Davide Rotondo - cp's OEIS frontend

A384917 Decimal expansion of 1/3645.

0, 0, 0, 2, 7, 4, 3, 4, 8, 4, 2, 2, 4, 9, 6, 5, 7, 0, 6, 4, 4, 7, 1, 8, 7, 9, 2, 8, 6, 6, 9, 4, 1, 0, 1, 5, 0, 8, 9, 1, 6, 3, 2, 3, 7, 3, 1, 1, 3, 8, 5, 4, 5, 9, 5, 3, 3, 6, 0, 7, 6, 8, 1, 7, 5, 5, 8, 2, 9, 9, 0, 3, 9, 7, 8, 0, 5, 2, 1, 2, 6, 2, 0, 0, 2

Offset: 0

Davide Rotondo, Jun 12 2025

			0.0002743484224965706447187928669410...

Cf. A002378, A021733.

RealDigits[1/3645, 10, 120, -1][[1]] (* Amiram Eldar, Jun 12 2025 *)

Equals Sum_{k>=1} (k*(k+1))/10^(k+3).

Equals A021733/5. - Hugo Pfoertner, Jun 12 2025

A381590 Primes with primitive root -100.

Original entry on oeis.org

3, 7, 19, 23, 31, 43, 47, 59, 67, 71, 83, 107, 131, 151, 163, 167, 179, 191, 199, 223, 227, 263, 283, 307, 311, 347, 359, 367, 379, 383, 419, 431, 439, 443, 467, 479, 487, 491, 499, 503, 523, 563, 571, 587, 599, 619, 631, 647, 659, 683, 719, 727, 743, 787, 811

Offset: 1

Davide Rotondo, Feb 28 2025

Union of long period primes (A006883) of the form 4k-1 and half period primes (A097443) of the form 4k-1.

Complement of A007349 in the union of A007348 and A001913. - Davide Rotondo, May 23 2025

Cf. A006883, A097443, A007349, A007348, A001913.

Select[Prime[Range[150]], MultiplicativeOrder[-100, #] == # - 1 &]  (* Amiram Eldar, Mar 02 2025 *)

is(n)=gcd(n,10)==1 && znorder(Mod(-100, n))==n-1 \\ Charles R Greathouse IV, Mar 01 2025

list(lim)=my(v=List([3])); forprime(p=7,lim, if(znorder(Mod(-100, p))==p-1, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Mar 01 2025

A380741 Decimal expansion of Sum_{k>=1} prime(k)/2^(k!).

Original entry on oeis.org

1, 8, 2, 8, 1, 2, 5, 4, 1, 7, 2, 3, 2, 5, 1, 3, 4, 2, 7, 7, 3, 4, 3, 7, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 2, 7, 5, 4, 8, 0, 2, 2, 9, 7, 8, 8, 9, 0, 4, 0, 5, 6, 0, 9, 9, 9, 0, 5, 2, 2, 2, 0, 4, 4, 6, 0, 9, 5, 7, 1, 8, 4, 3, 4, 0, 5, 5, 1, 9, 6, 7, 5, 4, 9, 6, 1, 5, 1, 4, 2, 1, 6, 1, 2, 0, 1, 2, 2, 0, 0, 0, 5

Offset: 1

Davide Rotondo, Jan 31 2025

Each prime can be recovered from this constant r by prime(k) = floor(r*2^(k!) - floor(r*2^((k-1)!)))*2^((k-1)*(k-1)!) when the binary positions 2^(k!) are far enough apart, which means k>=3.

Is this constant transcendental?

			1.828125417232513427734375...

Cf. A339764.

suminf(k=1, prime(k)/2^(k!)) \\ Michel Marcus, Feb 09 2025

More terms from Amiram Eldar, Jan 31 2025

A380509 Numbers of the form i+j+4ij for i,j >= 1 together with numbers of the form -i-j+4ij for i,j >= 2.

Original entry on oeis.org

6, 11, 12, 16, 19, 20, 21, 26, 29, 30, 31, 33, 36, 38, 40, 41, 42, 46, 47, 51, 52, 54, 55, 56, 61, 63, 65, 66, 68, 71, 72, 74, 75, 76, 81, 82, 83, 85, 86, 89, 90, 91, 92, 94, 96, 101, 103, 106, 107, 109, 110, 111, 116, 117, 118, 119, 120, 121, 123, 124, 126, 128, 129, 131, 132, 133

Offset: 1

Davide Rotondo, Jan 26 2025

nonn

			1+1+4*1*1 = 6; -2-2+4*2*2 = 12.

Cf. A054520, A380140, A380572 (complement), A380549, A380550.

L := 150: S := {}:
for i to L do for j to L do
if 4*i*j + i + j <= L then S := `union`(S, {4*i*j+i+j}, {4*i*j+3*i+3*j+2}) fi;
od; od;
S; # Peter Bala, Jan 30 2025

More terms from Stefano Spezia, Jan 26 2025

Missing terms 111, 121 and 126 added by Peter Bala, Jan 30 2025

A380572 Complement of A380509.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 13, 14, 15, 17, 18, 22, 23, 24, 25, 27, 28, 32, 34, 35, 37, 39, 43, 44, 45, 48, 49, 50, 53, 57, 58, 59, 60, 62, 64, 67, 69, 70, 73, 77, 78, 79, 80, 84, 87, 88, 93, 95, 97, 98, 99, 100, 102, 104, 105, 108, 111, 112, 113, 114, 115, 122

Offset: 1

Davide Rotondo, Jan 27 2025

nonn

4*a(n) + 1 or (4*a(n) + 1)/3 is a prime number.

Compare with A380550, numbers not of the form i + 3*j + 4*i*j. See also A006093, numbers not of the form i + j + i*j and A005097, numbers not of the form i + j + 2*i*j. - Peter Bala, Jan 30 2025

Peter Bala, 4*A380572(n) + 1 is either a prime or three times a prime

Cf. A005097, A006093, A380509, A380550.

S := {}:
for n from 1 to 150 do
  if isprime(4*n+1) then S := `union`(S, {n}) fi;
  if type((4*n+1)*(1/3), integer) then if isprime((4*n+1)*(1/3)) then S := `union`(S, {n}) fi; fi;
end do:
S; # Peter Bala, Jan 30 2025

A378698 Number of compositions of n into parts whose sizes are Fibonacci or Lucas numbers.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 31, 62, 123, 243, 481, 953, 1887, 3737, 7399, 14652, 29014, 57452, 113767, 225279, 446095, 883352, 1749201, 3463746, 6858864, 13581833, 26894570, 53256275, 105457382, 208825335, 413513204, 818833458, 1621443338, 3210760963, 6357907009

Offset: 0

Davide Rotondo, Dec 04 2024

nonn

a(n+1)/a(n) approximate the constant r = 1.9801869...

Cf. A000032, A000045, A116470.

Cf. A076739, A288039.

A116470(n) = if(n<6, n, if(n%2, fibonacci(n\2+3), fibonacci(n\2)+fibonacci(n\2+2)))
a(max_n) = {Vec(1/(1+sum(k=1,max_n-1, -1*x^A116470(k)))+O(x^max_n)); } \\ Thomas Scheuerle, Dec 04 2024

G.f.: 1/(1 - Sum_{k>=1} x^A116470(k)). - Thomas Scheuerle, Dec 04 2024

A372607 Let a(1) = 2, f(n) = a(1)a(2)...*a(n-1) for n >= 1 and a(n) = nextludicnumber(f(n)+1) - f(n) for n >= 2, where nextludicnumber(x) is the smallest ludic number > x.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 13, 25, 17

Offset: 1

Davide Rotondo, May 07 2024

Conjecture: every element is a ludic number.

This is the analog of Buss' conjecture (cf. A067836) for ludic numbers instead of primes, and similar to the idea of ludic Fortunate numbers (A376237) in analogy to the usual Fortunate numbers A005235. - M. F. Hasler, Nov 04 2024

Cf. A067836, A003309 (ludic numbers), A376237 (ludic Fortunate numbers).

A372607_upto(n=15, f=1)=vector(n,i,n=if(i>1, next_A003309(1+f*=n)-f,2)) \\ M. F. Hasler, Nov 02 2024

A347946 Products of nonprimitive roots of n, or 0 if n = 2 or has no primitive roots.

Original entry on oeis.org

0, 0, 1, 2, 4, 24, 48, 0, 4032, 17280, 5400, 0, 518400, 415134720, 0, 0, 1797120, 6467044147200, 39086530560, 0, 0, 1738201006080000, 10247897088, 0, 9632530575360000, 706822057112371200000, 569299069913333760000, 0, 54538738974720000, 0

Offset: 1

Davide Rotondo, Sep 26 2021

nonn

If n is a prime p, a(n) == -1 (mod p) for n > 3; if n is a composite c, a(n) == 0 (mod c) for n > 4.

			a(11) = 5400 because the primitive roots of 11 are {2,6,7,8} and therefore the nonprimitive roots of 11 are {1,3,4,5,9,10} and 1*3*4*5*9*10 = 5400.

Cf. A123475, A219027, A219028.

a[n_] := If[n == 2 || (p = PrimitiveRootList[n]) == {}, 0, (n - 1)!/Times @@ p]; Array[a, 30] (* Amiram Eldar, Sep 26 2021 *)

A345447 Numbers of the form i+j+2ij and 2+i+j+2ij for i,j >= 1.

Original entry on oeis.org

4, 6, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Davide Rotondo, Jun 19 2021

nonn

Except for 1 and 2 the complement sequence c is: 3, 5, 8, 11, 20, 23, 35, 41, 50, 53, 56, 65, ...; where 2*c(i) + 1 and 2*c(i) - 3 are a pair of cousin primes. This is a consequence of the sieve of Sundaram.

			For i,j = 1, 1+1+2*1*1 = 4 and 2+1+1+2*1*1 = 6.

Wikipedia, Sieve of Sundaram.

Cf. A046132, A023200.

Union of A047845 and A153043, except for 0 and 2.

def aupto(limit):
    aset = set()
    for i in range(1, limit//3):
        for j in range(i, limit//3):
            t = i + j + 2*i*j
            if t > limit: break
            aset.update([t, t+2])
    return sorted(an for an in aset if an <= limit)
print(aupto(80)) # Michael S. Branicky, Jul 05 2021

A341930 Decimal expansion of (A249270 + A340469)/2.

Original entry on oeis.org

2, 0, 6, 7, 4, 3, 5, 8, 9, 1, 4, 2, 0, 2, 3, 4, 2, 2, 9, 5, 1, 8, 8, 4, 7, 0, 7, 5, 6, 6, 7, 8, 9, 5, 2, 1, 1, 1, 3, 0, 9, 3, 9, 0, 7, 9, 7, 6, 6, 8, 4, 9, 5, 5, 5, 6, 7, 6, 5, 4, 8, 5, 7, 6, 2, 0, 0, 7, 7, 3, 8, 8, 1, 5, 5, 3, 7, 6, 4, 6, 9, 2, 3, 9, 7, 1, 1, 8, 5, 8, 6, 3, 2, 5, 5, 7, 0, 0, 0, 3, 9, 6, 8

Offset: 1

Davide Rotondo, Feb 23 2021

With this constant r(1) and using the formula r(n+1) = round(r(n))*(r(n) - round(r(n)) + 1.5) it is possible to obtain the sequence of prime numbers because round(r(n)) = prime(n).

			2.06743589142023422951884707566789...

Cf. A249270, A340469.

suminf(k=1, (prime(k)-1.5)/prod(i=1, k-1, prime(i))) \\ Michel Marcus, Feb 23 2021

r(1) = Sum_{k>=1} (prime(k)-1.5)/Product_{i=1..k-1} prime(i).

cp's OEIS Frontend

User: Davide Rotondo

A384917 Decimal expansion of 1/3645.

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A381590 Primes with primitive root -100.

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A380741 Decimal expansion of Sum_{k>=1} prime(k)/2^(k!).

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A380509 Numbers of the form i+j+4ij for i,j >= 1 together with numbers of the form -i-j+4ij for i,j >= 2.

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Maple

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A380572 Complement of A380509.

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Maple

A378698 Number of compositions of n into parts whose sizes are Fibonacci or Lucas numbers.

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A372607 Let a(1) = 2, f(n) = a(1)*a(2)*...*a(n-1) for n >= 1 and a(n) = nextludicnumber(f(n)+1) - f(n) for n >= 2, where nextludicnumber(x) is the smallest ludic number > x.

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A347946 Products of nonprimitive roots of n, or 0 if n = 2 or has no primitive roots.

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Mathematica

A345447 Numbers of the form i+j+2*i*j and 2+i+j+2*i*j for i,j >= 1.

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A372607 Let a(1) = 2, f(n) = a(1)a(2)...*a(n-1) for n >= 1 and a(n) = nextludicnumber(f(n)+1) - f(n) for n >= 2, where nextludicnumber(x) is the smallest ludic number > x.

A345447 Numbers of the form i+j+2ij and 2+i+j+2ij for i,j >= 1.