Albert Böschow - cp's OEIS frontend

A351404 Decimal expansion of Sum_{k>=1} A106400(k-1)/k.

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

Offset: 0

Albert Böschow and Julian Böschow, Feb 10 2022

Define S(j) = Sum_{k=1..2^j} A106400(k-1)/k; S(28) agrees with this constant for 104 digits. - Jon E. Schoenfield, Feb 22 2022

			0.39876108810841881240743054440027306033680...

Cf. A106400, A215016.

More digits from Jon E. Schoenfield, Feb 22 2022

A350777 Numbers k where phi(k) divides k - 3.

Original entry on oeis.org

1, 2, 3, 9, 195, 5187, 1141967133868035, 3658018932844533311864835

Offset: 1

Albert Böschow and Dennis Gruhlke, Jan 15 2022

Numbers in this sequence larger than 2 have to be odd, since phi(n) is even for n > 2, so n - 3 cannot be odd. Therefore n itself must be odd.

Terms having (k-3)/phi(k) = 2 are shared with A226105. - Max Alekseyev, Oct 26 2023

			phi(195) = 96, 195 - 3 = 192, and 96 divides 192.

Cf. A000010, A007694, A226105.

Select[Range[6000], Divisible[#-3, EulerPhi[#]] &] (* Amiram Eldar, Jan 19 2022 *)

isok(k) = !((k-3) % eulerphi(k)); \\ Michel Marcus, Jan 19 2022

from sympy import totient
print("1, 2", end=", ")
for k in range (3, 10**8, 2):
    if (k-3)%totient(k)==0:
        print(k, end=", ", flush=True) # Martin Ehrenstein, Mar 26 2022

a(7)-a(8) from Max Alekseyev, Nov 05 2023

A348183 a(n) is the determinant of the n X n matrix M = (m_{i,j}), i,j from 0 to n-1: m{i,j} = (i+j)^2 mod n.

Original entry on oeis.org

1, 0, -1, -2, 0, 250, 5616, -33614, 0, 204073344, -900000000, -9431790764, 0, 10752962364222, -1870899108384768, -36328974609375000, 0, 22899384412078526344, -111400529859275793629184, -43843094862278417487512, 0, 2870507605405055660542502550, 67015802375208384199755038720

Offset: 0

Dennis Gruhlke and Albert Böschow, Oct 05 2021

sign

It seems that for values of n divisible by 4 -> a(n) = 0 and rank(M) = n/2.

			a(2) =  |0^2 mod 2, 1^2 mod 2| = -1
        |1^2 mod 2, 2^2 mod 2|
--
        |0^2 mod 3, 1^2 mod 3, 2^2 mod 3|
a(3) =  |1^2 mod 3, 2^2 mod 3, 3^2 mod 3| = -2
        |2^2 mod 3, 3^2 mod 3, 4^2 mod 3|

Cf. A070896.

a[n_]:=Table[Mod[(i+j)^2,n],{i,0,n-1},{j,0,n-1}]; Join[{1},Table[Det[a[n]], {n, 22}]] (* Stefano Spezia, Oct 06 2021 *)

a(n) = matdet(matrix(n, n, i, j, i--; j--; (i+j)^2 % n)); \\ Michel Marcus, Oct 06 2021

from sympy import Matrix
def A348183(n): return Matrix(n,n,[pow(i+j,2,n) for i in range(n) for j in range(n)]).det() # Chai Wah Wu, Nov 24 2021

a(14)-a(17) corrected by and more terms from Stefano Spezia, Oct 06 2021.

A348190 Positive integers where each is chosen to be the second smallest number subject to the condition that no three terms a(j), a(j+k), a(j+2*k) (for any j and k) form an arithmetic progression.

Original entry on oeis.org

2, 2, 3, 2, 3, 3, 4, 2, 2, 5, 3, 4, 3, 5, 5, 7, 5, 2, 4, 2, 2, 5, 4, 6, 3, 2, 9, 5, 9, 3, 6, 10, 9, 9, 6, 5, 7, 4, 12, 11, 11, 2, 6, 4, 8, 3, 4, 6, 7, 13, 11, 5, 5, 6, 4, 8, 10, 9, 13, 4, 13, 4, 6, 6, 2, 11, 5, 4, 6, 11, 18, 9, 15, 2, 15, 12

Offset: 1

Albert Böschow, Oct 06 2021

The sequence seems to behave in a similar way as the "forest fire" A229037. The graph (up to n=5000) looks like it has a fractal structure, with each dense "pillar" approximately double the size of the previous one.

The terms of this sequence do not seem to be larger (on average) than those of A229037, despite the construction of this sequence.

			a(7) = 4, because 2 would form an arithmetic progression with a(1) = 2 and a(4) = 2 and 3 would form an arithmetic progression with a(5) = 3 and a(6) = 3. Therefore, 4 is the second smallest number which satisfies the condition (1 being the smallest).

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..8000
Rémy Sigrist, PARI program for A348190
Neal Gersh Tolunsky, Scatterplot for n=1...8000

Cf. A229037.

PARI
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User: Albert Böschow

A351404 Decimal expansion of Sum_{k>=1} A106400(k-1)/k.

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A350777 Numbers k where phi(k) divides k - 3.

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A348183 a(n) is the determinant of the n X n matrix M = (m_{i,j}), i,j from 0 to n-1: m{i,j} = (i+j)^2 mod n.

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Author

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Mathematica

PARI

Python

Extensions

A348190 Positive integers where each is chosen to be the second smallest number subject to the condition that no three terms a(j), a(j+k), a(j+2*k) (for any j and k) form an arithmetic progression.

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PARI