Jesiah Darnell - cp's OEIS frontend

User: Jesiah Darnell

Jesiah Darnell has authored 2 sequences.

A363285 a(n) is the smallest multiple k of 2^n such that |sigma(k) - 2*k| = 2^n, where sigma = A000203.

1, 10, 12, 56, 752, 992, 12224, 16256, 654848, 7337984, 10483712, 12580864, 167763968, 67100672, 2684321792, 38654574592, 584115027968, 17179738112, 206158168064, 274877382656, 149533572988928, 123145293922304, 9288674164342784, 34902896977903616

Offset: 0

Jesiah Darnell, May 25 2023

nonn

I conjecture that this sequence is infinite.

If 2^n*(2^(j+1) +- 1) - 1 is an odd prime p then k = 2^(n+j)*p is a multiple of 2^n such that |sigma(k) - 2*k| = 2^n. It is not known whether these solutions always include a(n), but this is the case for n = 1..23. - Andrew Howroyd, May 26 2023

			2^3 = 8, and the proper divisors of 56 are 1, 2, 4, 7, 8, 14, 28, which add up to 64, which is 8 more than 56, and since 56 is also divisible by 8 (and since there is no smaller number for which these things are true), a(3) = 56.
2^4 = 16, and the proper divisors of 752 are 1, 2, 4, 8, 16, 47, 94, 188, 376, which add up to 736, which is 16 less than 752, and since 752 is also divisible by 16 (and since there is no smaller number for which these things are true), a(4) = 752.

Cf. A000079, A000203, A292557, A292558.

a(n)=for(j=1, oo, my(k=2^n*j); if(abs(sigma(k)-2*k) == 2^n, return(k))) \\ Andrew Howroyd, May 25 2023

from itertools import count
from math import prod
from sympy import factorint
def A363285(n):
    m = 1<Chai Wah Wu, Jul 17 2023

a(10)-a(16) from Alois P. Heinz, May 25 2023

a(17)-a(23) from Andrew Howroyd, May 25 2023

A361980 a(n) is the n-th decimal digit of p(n)/q(n) where p(n) = A002260(n) and q(n) = A004736(n).

Original entry on oeis.org

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

Offset: 1

Jesiah Darnell, Apr 01 2023

Decimal digit positions are numbered 1 for the units, 2 for immediately after the decimal point, and so on.
This sequence can be interpreted as the decimal digits of a constant 1.5030006...
This constant shares an infinite number of decimal digits with any given rational r. This is since p,q go through all pairs of integers >= 1 and so p(n)/q(n) = r for infinitely many n.
This constant is irrational.

			p(1) = 1, q(1) = 1, p/q = 1/1 = 1,  a(1) = 1.
p(2) = 1, q(2) = 2, p/q = 1/2 = 0.5, a(2) = 5.
p(3) = 2, q(3) = 1, p/q = 2/1 = 2.00, a(3) = 0.
p(4) = 1, q(4) = 3, p/q = 1/3 = 0.333..., a(4) = 3.

Cf. A002260, A004736, A061480.

p(n) = n-binomial(floor(1/2+sqrt(2*n)), 2); \\ A002260
q(n) = binomial( floor(3/2 + sqrt(2*n)), 2) - n + 1; \\ A004736
a(n) = my(r = p(n)/q(n)); floor(r*10^(n-1)) % 10; \\ Michel Marcus, Apr 05 2023

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User: Jesiah Darnell

A363285 a(n) is the smallest multiple k of 2^n such that |sigma(k) - 2*k| = 2^n, where sigma = A000203.

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