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User: Max Z. Scialabba

Max Z. Scialabba has authored 2 sequences.

A368063 a(n) is the least number k such that sigma(sigma(k) * k) > n * sigma(k) * k.

1, 2, 3, 10, 160, 12155, 26558675

Offset: 0

Max Z. Scialabba, Dec 10 2023

Application of A134716 (sigma(k) / k > n) to A064987.
From Daniel Suteu, Dec 21 2023: (Start)
a(7) <= 114775357632650.
a(8) <= 272113056574982766111055794421. (End)

			For n = 4, the divisors of 160 sum to 378. 160 * 378 = 60480, whose divisors sum up to 243840 > 4 * 60480.

Cf. A064987, A134716.

public static void main(String[] args)
    {
        long max = 0;
        for (long c = 1; c < Math.pow(10, 8); c = c + 1)
        {
            if (factorSum(factorSum(c) * c) > max * factorSum(c) * c)
            {
                System.out.println(c + ": " + factorSum(c) * c);
                max = max + 1;
            }
        }
    }
    public static long factorSum(long n)
    {
        long sum = 0;
        for (long c = 1; c <= Math.sqrt(n); c = c + 1)
        {
            if (n % c == 0)
            {
                sum = sum + c;
                if (c != Math.sqrt(n))
                {
                    sum = sum + n / c;
                }
            }
        }
        return sum;
    }

a={}; For[n=0, n<=6, n++, k=1; While[DivisorSigma[1,DivisorSigma[1,k]k] <= n DivisorSigma[1,k] k, k++]; AppendTo[a,k]]; a (* Stefano Spezia, Dec 10 2023 *)

a(n) = my(k=1); while (sigma(sigma(k)*k) <= n * sigma(k) * k, k++); k; \\ Michel Marcus, Dec 10 2023

a(6) from Michel Marcus, Dec 10 2023

A351925 Squares which are the concatenation of two primes.

Original entry on oeis.org

25, 289, 361, 529, 729, 2401, 2601, 2809, 4761, 5329, 5929, 7569, 11449, 11881, 15129, 19881, 21609, 22801, 23409, 24649, 25281, 26569, 29241, 29929, 31329, 34969, 36481, 39601, 47961, 52441, 53361, 54289, 57121, 58081, 59049, 71289, 77841, 83521, 89401

Offset: 1

Max Z. Scialabba, Feb 25 2022

The first term that is the concatenation of two primes in more than one way is a(11) = 5929 = 5 | 929 = 59 | 29. - Robert Israel, Oct 01 2023

			25 is the concatenation of 2 and 5, both primes.
4761 is the concatenation of 47 and 61, both primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000290 (squares), A039686, A106582, inverse of A167535.

Cf. A038692, A225135.

L:= NULL: count:=0:
for x from 1 by 2 while count < 100 do
  xs:= x^2;
  for i from 1 to ilog10(xs) do
    a:= xs mod 10^i;
    if a > 10^(i-1) and isprime(a) then
      b:= (xs-a)/10^i;
      if isprime(b) then
        L:= L, xs; count:= count+1; break
      fi fi
od od:
L; # Robert Israel, Oct 01 2023

isb(n)={my(d=10); while(dAndrew Howroyd, Feb 26 2022

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    for k in count(1):
        s = str(k*k)
        if any(s[i] != '0' and isprime(int(s[:i])) and isprime(int(s[i:])) for i in range(1, len(s))):
            yield k*k
print(list(islice(agen(), 39))) # Michael S. Branicky, Feb 26 2022

Intersection of A106582 and A000290.

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User: Max Z. Scialabba

A368063 a(n) is the least number k such that sigma(sigma(k) * k) > n * sigma(k) * k.

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A351925 Squares which are the concatenation of two primes.

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