Simon Strandgaard - cp's OEIS frontend

A350493 a(n) = floor(sqrt(prime(n)))^2 mod n.

0, 1, 1, 0, 4, 3, 2, 0, 7, 5, 3, 0, 10, 8, 6, 1, 15, 13, 7, 4, 1, 20, 12, 9, 6, 22, 19, 16, 13, 10, 28, 25, 22, 19, 4, 0, 33, 30, 27, 9, 5, 1, 40, 37, 16, 12, 8, 4, 29, 25, 21, 17, 13, 9, 36, 32, 28, 24, 20, 16, 12, 41, 37, 33, 29, 25, 56, 52, 48, 44, 40, 36

Offset: 1

Simon Strandgaard, Jan 01 2022

nonn

			a(4) = A065730(4) mod 4 =  4 mod 4 = 0;
a(5) = A065730(5) mod 5 =  9 mod 5 = 4;
a(6) = A065730(6) mod 6 =  9 mod 6 = 3;
a(7) = A065730(7) mod 7 = 16 mod 7 = 2.

Cf. A000040, A048760, A065730.

Table[PowerMod[Floor[Sqrt[Prime[n]]],2,n],{n,72}] (* Stefano Spezia, Jan 02 2022 *)

a(n) = (sqrtint(prime(n))^2) % n;
vector(20,n,a(n))

from sympy import prime, integer_nthroot
def a(n): return (integer_nthroot(prime(n), 2)[0]**2)%n
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, Jan 02 2022

require 'prime'
values = []
Prime.first(20).each_with_index do |prime, i|
    values << ((Integer.sqrt(prime) ** 2) % (i + 1))
end
p values

a(n) = A065730(n) mod n.

A349487 a(n) = A132739((n-5)*(n+5)).

Original entry on oeis.org

11, 24, 39, 56, 3, 96, 119, 144, 171, 8, 231, 264, 299, 336, 3, 416, 459, 504, 551, 24, 651, 704, 759, 816, 7, 936, 999, 1064, 1131, 48, 1271, 1344, 1419, 1496, 63, 1656, 1739, 1824, 1911, 16, 2091, 2184, 2279, 2376, 99, 2576, 2679, 2784, 2891, 24, 3111

Offset: 6

Simon Strandgaard, Nov 19 2021

nonn

Shares 614 initial terms with A061043. First difference is A061043(620)=615 vs. a(620)=123.

			a(9)  = A132739(( 9-5)*( 9+5)) = A132739(56) = 56,
a(10) = A132739((10-5)*(10+5)) = A132739(75) = 3,
a(11) = A132739((11-5)*(11+5)) = A132739(96) = 96.

Cf. A061043, A098603, A132739.

Table[Last@Select[Divisors[(n - 5)*(n + 5)], Mod[#, 5] != 0 &], {n, 6,
   56}] (* Giorgos Kalogeropoulos, Nov 19 2021 *)
Table[(n - 5)*(n + 5)/5^IntegerExponent[(n - 5)*(n + 5), 5], {n, 6, 56}] (* Amiram Eldar, Nov 22 2021 *)

A132739(n)=n/5^valuation(n, 5);
a(n) = A132739((n-5)*(n+5));
[a(n)|n<-[6..25]]

def A349487(n):
    a, b = divmod(n*n-25, 5)
    while b == 0:
        a, b = divmod(a,5)
    return 5*a+b # Chai Wah Wu, Dec 05 2021

p (6..25).map { |n| x = (n-5)*(n+5); x /= 5 while (x % 5) == 0; x }

a(n) = A132739(A098603(n-5)).

A348762 a(n) = A000265((n-8)*(n+8)).

Original entry on oeis.org

17, 9, 57, 5, 105, 33, 161, 3, 225, 65, 297, 21, 377, 105, 465, 1, 561, 153, 665, 45, 777, 209, 897, 15, 1025, 273, 1161, 77, 1305, 345, 1457, 3, 1617, 425, 1785, 117, 1961, 513, 2145, 35, 2337, 609, 2537, 165, 2745, 713, 2961, 3, 3185, 825, 3417, 221, 3657

Offset: 9

Simon Strandgaard, Oct 31 2021

nonn

Shares 495 initial terms with A061049. First difference is A061049(504)=62 vs. a(504)=31.

			a( 9) = A000265(( 9-8)*( 9+8)) = A000265( 17) = 17,
a(10) = A000265((10-8)*(10+8)) = A000265( 36) = 9,
a(11) = A000265((11-8)*(11+8)) = A000265( 57) = 57,
a(12) = A000265((12-8)*(12+8)) = A000265( 80) = 5,
a(13) = A000265((13-8)*(13+8)) = A000265(105) = 105.

Cf. A061049, A069834, A000265, A098849.

a[n_] := (n - 8)*(n + 8)/2^IntegerExponent[(n - 8)*(n + 8), 2]; Array[a, 53, 9] (* Amiram Eldar, Nov 22 2021 *)

A000265(n) = n >> valuation(n, 2);
a(n) = A000265((n-8)*(n+8));
[a(n)|n<-[9..27]]

def A348762(n):
    a, b = divmod(n*n-64, 2)
    while b == 0:
        a, b = divmod(a,2)
    return 2*a+b # Chai Wah Wu, Dec 05 2021

p (9..27).map { |n| x = (n-8)*(n+8); x /= 2 while x.even?; x }

a(n) = A000265(A098849(n-8)).

A348287 Arrange nonzero digits of n in increasing order then append the zeros.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 12, 22, 23, 24, 25, 26, 27, 28, 29, 30, 13, 23, 33, 34, 35, 36, 37, 38, 39, 40, 14, 24, 34, 44, 45, 46, 47, 48, 49, 50, 15, 25, 35, 45, 55, 56, 57, 58, 59, 60, 16, 26, 36, 46, 56, 66

Offset: 0

Simon Strandgaard, Oct 10 2021

Shares 101 initial terms with A328447. First difference is A328447(101)=101 vs A348287(101)=110.

			a(1010) = 1100,
a(1234567890) = 1234567890,
a(9876543210) = 1234567890.

Cf. A004185, A328447.

a[n_] := 10^DigitCount[n, 10, 0] * FromDigits[Sort[IntegerDigits[n]]]; Array[a, 100, 0] (* Amiram Eldar, Oct 10 2021 *)

a(n) = fromdigits(vecsort(digits(n), x->if(x,x,10)));

def a(n): s = str(n); return int("".join(sorted(s)))*10**s.count('0')
print([a(n) for n in range(68)]) # Michael S. Branicky, Oct 10 2021

p Array.new(40) { |n|
  n.to_s.gsub('0','a').split(//).sort.join.gsub('a','0').to_i
}

A347342 a(n) = prime(n) mod floor(prime(n) / n).

Original entry on oeis.org

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

Offset: 1

Simon Strandgaard, Aug 27 2021

nonn

			a(1) =  2 mod floor( 2 / 1) =  2 mod 2 = 0,
a(2) =  3 mod floor( 3 / 2) =  3 mod 1 = 0,
a(3) =  5 mod floor( 5 / 3) =  5 mod 1 = 0,
a(4) =  7 mod floor( 7 / 4) =  7 mod 1 = 0,
a(5) = 11 mod floor(11 / 5) = 11 mod 2 = 1.

Simon Strandgaard, Plot of 1000 terms

Cf. A000040, A038605.

A347342[n_] := Mod[Prime[n] , Floor[Prime[n]/n]]; Table[A347342[n], {n, 1, 86}] (* Robert P. P. McKone, Aug 27 2021 *)

PARI
```
a(n) = prime(n) % (prime(n) \ n);
```

require 'prime'
values = []
Prime.first(30).each_with_index do |prime,i|
    values << prime % (prime/(i+1))
end
p values

a(n) = A000040(n) mod A038605(n).

A347112 a(n) = concat(prime(n+1),n) mod prime(n).

Original entry on oeis.org

1, 1, 3, 2, 3, 7, 10, 10, 0, 7, 22, 5, 8, 27, 4, 33, 40, 8, 17, 7, 37, 27, 42, 23, 37, 24, 15, 14, 102, 74, 50, 108, 96, 61, 86, 32, 9, 112, 138, 121, 62, 137, 52, 58, 48, 52, 192, 2, 22, 221, 185, 13, 89, 152, 141, 130, 257, 116, 182, 260, 212, 290, 156, 264

Offset: 1

Simon Strandgaard, Aug 18 2021

			a(1) = concat( 3,1) mod  2 = 1,
a(2) = concat( 5,2) mod  3 = 1,
a(3) = concat( 7,3) mod  5 = 3,
a(4) = concat(11,4) mod  7 = 2,
a(5) = concat(13,5) mod 11 = 3.

Simon Strandgaard, Plot of 10000 terms

Cf. A000040, A075110.

Array[Mod[#3*10^(1 + Floor[Log10[#1]]) + #1, #2] & @@ {#, Prime[#], Prime[# + 1]} &, 64] (* Michael De Vlieger, Aug 18 2021 *)

a(n) = eval(Str(prime(n+1),n)) % prime(n);

from sympy import prime
def a(n): return int(str(prime(n+1)) + str(n))%prime(n)
print([a(n) for n in range(1, 65)]) # Michael S. Branicky, Aug 18 2021

require 'prime'
values = []
primes = Prime.first(50)
primes.each_index do |n|
  next if n < 1
  values << (primes[n].to_s + n.to_s).to_i % primes[n-1]
end
p values

A345727 a(n) = (prime(n)+1) * prime(n+1).

Original entry on oeis.org

9, 20, 42, 88, 156, 238, 342, 460, 696, 930, 1184, 1558, 1806, 2068, 2544, 3186, 3660, 4154, 4828, 5256, 5846, 6640, 7476, 8730, 9898, 10506, 11128, 11772, 12430, 14478, 16768, 18084, 19182, 20860, 22650, 23864, 25754, 27388, 29064, 31146, 32580, 34762

Offset: 1

Simon Strandgaard, Jul 20 2021

nonn

			a(1) = (prime(1)+1) * prime(2) = 3 *  3 =  9,
a(2) = (prime(2)+1) * prime(3) = 4 *  5 = 20,
a(3) = (prime(3)+1) * prime(4) = 6 *  7 = 42,
a(4) = (prime(4)+1) * prime(5) = 8 * 11 = 88.

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

Cf. A000040, A006094, A008864, A123134, A180617.

A345727 := proc(n)
    (ithprime(n)+1)*ithprime(n+1) ;
end proc:
seq(A345727(n),n=1..10) ; # R. J. Mathar, Aug 16 2021

(Prime@#+1)Prime[#+1]&/@Range@50 (* Giorgos Kalogeropoulos, Jul 23 2021 *)
(#[[1]]+1)#[[2]]&/@Partition[Prime[Range[50]],2,1] (* Harvey P. Dale, Jan 08 2023 *)

for(n=1, 100, print1((prime(n)+1)*prime(n+1), ", "))

require 'prime'
values = []
primes = Prime.first(20)
primes.each_index do |n|
    next if n < 1
    values << (primes[n - 1] + 1) * primes[n]
end
p values

a(n) = A008864(n)*A000040(n+1).
a(n) = A180617(n)-A008864(n).
a(n) = A006094(n)+A000040(n+1).

A345366 a(n) = (p*q+1) mod (p+q) where p=prime(n) and q=prime(n+1).

Original entry on oeis.org

2, 0, 0, 6, 0, 12, 0, 18, 44, 0, 60, 36, 0, 42, 92, 104, 0, 120, 66, 0, 144, 78, 164, 78, 96, 0, 102, 0, 108, 192, 126, 260, 0, 264, 0, 300, 312, 162, 332, 344, 0, 348, 0, 192, 0, 170, 182, 222, 0, 228, 464, 0, 468, 500, 512, 524, 0, 540, 276, 0, 552, 552, 306

Offset: 1

Simon Strandgaard, Jun 16 2021

nonn

The graph of this function consists of three branches: the upper one corresponds to cases where q-p == 2 (mod 4) except the twin primes, the middle one to cases where q-p == 0 (mod 4), and the lower one (where a(n)=0) to cases where q-p = 2, the twin primes.

All terms are even.

			a(1) = ( 2* 3+1) mod ( 2+ 3) =   7 mod  5 = 2,
a(2) = ( 3* 5+1) mod ( 3+ 5) =  16 mod  8 = 0,
a(3) = ( 5* 7+1) mod ( 5+ 7) =  36 mod 12 = 0,
a(4) = ( 7*11+1) mod ( 7+11) =  78 mod 18 = 6,
a(5) = (11*13+1) mod (11+13) = 144 mod 24 = 0.

Cf. A000040, A212769, A029707 (indices of 0's).

Cf. A001043, A023523.

a:= n-> ((p, q)-> irem(p*q+1, p+q))(map(ithprime, [n, n+1])[]):
seq(a(n), n=1..63);  # Alois P. Heinz, Jul 03 2021

Mod[#1*#2 + 1, #1 + #2] & @@@ Partition[Select[Range[300], PrimeQ], 2, 1] (* Amiram Eldar, Jun 16 2021 *)

a(n)=my(p=prime(n), q=nextprime(p+1)); (p*q+1)%(p+q)

from sympy import nextprime
def aupton(nn):
    alst, p, q = [], 2, 3
    while len(alst) < nn: alst.append((p*q+1)%(p+q)); p, q = q, nextprime(q)
    return alst
print(aupton(62)) # Michael S. Branicky, Jun 16 2021

require 'prime'
values = []
Prime.first(21).each_cons(2) do |a, b|
    values << (a * b + 1) % (a + b)
end
p values

a(n) = A023523(n+1) mod A001043(n). - Michel Marcus, Jun 17 2021

A344348 a(n) = floor(frac(e * n) * n).

Original entry on oeis.org

0, 0, 0, 0, 3, 2, 1, 0, 5, 4, 1, 9, 7, 4, 0, 11, 7, 3, 16, 12, 7, 1, 17, 11, 5, 23, 17, 10, 3, 24, 16, 8, 31, 23, 14, 4, 30, 21, 11, 0, 29, 18, 7, 38, 26, 14, 1, 35, 22, 9, 45, 32, 18, 3, 42, 27, 12, 53, 38, 22, 5, 49, 33, 15, 62, 44, 26, 8, 57, 38, 19, 70

Offset: 0

Simon Strandgaard, May 15 2021

nonn

			x(n) = frac(e * n),
a(0) = floor(x(0) * 0) = floor(0.00 * 0) = 0,
a(1) = floor(x(1) * 1) = floor(0.72 * 1) = 0,
a(2) = floor(x(2) * 2) = floor(0.44 * 2) = 0,
a(3) = floor(x(3) * 3) = floor(0.15 * 3) = 0,
a(4) = floor(x(4) * 4) = floor(0.87 * 4) = 3.

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

Cf. A001113, A342730 (uses primes, where this uses positive integers).

Table[Floor[FractionalPart[E*n]*n],{n,0,80}] (* Harvey P. Dale, Jun 29 2025 *)

p (0..50).map {|n| (((Math::E * n) % 1) * n).floor }

A343496 First point of the straight lines in A340649.

Original entry on oeis.org

5, 31, 194, 1061, 6456, 40080, 251721, 1617206, 10553419, 69709769, 465769825

Offset: 1

Simon Strandgaard and Jamie Morken, Apr 17 2021

prime(a(n)+1) - prime(a(n)) = n*2. E.g., for n=4: prime(a(4)+1) - prime(a(4)) = 4*2, prime(1062) - prime(1061) = 4*2, 8521 - 8513 = 8.

			For n=1, consider k's with prime gap 1*2 = 2, i.e., k's such that A001223(k)=2. k=5 is the first place where A001223(k)=2 and A141042(k)=A340649(k), so a(1)=5.
For n=2, consider k's with prime gap 2*2 = 4, i.e., k's such that A001223(k)=4. k=31 is the first place where A001223(k)=4 and A141042(k)=A340649(k), so a(2)=31.
For n=3, consider k's with prime gap 3*2 = 6, i.e., k's such that A001223(k)=6. k=194 is the first place where A001223(k)=6 and A141042(k)=A340649(k), so a(3)=194.

Simon Strandgaard, Visualization of the first 5 terms.

Cf. A000040, A001223, A141042, A340649, A029707, A029709, A320701, A320702, A320703.

n = 1
last_prime = 2
find_gap = 2
result = []
Prime.each(10_000) do |prime|
    next if prime == 2
    gap = prime - last_prime
    if gap == find_gap
        value = (n * prime) % last_prime
        if value == n * gap
            result << n
            find_gap += 2
        end
    end
    n += 1
    last_prime = prime
end
p result

a(n) = smallest k that satisfies A001223(k) = 2*n and A340649(k) = A141042(k).

cp's OEIS Frontend

User: Simon Strandgaard

A350493 a(n) = floor(sqrt(prime(n)))^2 mod n.

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A349487 a(n) = A132739((n-5)*(n+5)).

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A348762 a(n) = A000265((n-8)*(n+8)).

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A348287 Arrange nonzero digits of n in increasing order then append the zeros.

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A347342 a(n) = prime(n) mod floor(prime(n) / n).

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A347112 a(n) = concat(prime(n+1),n) mod prime(n).

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A345727 a(n) = (prime(n)+1) * prime(n+1).

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