Infinity-touchable number

sweety439 · 2021-12-21, 23:57

An infinity-touchable number is a nonnegative integer n such that there exists such infinite sequence of nonnegative integers: s(a_1) = n, s(a_2) = a_1, s(a_3) = a_2, s(a_4) = a_3, ..., s(a_(i+1)) = a_i for all integer i>=1, where s(n) = sigma(n) - n (A001065)

Assuming the stronger version of Goldbach conjecture (i.e. all even numbers >6 can be written as sum of two distinct primes), every positive odd number (except 5) are infinity-touchable, and all numbers which are of the form p+1 or p+3 (or both) with prime p (except 5) are infinity-touchable.

Assuming the stronger version of Goldbach conjecture (i.e. all even numbers >6 can be written as sum of two distinct primes), there is a PARI/GP program to test whether a given nonnegative integer n is infinity-touchable: (using is(n) to test, maybe slow for some values of n, such as n=352)

Code:

a(n)=v=[];for(k=1,n^2,if(sigma(k)-k==n,v=concat(v,k)));v \\ row n of A152454
is(n)=if(n==0 || sigma(n)==2*n || (n%2 && n != 5),1,for(k=1,length(a(n)),if(is(a(n)[length(a(n))+1-k]),return(1)));0)

~~(I do not known why the code is(n)=if(n==0 || sigma(n)==2*n || (n%2 && n != 5),1,for(k=1,j=length(a(n)),if(is(a(n)[j+1-k]),return(1)));0) do not work, maybe there is a bug?~~

2021-12-21, 23:57	#1
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2⁵·5·23 Posts	Infinity-touchable number An infinity-touchable number is a nonnegative integer n such that there exists such infinite sequence of nonnegative integers: s(a_1) = n, s(a_2) = a_1, s(a_3) = a_2, s(a_4) = a_3, ..., s(a_(i+1)) = a_i for all integer i>=1, where s(n) = sigma(n) - n (A001065) Assuming the stronger version of Goldbach conjecture (i.e. all even numbers >6 can be written as sum of two distinct primes), every positive odd number (except 5) are infinity-touchable, and all numbers which are of the form p+1 or p+3 (or both) with prime p (except 5) are infinity-touchable. Assuming the stronger version of Goldbach conjecture (i.e. all even numbers >6 can be written as sum of two distinct primes), there is a PARI/GP program to test whether a given nonnegative integer n is infinity-touchable: (using is(n) to test, maybe slow for some values of n, such as n=352) Code: a(n)=v=[];for(k=1,n^2,if(sigma(k)-k==n,v=concat(v,k)));v \\ row n of A152454 is(n)=if(n==0 \|\| sigma(n)==2n \|\| (n%2 && n != 5),1,for(k=1,length(a(n)),if(is(a(n)[length(a(n))+1-k]),return(1)));0) ~~(I do not known why the code is(n)=if(n==0 \|\| sigma(n)==2n \|\| (n%2 && n != 5),1,for(k=1,j=length(a(n)),if(is(a(n)[j+1-k]),return(1)));0) do not work, maybe there is a bug?~~

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