respectively 51, 63 and 75 that are all composite; the first such AP that starts with A342309(22) = 7 is (7, 29).

Equals A359408 \ A040976.

Cf. A047235, A123556, A206037, A342309.

filter := d -> (irem(d, 2) = 0) and (irem(d, 3) <> 0) and not isprime(3+d) or isprime(3+d) and not isprime(3+2*d) : select(filter, [`$`(1 .. 270)]);
isA360735 := d -> isA047235(d) and not isA206037(d): # Peter Luschny, Mar 03 2023

Select[Range[2, 270, 2], Mod[#, 3] > 0 && Nand @@ PrimeQ[{# + 3, 2*# + 3}] &] (* Amiram Eldar, Mar 03 2023 *)

isok(d) = !(d%2) && (d%3) && !(isprime(d+3) && isprime(2*d+3)); \\ Michel Marcus, Mar 03 2023

If m is a term then A123556(m) = 2, but the converse is false: a counterexample is A123556(11) = 2 and 11 is not a term.

A360825 a(n) is the remainder after dividing n! by its least nondivisor.

Original entry on oeis.org

1, 1, 2, 2, 4, 1, 6, 2, 5, 1, 10, 1, 12, 3, 8, 1, 16, 1, 18, 4, 11, 1, 22, 22, 6, 5, 14, 1, 28, 1, 30, 33, 20, 31, 18, 1, 36, 7, 20, 1, 40, 1, 42, 8, 23, 1, 46, 19, 11, 9, 26, 1, 52, 30, 27, 10, 29, 1, 58, 1, 60, 43, 53, 56, 33, 1, 66, 12, 35, 1, 70, 1, 72, 27, 23

Offset: 0

Sebastian F. Orellana, Feb 22 2023

nonn

For every term besides a(3), the least nondivisor is the next prime after n.

			a(5) = 5! mod 7 = 120 mod 7 = 1.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000040, A000142, A006093, A040976, A151800, A213636, A275111, A360805.

a[n_] := Module[{f = n!, m = n + 1}, While[Divisible[f, m], m++]; Mod[f, m]]; Array[a, 100, 0] (* Amiram Eldar, Feb 22 2023 *)

a(n) = my(k=1, r); while(!(r=(n! % (n+k))), k++); r; \\ Michel Marcus, Feb 22 2023

from functools import reduce
from sympy import nextprime
def A360825(n):
    if n == 3: return 2
    m = nextprime(n)
    return reduce(lambda i, j: i*j%m,range(2,n+1),1)%m # Chai Wah Wu, Feb 22 2023

from functools import reduce
from sympy import nextprime
def A360825(n):
    if n == 3: return 2
    m = nextprime(n)
    return (m-1)*pow(reduce(lambda i,j:i*j%m,range(n+1,m),1),-1,m)%m # Chai Wah Wu, Feb 23 2023

a(n) = 1 <=> n in { A040976 } \ { 3 }.
a(n) = n <=> n in { A006093 }.
a(n) = n! mod A151800(n) for n > 3.
a(n) = A213636(n!) = A213636(A000142(n)).
a(A000040(n)) = A275111(n) for n >= 3.
a(n) > n <=> n in { A360805 }.

A090321 T(n,k) = prime(n+1) - prime(n-k+1), 1<=k<=n, triangular array read by rows.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 4, 6, 8, 9, 2, 6, 8, 10, 11, 4, 6, 10, 12, 14, 15, 2, 6, 8, 12, 14, 16, 17, 4, 6, 10, 12, 16, 18, 20, 21, 6, 10, 12, 16, 18, 22, 24, 26, 27, 2, 8, 12, 14, 18, 20, 24, 26, 28, 29, 6, 8, 14, 18, 20, 24, 26, 30, 32, 34, 35, 4, 10, 12, 18, 22, 24, 28, 30, 34, 36, 38

Offset: 1

Reinhard Zumkeller, Jan 26 2004

Row-reversed A086800. - R. J. Mathar, Sep 15 2012

			1;
2, 3;
2, 4, 5;
4, 6, 8, 9;
2, 6, 8, 10, 11;
4, 6, 10, 12, 14, 15;

Eric Weisstein's World of Mathematics, Prime Difference Function

Cf. A000040.

T(n,1) = A001223(n); T(n,n) = A040976(n).

A122585 Reciprocal of n modulo smallest prime greater than n.

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 8, 7, 5, 10, 6, 12, 4, 11, 8, 16, 9, 18, 17, 15, 11, 22, 24, 23, 7, 19, 14, 28, 15, 30, 6, 22, 9, 12, 18, 36, 10, 27, 20, 40, 21, 42, 35, 31, 23, 46, 44, 21, 13, 35, 26, 52, 49, 47, 44, 39, 29, 58, 30, 60, 11, 40, 50, 22, 33, 66, 53, 47, 35, 70, 36, 72, 13, 63, 59

Offset: 1

Franklin T. Adams-Watters, Oct 20 2006

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
Eric Weisstein's World of Mathematics, Modular Inverse

Cf. A006093, A040976, A124223, A151800, A362254.

a:= n-> n&^(-1) mod nextprime(n):
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 13 2023

Table[PowerMod[n,-1,NextPrime[n]],{n,80}] (* Harvey P. Dale, Apr 13 2023 *)

from sympy import nextprime
def A122585(n): return pow(n,-1,nextprime(n)) # Chai Wah Wu, Apr 13 2023

From Alois P. Heinz, Apr 13 2023: (Start)
a(n) = n <=> n in { A006093 }.
a(n) = (n+1)/2 <=> n in { A040976 } \ { 0 }. (End)

A123317 Smallest prime power m such that n+m is a prime number.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 4, 3, 2, 1, 8, 5, 4, 3, 2, 1, 2, 1, 16, 5, 4, 3, 2, 1, 4, 3, 2, 1, 2, 1, 4, 3, 2, 1, 32, 5, 4, 3, 2, 1, 8, 5, 4, 3, 2, 1, 2, 1, 256, 5, 4, 3, 2, 1, 4, 3, 2, 1, 2, 1, 16, 5, 4, 3, 2, 1, 4, 3, 2, 1, 128, 5, 4, 3, 2, 1, 8, 7, 16, 5, 4, 3, 2, 1, 4, 3, 2, 1, 2, 1

Offset: 1

Reinhard Zumkeller, Sep 27 2006

nonn

			n=23: 23+1=3*2^3, 23+2=5^2, 23+3=13*2, 23+2^2=3^3, 23+5=7*2^2, 23+7=5*3*2, but 23+8=31=A000040(11), therefore a(23)=8;
n=24: 24+1=5^2, 24+2=13*2, 24+3=3^3, 24+2^2=7*2^2, but 24+5=29=A000040(10), therefore a(24)=5;
the smallest occurring proper odd prime power is 9=3^2:
n=118: 118+1=17*7, 118+2=5*3*2^3, 118+3=11^2, 118+2^2=61*2, 118+5=41*3, 118+7=5^3, 118+2^3=7*2*3^2, but 118+3^2=127=A000040(31), therefore a(118)=9.

Cf. A013632, A000961.

A123317 := proc(n)
    local m ;
    m :=1 ;
    if isprime(n+m) then
        return m ;
    end if;
    for m from 2 do
        if nops(numtheory[factorset](m)) = 1 then
            if isprime(n+m) then
                return m;
            end if;
        end if;
    end do:
end proc:
seq(A123317(n),n=1..102) ; # R. J. Mathar, Aug 09 2019

A123318(n) = n + a(n);

a(A006093(n)) = 1; a(A040976(n)) = 2 for n>2.

cp's OEIS Frontend

A244797 Number of moduli m such that (prime(n) mod m) = 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A249426 Record values in A249150.

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

A140777 a(n) = 2*prime(n) - 4.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A215747 a(n) = (-2)^n mod n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

A359262 a(n) is the largest number m such that prime(n)^m is in A359260.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A360735 Even integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has only two elements.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A360825 a(n) is the remainder after dividing n! by its least nondivisor.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs