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 1-7 of 7 results.

A340232 a(n) is the least number with exactly 2*n bi-unitary divisors.

Original entry on oeis.org

2, 6, 32, 24, 512, 96, 8192, 120, 131072, 1536, 2097152, 480, 33554432, 24576, 536870912, 840, 8589934592, 7776, 137438953472, 7680, 2199023255552, 6291456, 35184372088832, 3360, 562949953421312, 100663296, 9007199254740992, 122880, 144115188075855872, 124416
Offset: 1

Views

Author

Amiram Eldar, Jan 01 2021

Keywords

Comments

Every integer except 1 has an even number of bi-unitary divisors.

Examples

			a(1) = 2 since 2 is the least number with 2*1 = 2 bi-unitary divisors, 1 and 2.
a(2) = 6 since 6 is the least number with 2*2 = 4 bi-unitary divisors, 1, 2, 3 and 6.

Links

Crossrefs

Subsequence of A025487.
Cf. A222266, A286324, A293185.
Similar sequences: A005179 (all divisors), A038547 (odd divisors), A085629 (coreful divisors), A309181 (nonunitary), A340233 (exponential).

Programs

  • Mathematica
    f[p_, e_] := If[OddQ[e], e + 1, e]; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]);  max = 10; s = Table[0, {max}]; c = 0; n = 2;  While[c < max, i = d[n]/2; If[i <= max && s[[i]] == 0, c++; s[[i]] = n]; n++]; s

Formula

A286324(a(n)) = 2*n and A286324(k) != 2*n for all k < a(n).

A340233 a(n) is the least number with exactly n exponential divisors.

Original entry on oeis.org

1, 4, 16, 36, 65536, 144, 18446744073709551616, 576, 1296, 589824
Offset: 1

Views

Author

Amiram Eldar, Jan 01 2021

Keywords

Comments

a(11) = 2^(2^10) has 309 digits and is too large to be included in the data section.
See the link for more values of this sequence.

Examples

			a(2) = 4 since 4 is the least number with 2 exponential divisors, 2 and 4.

Links

Crossrefs

Subsequence of A025487.
Cf. A049419, A318278, A322791.
Similar sequences: A005179 (all divisors), A038547 (odd divisors), A085629 (coreful divisors), A309181 (nonunitary), A340232 (bi-unitary).

Programs

  • Mathematica
    f[p_, e_] := DivisorSigma[0, e]; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]);  max = 6; s = Table[0, {max}]; c = 0; n = 1;  While[c < max, i = d[n]; If[i <= max && s[[i]] == 0, c++; s[[i]] = n]; n++]; s (* ineffective for n > 6 *)

Formula

A049419(a(n)) = n and A049419(k) != n for all k < a(n).

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

Views

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};

A344315 a(n) is the least number k such that A048105(k) = A048105(k+1) = 2*n, and 0 if it does not exist.

Original entry on oeis.org

1, 27, 135, 2511, 2295, 6975, 5264, 12393728, 12375, 2200933376, 108224, 257499, 170624, 3684603215871, 4402431, 2035980763136, 126224, 41680575, 701443071, 46977524, 1245375, 2707370000, 4388175, 3129761024, 1890944
Offset: 0

Views

Author

Amiram Eldar, May 14 2021

Keywords

Comments

There are no two consecutive numbers with an odd number of non-unitary divisors, since A048105(k) is odd only if k is a perfect square.
a(25) <= 1965640805422351777791, a(26) <= 3127059999. In general, a(n) <= A215199(n+1). - Daniel Suteu, May 20 2021

Examples

			a(0) = 1 since A048105(1) = A048105(2) = 0.
a(1) = 27 since A048105(27) = A048105(28) = 2.

Crossrefs

Cf. A048105, A075036, A093548, A309181, A344314.

Programs

  • Mathematica
    nd[n_] := DivisorSigma[0, n] - 2^PrimeNu[n]; seq[max_] := Module[{s = Table[0, {max}], k = 2, c = 0, nd1 = 0}, While[c < max, If[(nd2 = nd[k]) == nd1 && nd2 < 2*max && s[[nd2/2 + 1]] == 0, c++; s[[nd2/2 + 1]] = k - 1]; nd1 = nd2; k++]; s]; seq[7]
  • PARI
    A048105(n) = numdiv(n) - 2^omega(n);
isok(n,k) = A048105(k) == 2*n && A048105(k+1) == 2*n;
a(n) = for(k=1, oo, if(isok(n, k), return(k))); \\ Daniel Suteu, May 16 2021

Extensions

a(13)-a(24) confirmed by Martin Ehrenstein, May 20 2021

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};

A381731 a(n) is the least number k with squarefree neighbors such that the number of non-unitary divisors of k (A048105) is equal to n, or 0 if no such k exists.

Original entry on oeis.org

2, 4, 12, 16, 32, 36, 112, 256, 72, 0, 180, 144, 216, 16384, 768, 65536, 432, 1600, 3072, 900, 864, 1296, 720, 12544, 1080, 67108864, 2592, 268435456, 1440, 9216, 196608, 5184, 2160, 17179869184, 2880, 36864, 10368, 3600, 6300
Offset: 0

Views

Author

Juri-Stepan Gerasimov, Mar 05 2025

Keywords

Comments

From Amiram Eldar, Mar 06 2025: (Start)
For odd k a(k) is a square. a(9) = 0 because for a square m we have tau(m) >= 3^omega(m). Since A048105(m) = tau(m) - 2^omega(m) = 9, we have 2^omega(m) + 9 >= 3^omega(m) so omega(m) = 1.
Because m^2-1 is squarefree, m must be even, so with omega(m) = 1, we have m = 2^k and with tau(2^k) = 2^1 + 9 = 11 we get k = 10, m = 1024. But 1025 is not squarefree. Therefore a(9) = 0. (End)

Crossrefs

Cf. A048105, A280892, A309181.

Extensions

a(25), a(27) and more terms from Amiram Eldar, Mar 06 2025
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.