Aliquot-like sequences

[sweety439]

* "unitary Aliquot sequence" (x --> A063919(x), see A003062, A063991, and A097030)
* "Aliquot-like sequence based on Dedekind psi function" (x --> A306927(x), see A323327)
* "exponential Aliquot sequence" (x --> A126164(x), see A126165 and A054979)
* "infinitary Aliquot sequence" (x --> A126168(x), see A126169)
* "coreful Aliquot sequence" (x --> A336563(x), see A307958)
* "bi-unitary Aliquot sequence" (x --> A331970(x), see A292981)
* "totient Aliquot sequence" (x --> A092693(x), see A286233 for "totient amicable numbers" and see A343243 for "totient sociable numbers")

Also use the sense of http://www.aliquotes.com/autres_proc...iteratifs.html (x --> A001065(x)+k with fixed integer k) to generate (the range of k is from -100 to +100 inclusive), i.e. x --> A063919(x)+k, x --> A306927(x)+k, etc.)

Now we have 1608 sequences (x --> A001065(x)+k, x --> A063919(x)+k, x --> A306927(x)+k, x --> A126164(x)+k, x --> A126168(x)+k, x --> A336563(x)+k, x --> A331970(x)+k, x --> A092693(x)+k, all with integer -100<=k<=100), all of these sequences have “perfect” numbers, “amicable” numbers, “sociable” numbers (see this page), as well as “Aliquot” sequences, there may be some n whose “Aliquot” sequence is still open, like the original Aliquot sequence for 276 (usually, when we reach a prime or a number <= 1 (including all negative numbers, this situation exists for k <= -4), then we stop), also the “untouchable” numbers

Are there any interest for these Aliquot-like sequences?

[sweety439]

PARI/GP program for this:

a1(n) = sigma(n) - n
usigma(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d))
a2(n) = usigma(n) - n
a3(n) = n*(sumdivmult(n, d, issquarefree(d)/d) - 1)
b4(n) = { my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2], d, f[i, 1]^d)); };
a4(n) = b4(n) - n
b5(n) = {my(b, f=factorint(n)); prod(k=1, #f[, 2], b = binary(f[k, 2]); prod(j=1, #b, if(b[j], 1+f[k, 1]^(2^(#b-j)), 1)))}
a5(n) = b5(n) - n
b6(n) = {my(f = factor(n)); for (i=1, #f~, f[i, 2]=1); my(pp = factorback(f)); sumdiv(n, d, if (! (d % pp), d, 0)); }
a6(n) = b6(n) - n
b7(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i, 2] = 1; ); factorback(f); }
a7(n) = b7(n) - n
a8(n) = my(k); while(n>1, k+=n=eulerphi(n)); k
aliquot(n,r,m) = print1(n, ", ");for(k=1, 1000, if(n>1 && !isprime(n),n=[a1(n),a2(n),a3(n),a4(n),a5(n),a6(n),a7(n),a8(n)][r]+m; print1(n, ", "), return()))

programs after the PARI/GP programs in the OEIS sequences pages

use aliquot(n,r,m) for x --> a_r(x) + m (r = 1, 2, 3, 4, 5, 6, 7, 8, m is (positive or negative or 0) integer), start with n

a_1: standard Aliquot sequence
a_2: unitary Aliquot sequence
a_3: Aliquot-like sequence based on Dedekind psi function
a_4: exponential Aliquot sequence
a_5: infinitary Aliquot sequence
a_6: coreful Aliquot sequence
a_7: bi-unitary Aliquot sequence
a_8: totient Aliquot sequence

if we reach a prime or a number <= 1 (including all negative numbers, this situation exists for m <= -4), then we stop