Displaying 1-10 of 11 results found.
1, 1, 2, 1, 4, 0, 6, 1, 1, 2, 10, 2, 12, 4, 6, 1, 16, 3, 18, 2, 10, 8, 22, 6, 1, 10, 2, 14, 28, 12, 30, 1, 18, 14, 22, 1, 36, 16, 22, 10, 40, 12, 42, 2, 3, 20, 46, 14, 1, 1, 30, 2, 52, 12, 38, 2, 34, 26, 58, 6, 60, 28, 1, 1, 46, 12, 66, 2, 42, 4, 70, 3, 72, 34, 1, 2, 58, 12, 78, 2, 1, 38, 82, 14, 62, 40, 54, 2, 88
PROG
(PARI)
A007947(n) = factorback(factorint(n)[, 1]);
CROSSREFS
Differs from A326144 at the positions given by A336555, for the first time at n=45, where a(45) = 3, while A326144(45) = 6.
a(n) is the number of possible values of k such that the sum of aliquot coreful divisors of k ( A336563) is n.
+20
4
0, 1, 1, 0, 1, 3, 1, 0, 0, 2, 1, 1, 1, 3, 2, 0, 1, 1, 1, 0, 2, 2, 1, 1, 0, 2, 0, 0, 1, 6, 1, 0, 2, 2, 2, 1, 1, 2, 3, 0, 1, 5, 1, 0, 0, 2, 1, 0, 0, 0, 2, 0, 1, 0, 2, 1, 2, 2, 1, 2, 1, 3, 0, 0, 2, 4, 1, 0, 2, 4, 1, 0, 1, 2, 0, 0, 2, 5, 1, 1, 0, 2, 1, 1, 2, 2, 2
COMMENTS
A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958).
FORMULA
a(n) = 0 if and only if n is in A372740. a(n) = 1 if and only if n is in A372742.
EXAMPLE
a(2) = 1 since there is 1 possible value of k, k = 4, such that A336563(k) = 2. a(6) = 3 since there are 3 possible values of k, k = 8, 12 and 18, such that A336563(k) = 6.
MATHEMATICA
f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; seq[max_] := Module[{v = Table[0, {max}], i}, Do[i = s[k]; If[0 < i <= max, v[[i]]++], {k, 1, max^2}]; v]; seq[100]
PROG
(PARI) s(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2] + 1) - 1)/(f[i, 1] - 1) - 1) - n; }
lista(nmax) = {my(v = vector(nmax), i); for(k = 1, nmax^2, i = s(k); if(i > 0 && i <= nmax, v[i]++)); v; }
Numbers k such that there is a unique number m for which the sum of the aliquot coreful divisors of m ( A336563) is k.
+20
4
2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 23, 24, 29, 31, 36, 37, 41, 43, 47, 53, 56, 59, 61, 67, 71, 73, 79, 80, 83, 84, 89, 96, 97, 98, 101, 103, 107, 109, 112, 113, 127, 131, 135, 137, 139, 140, 149, 150, 151, 156, 157, 163, 167, 173, 179, 181, 191, 193, 197, 198
COMMENTS
A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958). Numbers k such that A372739(k) = 1. The corresponding values of m are in A372743. Includes all the primes numbers.
MATHEMATICA
f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; seq[max_] := Module[{v = Table[0, {max}], i}, Do[i = s[k]; If[1 <= i <= max, v[[i]]++], {k, 1, max^2}]; Position[v, 1] // Flatten]; seq[200]
PROG
(PARI) s(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2] + 1) - 1)/(f[i, 1] -1) - 1) - n; }
lista(nmax) = {my(v = vector(nmax), i); for(k=1, nmax^2, i = s(k); if(i > 0 && i <= nmax, v[i]++)); for(k = 1, nmax, if(v[k] == 1, print1(k, ", "))); }
Coreful untouchable numbers: numbers that are not the sum of aliquot coreful divisors ( A336563) of any number.
+20
3
1, 4, 8, 9, 16, 20, 25, 27, 28, 32, 40, 44, 45, 48, 49, 50, 52, 54, 63, 64, 68, 72, 75, 76, 81, 88, 92, 99, 100, 104, 108, 116, 117, 121, 124, 125, 128, 136, 144, 147, 148, 152, 153, 160, 162, 164, 169, 171, 172, 175, 176, 184, 188, 189, 192, 196, 200, 207, 208
COMMENTS
A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958). Numbers k such that A372739(k) = 0. Numbers that are not in the range of A336563. Except for 1, all the terms are not squarefree ( A013929), because if k is squarefree ( A005117), and there is a prime p such that p|k, then A336563(p*k) = k. Includes all the squares of primes ( A001248). The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are , 4, 29, 281, 2762, 27690, ... . Apparently, the asymptotic density of this sequence exists and equals 0.27... .
MATHEMATICA
f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; seq[max_] := Module[{v = Table[0, {max}], i}, Do[i = s[k]; If[0 < i <= max, v[[i]]++], {k, 1, max^2}]; Position[v, _?(# == 0 &)] // Flatten]; seq[200]
PROG
(PARI) s(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2] + 1) - 1)/(f[i, 1] - 1) - 1) - n; }
lista(nmax) = {my(v = vector(nmax), i); for(k = 1, nmax^2, i = s(k); if(i > 0 && i <= nmax, v[i]++)); for(k = 1, nmax, if(v[k] == 0, print1(k, ", "))); }
4, 9, 25, 49, 121, 27, 169, 289, 24, 361, 529, 54, 841, 961, 36, 1369, 1681, 1849, 2209, 2809, 343, 3481, 3721, 4489, 5041, 5329, 6241, 100, 6889, 189, 7921, 72, 9409, 112, 10201, 10609, 11449, 11881, 686, 12769, 16129, 17161, 225, 18769, 19321, 196, 22201, 160
COMMENTS
Includes all the squares of primes ( A001248).
MATHEMATICA
f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; seq[max_] := Module[{v = w = Table[0, {max}], i}, Do[i = s[k]; If[1 <= i <= max, v[[i]]++; w[[i]] = k], {k, 1, max^2}]; w[[Position[v, 1] // Flatten]]]; seq[200]
PROG
(PARI) s(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2] + 1) - 1)/(f[i, 1] -1) - 1) - n; }
lista(nmax) = {my(v = w = vector(nmax), i); for(k = 1, nmax^2, i = s(k); if(i > 0 && i <= nmax, v[i]++; w[i] = k)); for(k = 1, nmax, if(v[k] == 1, print1(w[k], ", "))); }
Coreful highly touchable numbers: numbers m > 0 such that a record number of numbers k have m as the sum of the aliquot coreful divisors ( A336563) of k.
+20
0
1, 2, 6, 30, 210, 930, 2310, 2730, 30030, 71610, 84630
COMMENTS
A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958). The corresponding record values are 0, 1, 3, 6, 8, 9, 11, 12, 15, 16, 17, ... .
a(12) > 2*10^5.
EXAMPLE
a(1) = 1 since it is the least number that is not the sum of aliquot coreful divisors of any number.
a(2) = 2 since it is the least number that is the sum of aliquot coreful divisors of one number: 2 = A336563(4). a(3) = 6 since it is the least number that is the sum of aliquot coreful divisors of 3 numbers: 6 = A336563(8) = A336563(12) = A336563(18), and there is no number between 2 and 6 that is the sum of aliquot coreful divisors of exactly 2 numbers.
MATHEMATICA
f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; seq[m_] := Module[{v = Table[0, {m}], vm = -1, w = {}, i}, Do[i = s[k]; If[1 <= i <= m, v[[i]]++], {k, 1, m^2}]; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[w, k]], {k, 1, m}]; w]; seq[1000]
PROG
(PARI) s(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2] + 1) - 1)/(f[i, 1] - 1) - 1) - n; }
lista(nmax) = {my(v = vector(nmax), vmax = -1, i); for(k = 1, nmax^2, i = s(k); if(i > 0 && i <= nmax, v[i]++)); for(k = 1, nmax, if(v[k] > vmax, vmax = v[k]; print1(k, ", "))); }
a(n) = n - A308135(n), where A308135(n) is the sum of non-coreful divisors of n.
+10
5
1, 1, 2, 3, 4, 0, 6, 7, 8, 2, 10, 2, 12, 4, 6, 15, 16, 3, 18, 8, 10, 8, 22, 6, 24, 10, 26, 14, 28, -12, 30, 31, 18, 14, 22, 17, 36, 16, 22, 20, 40, -12, 42, 26, 27, 20, 46, 14, 48, 17, 30, 32, 52, 12, 38, 34, 34, 26, 58, -18, 60, 28, 43, 63, 46, -12, 66, 44, 42, -4, 70, 45, 72, 34, 41, 50, 58, -12, 78, 44, 80, 38, 82, -14, 62
MATHEMATICA
f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; a[1] = 1; a[n_] := n - Times @@ f @@@ (fct = FactorInteger[n]) + Times @@ fc @@@ fct; Array[a, 100] (* Amiram Eldar, Dec 08 2023 *)
PROG
(PARI)
A007947(n) = factorback(factorint(n)[, 1]);
6, 28, 234, 496, 588, 600, 1521, 1638, 6552, 8128, 55860, 89376, 33550336, 168836850
COMMENTS
Numbers k such that either both A336563(k) and A336564(k) are zero (in which case k is squarefree), or A336563(k) divides A336564(k), in which case k is not squarefree. Question: Are there any other odd terms apart from 1521 = 39^2 ?
PROG
(PARI)
A007947(n) = factorback(factorint(n)[, 1]);
isA336565(n) = { my(b= A057723(n), c=(sigma(n)-b), d=(b-n)); (gcd(d, (n-c))==d); };
Sum of proper divisors of {n divided by its largest squarefree divisor}.
+10
3
0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 1, 0, 0, 0, 7, 0, 1, 0, 1, 0, 0, 0, 3, 1, 0, 4, 1, 0, 0, 0, 15, 0, 0, 0, 6, 0, 0, 0, 3, 0, 0, 0, 1, 1, 0, 0, 7, 1, 1, 0, 1, 0, 4, 0, 3, 0, 0, 0, 1, 0, 0, 1, 31, 0, 0, 0, 1, 0, 0, 0, 16, 0, 0, 1, 1, 0, 0, 0, 7, 13, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 15, 0, 1, 1, 8, 0, 0, 0, 3, 0
PROG
(PARI)
A007947(n) = factorback(factorint(n)[, 1]);
0, 1, 1, 3, 1, 6, 1, 7, 8, 8, 1, 10, 1, 10, 9, 15, 1, 15, 1, 18, 11, 14, 1, 18, 24, 16, 25, 14, 1, 18, 1, 31, 15, 20, 13, 35, 1, 22, 17, 30, 1, 30, 1, 42, 42, 26, 1, 34, 48, 49, 21, 50, 1, 42, 17, 54, 23, 32, 1, 54, 1, 34, 62, 63, 19, 54, 1, 66, 27, 66, 1, 69, 1, 40, 74, 74, 19, 66, 1, 78, 80, 44, 1, 70, 23, 46, 33, 86
COMMENTS
Some terms, for example a(600) and a(6552), are negative. - Georg Fischer, Jul 31 2020
PROG
(PARI)
A007947(n) = factorback(factorint(n)[, 1]);
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