William C. Laursen - cp's OEIS frontend

A374578 Pierpont primes are primes of the form 2^t*3^u + 1; this sequence gives the u's in order.

0, 0, 0, 1, 1, 0, 2, 2, 2, 1, 3, 4, 1, 0, 3, 5, 2, 1, 2, 4, 6, 4, 6, 3, 5, 4, 1, 7, 2, 9, 8, 0, 7, 2, 8, 4, 10, 9, 6, 1, 8, 5, 2, 6, 3, 5, 4, 9, 4, 12, 11, 3, 14, 3, 15, 5, 7, 16, 13, 3, 10, 4, 17, 10, 11, 12, 3, 4, 1, 8, 5, 8, 4, 11, 7, 15, 12, 2, 10, 1, 22, 4

Offset: 1

William C. Laursen, Jul 11 2024

nonn

This sequence gives the exponents of 3's in the Pierpont primes, A374577 gives the exponents of 2's.

			a(1) = 0, because the first Pierpont prime is 2 = 2^0 * 3^0 + 1.
a(6) = 0, because the sixth Pierpont prime is 17 = 2^4 * 3^0 + 1.
a(7) = 2, because the seventh Pierpont prime is 19 = 2^1 * 3^2 + 1.

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

Cf. A005109, A007949, A374577.

With[{lim = 10^12}, IntegerExponent[Select[Sort@ Flatten@Table[2^i*3^j + 1, {i, 0, Log2[lim]}, {j, 0, Log[3, lim/2^i]}], PrimeQ] - 1, 3]] (* Amiram Eldar, Sep 02 2024 *)

lista(lim) = {my(s = List()); for(i = 0, logint(lim, 2), for(j = 0, logint(lim >> i, 3), listput(s, 2^i * 3^j + 1))); s = Set(s); for(i = 1, #s, if(isprime(s[i]), print1(valuation(s[i] - 1, 3), ", ")));} \\ Amiram Eldar, Sep 02 2024

a(n) = A007949(A005109(n)-1).

More terms from Stefano Spezia, Jul 12 2024

A374577 Pierpont primes are primes of the form 2^t*3^u + 1; this sequence gives the t's in order.

Original entry on oeis.org

0, 1, 2, 1, 2, 4, 1, 2, 3, 5, 2, 1, 6, 8, 4, 1, 6, 8, 7, 4, 1, 5, 2, 7, 4, 7, 12, 3, 11, 1, 3, 16, 6, 14, 5, 12, 3, 5, 10, 18, 7, 12, 17, 11, 16, 13, 15, 8, 16, 4, 6, 19, 2, 20, 2, 18, 15, 1, 6, 22, 11, 21, 1, 13, 12, 11, 26, 25, 30, 19, 24, 20, 27, 16, 23, 11

Offset: 1

William C. Laursen, Jul 11 2024

nonn

This sequence gives the exponents of 2's in the Pierpont primes, A374578 gives the exponents of 3's.

			a(1) = 0, because the first Pierpont prime is 2 = 2^0 * 3^0 + 1.
a(6) = 4, because the sixth Pierpont prime is 17 = 2^4 * 3^0 + 1.
a(7) = 1, because the seventh Pierpont prime is 19 = 2^1 * 3^2 + 1.

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

Cf. A005109, A007814, A374578.

With[{lim = 10^11}, IntegerExponent[Select[Sort@ Flatten@Table[2^i*3^j + 1, {i, 0, Log2[lim]}, {j, 0, Log[3, lim/2^i]}], PrimeQ] - 1, 2]] (* Amiram Eldar, Sep 02 2024 *)

lista(lim) = {my(s = List()); for(i = 0, logint(lim, 2), for(j = 0, logint(lim >> i, 3), listput(s, 2^i * 3^j + 1))); s = Set(s); for(i = 1, #s, if(isprime(s[i]), print1(valuation(s[i] - 1, 2), ", ")));} \\ Amiram Eldar, Sep 02 2024

a(n) = A007814(A005109(n)-1).

More terms from Stefano Spezia, Jul 12 2024

A343575 a(n) = floor((2+sqrt(5))^n - 2^(n+1)) mod (20*n).

Original entry on oeis.org

0, 9, 0, 49, 0, 9, 0, 129, 60, 49, 0, 49, 0, 9, 100, 129, 0, 249, 0, 49, 340, 9, 0, 449, 0, 9, 240, 289, 0, 249, 0, 129, 60, 9, 600, 49, 0, 9, 580, 449, 0, 609, 0, 289, 700, 9, 0, 449, 700, 249, 60, 289, 0, 969, 200, 129, 60, 9, 0, 49, 0, 9, 1240, 769, 0, 369, 0

Offset: 1

William C. Laursen, Apr 20 2021

Whenever n is an odd prime, a(n) is 0 (see M. Penn).

Jianing Song, Table of n, a(n) for n = 1..10000
M. Penn, Hello, old friend..., YouTube video.

Cf. A345031.

Table[Mod[Floor[(2+Sqrt[5])^n-2^(n+1)],20n],{n,67}] (* Stefano Spezia, Apr 21 2021 *)

a(n) = my(M = [6, -7, -2; 1, 0, 0; 0, 1, 0]); 10*((M^n)[3, 1] % (2*n)) - !(n%2) \\ Jianing Song, Jun 07 2021

From Jianing Song, Jun 07 2021: (Start)
For even n, a(n) = 10*(A345031(n) mod (2*n)) - 1;
For odd n, a(n) = 10*(A345031(n) mod (2*n)). (End)

More terms from Stefano Spezia, Apr 21 2021

A342175 a(n) is the difference between the n-th composite number and the smallest larger composite to which it is relatively prime.

Original entry on oeis.org

5, 19, 1, 1, 11, 13, 1, 1, 5, 7, 1, 1, 3, 1, 1, 1, 1, 5, 19, 1, 1, 1, 1, 13, 1, 1, 9, 13, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 5, 17, 1, 1, 1, 1, 19, 1, 1, 11, 5, 1, 1, 1, 1, 7, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 19, 1, 1, 11, 13, 1, 1, 5, 7, 1, 1, 3, 1

Offset: 1

William C. Laursen, Mar 04 2021

nonn

Conjecture: The only nonprime terms are squares (based on checking the first 2 million terms). - Ivan N. Ianakiev, Mar 28 2021

The conjecture above is false (see A353203 for counterexamples). - Ivan N. Ianakiev, Jul 04 2022

			The first composite number is 4, and the smallest larger composite to which it is coprime is 9, so a(1) = 9 - 4 = 5.
The second composite number is 6, and the smallest larger composite to which it is coprime is 25, so a(2) = 25 - 6 = 19.

Cf. A002808, A113496, A353203.

Table[Block[{k = 1}, While[Nand[GCD[#, k] == 1, CompositeQ[# + k]], k++]; k] &@ FixedPoint[n + PrimePi@ # + 1 &, n + PrimePi@ n + 1], {n, 83}] (* Michael De Vlieger, Mar 19 2021 *)

lista(nn) = {forcomposite(c=1, nn, my(x=c+1); while (isprime(x) || (gcd(x,c) != 1), x++); print1(x - c, ", "););} \\ Michel Marcus, Mar 04 2021

from sympy import isprime, gcd, composite
def A342175(n):
    m = composite(n)
    k = m+1
    while gcd(k,m) != 1 or isprime(k):
        k += 1
    return k-m # Chai Wah Wu, Mar 28 2021

a(n) = A113496(n) - A002808(n). - Jon E. Schoenfield, Mar 04 2021

A339401 a(n) = numerator of (1/e)^n * Sum_{k>=0}(n^kk^n)/(n!k!).

Original entry on oeis.org

1, 1, 3, 19, 63, 322, 44683, 941977, 4677605, 668520163, 21622993111, 9759873853, 31135480907413, 194137920764803, 64440211018897379, 3298807094967155971, 181322497435007616497, 532556590750629416219, 665881649529214120845679, 2596711638295426703997397, 1031081559092352146579024047

Offset: 0

William C. Laursen, Dec 03 2020

Cf. A242817, A339402 (denominators).

A:= proc(n, k) option remember; `if`(n=0, 1, (1+
      add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
    end:
a:= n-> numer(A(n$2)/n!):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 07 2020

a[n_] := BellB[n, n]/n! // Numerator;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 27 2022 *)

a(n)/A339402(n) = A242817(n)/n!. - Pontus von Brömssen, Dec 03 2020

a(n) = numerator([x^n] exp(n*(exp(x)-1))). - Alois P. Heinz, Dec 07 2020

A339402 a(n) = denominator of (1/e)^n * Sum_{k>=0}(n^kk^n)/(n!k!).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 120, 720, 1008, 40320, 362880, 45360, 39916800, 68428800, 6227020800, 87178291200, 1307674368000, 1046139494400, 355687428096000, 376610217984000, 40548366802944000, 2432902008176640000, 5676771352412160000, 40142883134914560000, 25852016738884976640000

Offset: 0

William C. Laursen, Dec 03 2020

Cf. A339401 for numerators and relation to A242817.

A:= proc(n, k) option remember; `if`(n=0, 1, (1+
      add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
    end:
a:= n-> denom(A(n$2)/n!):
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 07 2020

a[n_] := BellB[n, n]/n! // Denominator;
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 27 2022 *)

A339401(n)/a(n) = A242817(n)/n!. - Pontus von Brömssen, Dec 03 2020

a(n) = denominator([x^n] exp(n*(exp(x)-1))). - Alois P. Heinz, Dec 07 2020

A336784 Number of steps in Conway's Game of Life for a block and row of n cells to stabilize.

Original entry on oeis.org

0, 1, 2, 551, 4, 74, 77, 38, 15, 16, 16, 15, 1185, 41, 17, 84, 273, 21, 25, 20, 1342, 164, 19, 51, 66, 55, 62, 65, 78, 93, 34, 79, 141, 105, 56, 133, 357, 2621, 100, 119, 799, 278, 149, 305, 305, 126, 99, 227, 387, 272, 274, 465, 714, 580, 689, 172, 282, 2163

Offset: 0

William C. Laursen, Aug 04 2020

nonn

			As a block by itself is stable, a(0)=0.
. . . .
. o o .
. o o .
. . . .
A block with a single square adjacent will turn into a boat on the next tick, which is stable.
. . . . .|. . . . .
. o o . .|. o o . .
. o o . .|. o . o .
. . . o .|. . o . .
. . . . .|. . . . .
A block with a row of two squares will take two generations to turn into a boat.
. . . . . .|. . . . .|. . . . .
. o o . . .|. o o . .|. o o . .
. o o . . .|. o . . .|. o . o .
. . . o o .|. . o o .|. . o . .
. . . . . .|. . . . .|. . . . .
A block with a row of three squares is known as a methuselah (see Wikipedia link), taking 551 generations to stabilize. The final configuration has two escaped gliders, one blinker, eight blocks, two boats, one ship, two beehives, one loaf, and one fleet. Only the initial configuration is shown below.
. . . . . . .
. o o . . . .
. o o . . . .
. . . o o o .
. . . . . . .

Wikipedia, Conway's Game of Life.
Wikipedia, Methuselah (cellular automaton).

A333451 Expansion of golden ratio (1 + sqrt(5))/2 in base 3.

Original entry on oeis.org

1, 1, 2, 1, 2, 0, 0, 1, 1, 2, 2, 0, 2, 1, 2, 1, 0, 2, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 2, 0, 1, 1, 2, 2, 1, 0, 2, 1, 2, 0, 0, 2, 2, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 2, 2, 1, 1, 2, 0, 0, 2, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 0, 2, 1, 1, 1, 2, 2, 2, 0, 2, 2, 0, 0, 2, 0, 2, 1, 1, 0, 2, 2, 1, 0

Offset: 1

William C. Laursen, Mar 21 2020

			1.12120011220212102001021001...

Cf. A001622, A068432, A164629, A333452, A333454.

RealDigits[ (1+Sqrt[5])/2, 3, 1000][[1]]

a(n) = floor(quadgen(5)*3^(n-1))%3 \\ Chittaranjan Pardeshi, Feb 06 2023

A333452 Expansion of golden ratio (1 + sqrt(5))/2 in base 4.

Original entry on oeis.org

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

Offset: 1

William C. Laursen, Mar 21 2020

			1.213203131321232113331022...

Cf. A001622, A068432, A164629, A333451, A333454.

RealDigits[ (1+Sqrt[5])/2, 4, 1000][[1]]

More terms from Jinyuan Wang, Mar 21 2020

A333454 Expansion of golden ratio (1 + sqrt(5))/2 in base 6.

Original entry on oeis.org

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

Offset: 1

William C. Laursen, Mar 21 2020

			1.34125455435343145134223514015012004525012404411401...

Cf. A001622, A068432, A164629, A333451, A333452.

RealDigits[ (1+Sqrt[5])/2, 6, 1000][[1]]

More terms from Jinyuan Wang, Mar 21 2020

cp's OEIS Frontend

User: William C. Laursen

A374578 Pierpont primes are primes of the form 2^t*3^u + 1; this sequence gives the u's in order.

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A374577 Pierpont primes are primes of the form 2^t*3^u + 1; this sequence gives the t's in order.

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A343575 a(n) = floor((2+sqrt(5))^n - 2^(n+1)) mod (20*n).

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A342175 a(n) is the difference between the n-th composite number and the smallest larger composite to which it is relatively prime.

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Python

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A339401 a(n) = numerator of (1/e)^n * Sum_{k>=0}(n^k*k^n)/(n!*k!).

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Maple

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Formula

A339402 a(n) = denominator of (1/e)^n * Sum_{k>=0}(n^k*k^n)/(n!*k!).

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Maple

Mathematica

Formula

A336784 Number of steps in Conway's Game of Life for a block and row of n cells to stabilize.

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Examples

Links

A333451 Expansion of golden ratio (1 + sqrt(5))/2 in base 3.

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Author

Keywords

A339401 a(n) = numerator of (1/e)^n * Sum_{k>=0}(n^kk^n)/(n!k!).

A339402 a(n) = denominator of (1/e)^n * Sum_{k>=0}(n^kk^n)/(n!k!).