A372742 -id:A372742 - cp's OEIS frontend

A372743 a(n) is the unique number m such that A336563(m) = A372742(n).

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

Offset: 1

Amiram Eldar, May 12 2024

nonn

Includes all the squares of primes (A001248).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A336563, A372742.
A001248 is a subsequence.
Similar sequences: A357313, A357325, A361420.

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]

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], ", ")));}

A336563(a(n)) = A372742(n).

A372739 a(n) is the number of possible values of k such that the sum of aliquot coreful divisors of k (A336563) is n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 12 2024

nonn

A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958).

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A307958, A336563, A372740, A372742.

Similar sequences: A048138, A324938, A331971, A331973.

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]

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;}

a(n) = 0 if and only if n is in A372740.

a(n) = 1 if and only if n is in A372742.

A373743 Expansion of e.g.f. exp(x^3/6 * (1 + x)^2).

Original entry on oeis.org

1, 0, 0, 1, 8, 20, 10, 280, 3360, 20440, 67200, 462000, 7407400, 73673600, 482081600, 3364761400, 47311264000, 657536880000, 6586994814400, 58707179731200, 740032028736000, 11832726841936000, 161121297104768000, 1857897194273120000, 23875495204536976000

Offset: 0

Seiichi Manyama, Jun 16 2024

nonn

Cf. A361568, A372742.
Cf. A361278, A361567.
Cf. A264622.

a(n) = n!*sum(k=0, n\3, binomial(2*k,n-3*k)/(6^k*k!));

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(2*k,n-3*k)/(6^k * k!).

a(n) = (n-1)*(n-2)/6 * (3*a(n-3) + 8*(n-3)*a(n-4) + 5*(n-3)*(n-4)*a(n-5)).

cp's OEIS Frontend

A372743 a(n) is the unique number m such that A336563(m) = A372742(n).

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A372739 a(n) is the number of possible values of k such that the sum of aliquot coreful divisors of k (A336563) is n.

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A373743 Expansion of e.g.f. exp(x^3/6 * (1 + x)^2).

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Author

Keywords

Crossrefs

Programs

PARI

Formula