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 1-5 of 5 results.

A358252 a(n) is the least number with exactly n non-unitary square divisors.

Original entry on oeis.org

1, 8, 32, 128, 288, 864, 1152, 2592, 4608, 13824, 10368, 20736, 28800, 41472, 64800, 279936, 115200, 331776, 345600, 663552, 259200, 1679616, 518400, 1620000, 1166400, 4860000, 1036800, 17915904, 2073600, 15552000, 6998400, 26873856, 4147200, 53747712, 8294400
Offset: 0

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Author

Amiram Eldar, Nov 05 2022

Keywords

Comments

a(n) is the least number k such that A056626(k) = n.
Since A056626(k) depends only on the prime signature of k, all the terms of this sequence are in A025487.

Examples

			a(1) = 8 since 8 is the least number that has exactly one non-unitary square divisor, 4.

Links

Crossrefs

Cf. A056626, A025487, A358253.
Similar sequences: A005179 (all divisors), A038547 (odd divisors), A085629 (coreful divisors), A130279 (square), A187941 (even), A309181 (non-unitary), A340232 (bi-unitary), A340233 (exponential), A357450 (odd square).

Programs

  • Mathematica
    f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^(1 - Mod[e, 2]); f[1] = 0; f[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[21, 10^6]
  • PARI
    s(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + floor(f[i,2]/2)) - 2^sum(i = 1, #f~, 1 - f[i,2]%2);}
lista(len, nmax) = {my(v = vector(len), c = 0, n = 1, i); while(c < len && n < nmax, i = s(n) + 1; if(i <= len && v[i] == 0, c++; v[i] = n); n++); v};

A358262 a(n) is the least number with exactly n noninfinitary square divisors.

Original entry on oeis.org

1, 16, 144, 256, 3600, 1296, 2304, 65536, 129600, 16777216, 32400, 20736, 57600, 331776, 589824, 4294967296, 6350400, 1099511627776, 150994944, 810000, 1587600, 1679616, 518400, 5308416, 2822400, 84934656, 8294400, 26873856, 14745600, 21743271936, 38654705664
Offset: 0

Views

Author

Amiram Eldar, Nov 06 2022

Keywords

Comments

a(n) is the least number k such that A358261(k) = n.
Since A358261(k) depends only on the prime signature of k, all the terms of this sequence are in A025487.

Examples

			a(1) = 16 since 16 is the least number with exactly one noninfinitary divisor, 4.

Links

Crossrefs

Cf. A025487, A358261, A358263.
Similar sequences: A005179 (all divisors), A038547 (odd divisors), A085629 (coreful divisors), A130279 (square), A187941 (even), A309181 (non-unitary), A340232 (bi-unitary), A340233 (exponential), A357450 (odd square), A358252 (non-unitary square).

Programs

  • Mathematica
    f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^DigitCount[If[OddQ[e], e - 1, e], 2, 1]; f[1] = 0; f[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[15, 2*10^7]
  • PARI
    s(n) = {my(f = factor(n));  prod(i=1, #f~, 1+f[i,2]\2) - prod(i=1, #f~, 2^hammingweight(if(f[i,2]%2, f[i,2]-1, f[i,2])))};
lista(len, nmax) = {my(v = vector(len), c = 0, n = 1, i); while(c < len && n < nmax, i = s(n) + 1; if(i <= len && v[i] == 0, c++; v[i] = n); n++); v};

A358345 a(n) is the number of even square divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2
Offset: 1

Views

Author

Amiram Eldar, Nov 11 2022

Keywords

Comments

First differs from A235127 at n = 36.
The first position of k >= 0 in this sequence is A187941(k)^2.

Links

Crossrefs

Cf. A016742, A046951, A187941, A222171, A235127, A298735.

Programs

  • Mathematica
    f1[p_, e_] := Floor[e/2] + 1; f2[p_, e_] := If[p == 2, 1, Floor[e/2] + 1]; a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i=1, #f~, 1+f[i,2]\2) - prod(i=1, #f~, if(f[i,1] == 2, 1, 1+f[i,2]\2))};
  • PARI
    a(n) = sumdiv(n, d, if (issquare(d) && !(d%2), 1)); \\ Michel Marcus, Nov 11 2022

Formula

a(n) = A046951(n) - A298735(n).
a(n) = 2 * A046951(n) - A046951(4*n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/24 (A222171).

A361418 a(n) is the least number with exactly n noninfinitary divisors.

Original entry on oeis.org

1, 4, 12, 16, 60, 36, 48, 256, 360, 4096, 180, 144, 240, 576, 768, 65536, 2520, 1048576, 12288, 900, 1260, 1296, 720, 2304, 1680, 9216, 2880, 5184, 3840, 147456, 196608, 36864, 27720, 46656, 3145728, 4398046511104, 61440, 3600, 6300, 18014398509481984, 10080, 20736
Offset: 0

Views

Author

Amiram Eldar, Mar 11 2023

Keywords

Comments

a(n) is the least number k such that A348341(k) = n.
Since A348341(k) depends only on the prime signature of k, all the terms of this sequence are in A025487.

Examples

			a(1) = 4 since 4 is the least number with exactly one noninfinitary divisor, 2.

Links

Crossrefs

Cf. A025487, A348341, A348342.
Similar sequences: A005179 (all divisors), A038547 (odd divisors), A085629 (coreful divisors), A130279 (square), A187941 (even), A309181 (non-unitary), A340232 (bi-unitary), A340233 (exponential), A357450 (odd square), A358252 (non-unitary square).

Programs

  • Mathematica
    f[1] = 0; f[n_] := DivisorSigma[0, n] - Times @@ Flatten[2^DigitCount[#, 2, 1] & /@ FactorInteger[n][[;; , 2]]];
seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s];
seq[35, 10^7]
  • PARI
    s(n) = {my(f = factor(n)); numdiv(f) - prod(i = 1, #f~, 2^hammingweight(f[i,2]));}
lista(len, nmax) = {my(v = vector(len), c = 0, n = 1, i); while(c < len && n < nmax, i = s(n) + 1; if(i <= len && v[i] == 0, c++; v[i] = n); n++); v};

A305183 Least k such that A048272(k) = -n.

Original entry on oeis.org

2, 4, 8, 16, 24, 64, 48, 256, 96, 144, 192, 4096, 240, 16384, 768, 576, 480, 262144, 720, 1048576, 960, 2304, 12288, 16777216, 1440, 5184, 49152, 3600, 3840, 1073741824, 2880, 4294967296, 3360, 36864, 786432, 20736, 5040, 274877906944, 3145728, 147456, 6720, 4398046511104, 11520, 17592186044416
Offset: 0

Views

Author

Seiichi Manyama, May 26 2018

Keywords

Comments

All terms are even.

Crossrefs

Cf. A005179, A048272, A187941.
Cf. A038547 (similar but with n instead of -n).

Programs

  • PARI
    a048272(n) = sumdiv(n, d, if (d%2, 1, -1));
a(n) = {my(k=1); while (a048272(k) != -n, k++); k;} \\ Michel Marcus, May 27 2018

Formula

a(n) = 2 * A187941(n).
a(n) = 4 * A005179(n) for n > 0.
Showing 1-5 of 5 results.

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