cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A331974 Infinitary highly touchable numbers: numbers m > 1 such that a record number of numbers k have m as the sum of the proper infinitary divisors of k.

Original entry on oeis.org

2, 6, 8, 17, 21, 37, 49, 55, 67, 79, 85, 91, 121, 151, 175, 181, 211, 295, 301, 361, 391, 421, 481, 511, 571, 631, 781, 841, 991, 1051, 1231, 1261, 1471, 1561, 1651, 1681, 1891, 2101, 2311, 2731, 3151, 3361, 3571, 3991, 4201, 4291, 4411, 4621, 5251, 5461, 6091
Offset: 1

Views

Author

Amiram Eldar, Feb 03 2020

Keywords

Comments

The corresponding record values are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...
The infinitary version of A238895.

Examples

			a(1) = 2 since it is the first number which is not the sum of proper infinitary divisors of any number.
a(2) = 6 since it is the least number which is the sum of proper infinitary divisors of one number: 6 = A126168(6).
a(3) = 8 since it is the least number which is the sum of proper infinitary divisors of 2 numbers: 8 = A126168(10) = A126168(12).

Links

Crossrefs

Cf. A049417, A077609, A126168, A238895, A324277, A325177, A331972, A331973.

Programs

  • Mathematica
    fun[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ (fun @@@ FactorInteger[n]); is[n_] := isigma[n] - n; m = 300; v = Table[0, {m}]; Do[i = is[k]; If[2 <= i <= m, v[[i]]++], {k, 1, m^2}]; s = {}; vm = -1; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[s, k]], {k, 2, m}]; s

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

Views

Author

Amiram Eldar, May 12 2024

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A307958, A336563, A372740, A372742.
Similar sequences: A048138, A324938, A331971, A331973.

Programs

  • 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]
  • 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;}

Formula

a(n) = 0 if and only if n is in A372740.
a(n) = 1 if and only if n is in A372742.

A331971 a(n) is the number of values of m such that the sum of proper bi-unitary divisors of m (A331970) is n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 1, 3, 2, 3, 1, 3, 1, 3, 2, 4, 1, 6, 1, 4, 2, 4, 2, 5, 0, 3, 1, 4, 2, 5, 1, 4, 2, 4, 1, 6, 2, 5, 2, 5, 2, 8, 1, 6, 1, 4, 2, 7, 1, 5, 3, 5, 2, 8, 0, 5, 1, 6, 1, 8, 2, 5, 3, 6, 3, 9, 0, 6, 2, 5, 1, 9, 1, 7, 1
Offset: 2

Views

Author

Amiram Eldar, Feb 03 2020

Keywords

Comments

The bi-unitary version of A048138.
The offset is 2 as in A048138 since there are infinitely many numbers k (the primes and squares of primes) for which A331970(k) = 1.

Examples

			a(8) = 2 since 8 is the sum of the proper bi-unitary divisors of 2 numbers: 10 (1 + 2 + 5) and 12 (1 + 3 + 4).

Links

Crossrefs

Cf. A048138, A188999, A222266, A324938, A331970, A331973.

Programs

  • Mathematica
    fun[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), (p^(e + 1) - 1)/(p - 1) - p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); bs[n_] := bsigma[n] - n; m = 300; v = Table[0, {m}]; Do[b = bs[k]; If[2 <= b <= m, v[[b]]++], {k, 1, m^2}]; Rest @ v

A361419 Numbers k such that there is a unique number m for which the sum of the aliquot infinitary divisors of m (A126168) is k.

Original entry on oeis.org

0, 6, 7, 9, 11, 18, 32, 44, 56, 62, 72, 82, 94, 96, 98, 102, 104, 110, 116, 122, 132, 136, 138, 146, 150, 152, 178, 180, 182, 210, 222, 226, 230, 236, 238, 242, 248, 252, 264, 272, 284, 292, 296, 304, 322, 332, 338, 342, 350, 356, 360, 374, 382, 390, 392, 404
Offset: 1

Views

Author

Amiram Eldar, Mar 11 2023

Keywords

Comments

Numbers k such that A331973(k) = 1.

Links

Crossrefs

Cf. A126168, A324277, A331974, A331973, A361420.
Similar sequences: A057709, A357324.

Programs

  • Mathematica
    f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; is[1] = 0; is[n_] := Times @@ f @@@ FactorInteger[n] - n;
seq[max_] := Module[{v = Table[0, {max}], i}, Do[i = is[k] + 1; If[i <= max, v[[i]]++], {k, 1, max^2}]; -1 + Position[v, 1] // Flatten];
seq[500]
  • PARI
    s(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k)) + 1, 1))) - n; }
lista(nmax) = {my(v = vector(nmax+1)); for(k=1, nmax^2, i = s(k) + 1; if(i <= nmax+1, v[i] += 1)); for(i = 1, nmax+1, if(v[i] == 1, print1(i-1, ", "))); }

Formula

a(n) = A126168(A361420(n)).
Showing 1-4 of 4 results.

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