Martin Ehrenstein - cp's OEIS frontend

A373970 Squares equal to the sum of a cube and a factorial number.

1, 9, 25, 121, 784, 1024, 5041, 363609, 542939080336, 160351569000000, 51312167754301440000, 65249379457597440000, 449519240612413440000, 957230928468541440000, 4797276365676433637376, 16367773504928806600704, 308090303428827709440000

Offset: 1

Martin Ehrenstein, Jun 24 2024

			1 = 0^3 + 1!.
25 = 1^3 + 4!.
784 = 4^3 + 6!.
5041 = 1^3 + 7!.

Cf. A000142, A000290, A000578, A373218.

A085692 is a subsequence.

is1(n) = {my(sq = n^2, f = 1, k = 1); while(f <= sq && !ispower(sq - f, 3), k++; f *= k); f <= sq;}
lista(kmax) = for(k = 1, kmax, if(is1(k), print1(k^2, ", "))); \\ Amiram Eldar, Jun 24 2024

a(10) from Amiram Eldar, Jun 24 2024
a(11)-a(16) from David A. Corneth, Jun 24 2024
a(17) from Michael S. Branicky, Jun 26 2024

A364785 Primes whose binary representation has more 1-bits than its cube.

Original entry on oeis.org

445317119867, 28498383073019, 114304774692347, 7594322375176157

Offset: 1

Martin Ehrenstein, Aug 07 2023

			445317119867 is a term because it is a prime number and A000120(445317119867) = 31, while A192085(445317119867) = A000120(445317119867^3) = 30.

Chris K. Caldwell and G. L. Honaker, Jr., 445317119867, Prime Curios!

Intersection of A363799 and A000040.

Cf. A000120, A192085, A138597.

A352631 Minimum number of zeros in a binary n-digit perfect square, or -1 if there are no such numbers.

Original entry on oeis.org

0, -1, 2, 2, 2, 3, 2, 4, 3, 4, 3, 4, 4, 5, 2, 5, 4, 6, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 6, 8, 8, 6, 7, 7, 8, 8, 9, 8, 9, 9, 8, 9, 10, 9, 9, 10, 9, 9, 9, 9, 10, 10, 10, 10, 11, 10, 11, 11, 11, 9, 9, 11, 11, 11, 12, 11, 12, 11, 12

Offset: 1

Martin Ehrenstein, Mar 25 2022

Is there a formula that is easy to compute?

			a(6) = 3, because there are two 6-bit squares 36 = 100100_2 and 49 110001_2 with 4 and 3 zeros, respectively.
a(2) = -1, because the first two perfect squares 1 = 1_2 and 4 = 100_2 have 1 and 3 bits, respectively.

Cf. A000290, A023416, A070939.

Cf. A357658 (maximum 1's).

from gmpy2 import is_square, popcount
for n in range(1, 33):
    m=n+1
    for k in range(2**(n-1), 2**n):
        if is_square(k):
            m=min(m, n-popcount(k))
    print(n, -1 if m>n else m)

from math import isqrt
def A352631(n): return -1 if n == 2 else min(n-(k**2).bit_count() for k in range(1+isqrt(2**(n-1)-1),1+isqrt(2**n))) # Chai Wah Wu, Mar 28 2022

a(43)-a(71) from Pontus von Brömssen, Mar 26 2022

a(72)-a(80) from Chai Wah Wu, Apr 01 2022

A352599 Number of ways of placing A352426(n) nonattacking white-square queens on an n X n board.

Original entry on oeis.org

1, 2, 4, 6, 2, 6, 64, 112, 68, 80, 32, 36, 28, 1414, 576, 384, 40, 70396, 40896, 22468, 9808, 1152, 252, 2246138, 482272, 185932

Offset: 1

Martin Ehrenstein, Mar 22 2022

Equivalently the number of ways of placing the maximal number of nonattacking black-square queens on an inverted n X n chessboard, that is a board with the a1 square white, the a2 and b1 squares black, etc.

Cf. A352241 (maximal number for black-squares), A352325 (black-squares counts), A352426 (maximal number for white-squares), this sequence (white-squares counts).

Python
```
See A352426.
```

a(2n) = A352325(2n).

a(21)-a(24) from Vaclav Kotesovec, Mar 23 2022

a(25)-a(26) from Vaclav Kotesovec, Apr 01 2022

A352426 Maximal number of nonattacking white-square queens on an n X n chessboard.

Original entry on oeis.org

0, 1, 1, 2, 4, 4, 4, 5, 6, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15, 16, 17, 18, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 26, 27, 28, 29, 29, 30, 31, 32, 33, 33, 34, 35, 36, 36, 37, 38, 39, 40, 40, 41, 42, 43, 44, 44, 45, 46, 47, 48, 48, 49, 50, 51, 51, 52, 53

Offset: 1

Martin Ehrenstein, Mar 16 2022

nonn

Equivalently the maximal number of nonattacking black-square queens on an inverted n X n chessboard, that is a board with the a1 square white, the a2 and b1 squares black, etc.

Andy Huchala, Table of n, a(n) for n = 1..92
Andy Huchala, Python program.
Math StackExchange, Black queens on n X n board, 2022.

Cf. A352241, A274616.

def fill(rows, queens, leftattack, notdownattack, rightattack, color):
    global c
    available = ~leftattack & notdownattack & ~rightattack & color
    if rows==1:
        if available==0:
            c[queens] = c.get(queens, 0) + 1
        else:
            c[queens+1] = c.get(queens+1, 0) + bin(available).count('1')
        return
    while available:
        attack = available & -available
        fill(rows-1, queens+1, (leftattack|attack)<<1, notdownattack&~attack, (rightattack|attack)>>1, ~color)
        available &= available - 1
    fill(rows-1, queens, leftattack<<1, notdownattack, rightattack>>1, ~color)
print(' n a(n)    count')
for n in range(1, 32):
    c=dict()
    fill(n, 0, 0, (1<

a(2n) = A352241(2n).

a(17)-a(24) from Vaclav Kotesovec, Mar 17 2022
a(25)-a(26) from Vaclav Kotesovec, Mar 20 2022
a(27) onwards from Andy Huchala, Mar 27 2024

A340712 Primes p such that p divides (2 + product of primes < p).

Original entry on oeis.org

557, 248137, 4085791, 519807973

Offset: 1

Martin Ehrenstein, Jan 16 2021

			557 is in the sequence because 2 + A034386(557 - 1) = 557 * 4627335992...5904782776 (220 digits).

Norman Luhn, Prime Curios! 557

Cf. A338543, A062347, A034386.

from sympy import nextprime
def aupto(limit):
  p, psharp = 3, 2
  while p <= limit:
    if (psharp+2)%p == 0: print(p, end=", ")
    psharp, p = psharp*p, nextprime(p)
aupto(500000) # Michael S. Branicky, Mar 24 2021

cp's OEIS Frontend

User: Martin Ehrenstein

A373970 Squares equal to the sum of a cube and a factorial number.

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A364785 Primes whose binary representation has more 1-bits than its cube.

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A352631 Minimum number of zeros in a binary n-digit perfect square, or -1 if there are no such numbers.

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A352599 Number of ways of placing A352426(n) nonattacking white-square queens on an n X n board.

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A352426 Maximal number of nonattacking white-square queens on an n X n chessboard.

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Python

Formula

Extensions

A340712 Primes p such that p divides (2 + product of primes < p).

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Python