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.

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A355391 Position of first appearance of n in A181591 = binomial(bigomega(n), omega(n)).

Original entry on oeis.org

1, 4, 8, 16, 32, 24, 128, 256, 512, 48, 2048, 4096, 8192, 16384, 96, 65536, 131072, 262144, 524288, 240, 192, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 384, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 480, 768, 137438953472
Offset: 1

Views

Author

Gus Wiseman, Jul 04 2022

Keywords

Comments

The statistic omega = A001221 counts distinct prime factors (without multiplicity).
The statistic bigomega = A001222 counts prime factors with multiplicity.
We have A181591(2^k) = k, so the sequence is fully defined. Positions meeting this maximum are A185024, complement A006987.

Examples

			The terms together with their prime indices begin:
       1: {}
       4: {1,1}
       8: {1,1,1}
      16: {1,1,1,1}
      32: {1,1,1,1,1}
      24: {1,1,1,2}
     128: {1,1,1,1,1,1,1}
     256: {1,1,1,1,1,1,1,1}
     512: {1,1,1,1,1,1,1,1,1}
      48: {1,1,1,1,2}
    2048: {1,1,1,1,1,1,1,1,1,1,1}
    4096: {1,1,1,1,1,1,1,1,1,1,1,1}
    8192: {1,1,1,1,1,1,1,1,1,1,1,1,1}
   16384: {1,1,1,1,1,1,1,1,1,1,1,1,1,1}
      96: {1,1,1,1,1,2}
   65536: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  131072: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  262144: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  524288: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
     240: {1,1,1,1,2,3}
     192: {1,1,1,1,1,1,2}

Links

Crossrefs

Positions of powers of 2 are A185024, complement A006987.
Counting multiplicity gives A355386.
The sorted version is A355392.
A000005 counts divisors.
A001221 counts prime factors without multiplicity.
A001222 count prime factors with multiplicity.
A070175 gives representatives for bigomega and omega, triangle A303555.
Cf. A022811, A056239 , A071625, A118914, A181819, A323014, A323023, A355383 (with multiplicity A339006), A355384.

Programs

  • Mathematica
    s=Table[Binomial[PrimeOmega[n],PrimeNu[n]],{n,1000}];
Table[Position[s,k][[1,1]],{k,Select[Union[s],SubsetQ[s,Range[#]]&]}]
  • PARI
    f(n) = binomial(bigomega(n), omega(n)); \\ A181591
a(n) = my(k=1); while (f(k) != n, k++); k; \\ Michel Marcus, Jul 10 2022

Formula

binomial(bigomega(a(n)), omega(a(n))) = n.

Extensions

a(22)-a(28) from Michel Marcus, Jul 10 2022
a(29)-a(37) from Amiram Eldar, Jul 10 2022

A355392 Sorted positions of first appearances in A181591 = binomial(bigomega(n), omega(n)).

Original entry on oeis.org

1, 4, 8, 16, 24, 32, 48, 96, 128, 192, 240, 256, 384, 480, 512, 768, 960, 1536, 1920, 2048, 3072, 3360, 3840, 4096, 6144, 6720, 7680, 8192, 12288, 13440, 15360, 16384, 24576, 26880, 30720, 49152, 53760, 61440, 65536, 73920, 107520, 122880, 131072, 147840, 196608
Offset: 1

Views

Author

Gus Wiseman, Jul 04 2022

Keywords

Comments

These are the positions of terms in A181591 that are different from all prior terms.
The statistic omega = A001221 counts distinct prime factors (without multiplicity).
The statistic bigomega = A001222 counts prime factors with multiplicity.
We have A181591(2^k) = k, so the image under A181591 is a permutation of the positive integers. It begins: 1, 2, 3, 4, 6, 5, 10, 15, 7, 21, 20, ...

Examples

			The terms together with their prime indices begin:
    1: {}
    4: {1,1}
    8: {1,1,1}
   16: {1,1,1,1}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   48: {1,1,1,1,2}
   96: {1,1,1,1,1,2}
  128: {1,1,1,1,1,1,1}
  192: {1,1,1,1,1,1,2}
  240: {1,1,1,1,2,3}
  256: {1,1,1,1,1,1,1,1}
  384: {1,1,1,1,1,1,1,2}
  480: {1,1,1,1,1,2,3}
  512: {1,1,1,1,1,1,1,1,1}
  768: {1,1,1,1,1,1,1,1,2}
  960: {1,1,1,1,1,1,2,3}

Links

Crossrefs

The unsorted version with multiplicity is A355386.
This is the sorted version of A355391.
A000005 counts divisors.
A001221 counts prime indices without multiplicity.
A001222 count prime indices with multiplicity.
A070175 gives representatives for bigomega and omega, triangle A303555.
Cf. A000712, A022811, A056239, A071625, A118914, A181819, A323023, A355384.

Programs

  • Mathematica
    s=Table[Binomial[PrimeOmega[n],PrimeNu[n]],{n,1000}];
Select[Range[Length[s]],FreeQ[Take[s,#-1],s[[#]]]&]

Extensions

a(41)-a(45) from Amiram Eldar, Jul 10 2022

A378630 Numbers that set records in A376567.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 30, 48, 60, 120, 210, 240, 420, 840, 1680, 3360, 6720, 9240, 13440, 18480, 26880, 36960, 73920, 147840, 240240, 295680, 480480, 591360, 960960, 1921920, 3843840, 7687680, 15375360, 30750720, 32672640, 61501440, 65345280, 123002880, 130690560
Offset: 1

Views

Author

Michael De Vlieger, Dec 02 2024

Keywords

Comments

Numbers n that set records for binomial(bigomega(n)+omega(n), omega(n)), where bigomega = A001222 and omega = A001221.
a(n) is of the form 2^k * P(i), k >= 0, where primorial P = A002110.
Proper subset of A070175.

Links

Crossrefs

Cf. A000079, A001221, A001222, A002110, A007947, A070175, A376567.

Programs

  • Mathematica
    f[x_] := Block[{i, k, m, nn, p}, nn = Product[Prime[j], {j, x}]; Set[{k, i, p}, Range[0, 2]]; {1}~Join~Union@ Reap[Until[i > x, While[Set[m, 2^k*p] <= nn, Sow[m]; k++]; k = 0; i++; p *= Prime[i] ] ][[-1, 1]] ] (* generate A070175 *);
r = 0; Reap[Do[If[# > r, r = #; Sow[n]] &@ Binomial[#2 + #1, #1] & @@ {PrimeNu[n], PrimeOmega[n]}, {n, f[10]}] ][[-1, 1]]

A378631 Numbers that set records in A377071.

Original entry on oeis.org

1, 6, 12, 24, 30, 60, 120, 210, 420, 840, 1680, 2310, 3360, 4620, 6720, 9240, 18480, 36960, 60060, 73920, 120120, 147840, 240240, 480480, 960960, 1921920, 3843840, 7687680, 8168160, 15375360, 16336320, 30750720, 32672640, 65345280, 130690560, 261381120, 310390080
Offset: 1

Views

Author

Michael De Vlieger, Dec 02 2024

Keywords

Comments

Numbers n that set records for binomial(bigomega(n)+omega(n)-1, omega(n)-1), where bigomega = A001222 and omega = A001221.
a(n) is of the form 2^k * P(i), k >= 0, where primorial P = A002110.
Proper subset of A070175.

Links

Crossrefs

Cf. A000079, A001221, A001222, A002110, A007947, A070175, A377071.

Programs

  • Mathematica
    f[x_] := Block[{i, k, m, nn, p}, nn = Product[Prime[j], {j, x}]; Set[{k, i, p}, Range[0, 2]]; {1}~Join~Union@ Reap[Until[i > x, While[Set[m, 2^k*p] <= nn, Sow[m]; k++]; k = 0; i++; p *= Prime[i] ] ][[-1, 1]] ] (* generate A070175 *);
r = 0; Reap[Do[If[# > r, r = #; Sow[n]] &@ Binomial[#2 + #1 - 1, #1 - 1] & @@ {PrimeNu[n], PrimeOmega[n]}, {n, f[10]}] ][[-1, 1]]

A378632 Numbers that set records in A378181.

Original entry on oeis.org

1, 4, 6, 12, 24, 48, 60, 96, 120, 240, 420, 480, 840, 1680, 3360, 6720, 13440, 18480, 26880, 36960, 53760, 73920, 147840, 295680, 480480, 591360, 960960, 1182720, 1921920, 3843840, 7687680, 15375360, 30750720, 61501440, 65345280, 123002880, 130690560, 246005760
Offset: 1

Views

Author

Michael De Vlieger, Dec 02 2024

Keywords

Comments

Numbers n that set records for binomial(bigomega(n)+omega(n)-1, omega(n)), where bigomega = A001222 and omega = A001221.
a(n) is of the form 2^k * P(i), k >= 0, where primorial P = A002110.
Proper subset of A070175.

Links

Crossrefs

Cf. A000079, A001221, A001222, A002110, A007947, A070175, A378181.

Programs

  • Mathematica
    f[x_] := Block[{i, k, m, nn, p}, nn = Product[Prime[j], {j, x}]; Set[{k, i, p}, Range[0, 2]]; {1}~Join~Union@ Reap[Until[i > x, While[Set[m, 2^k*p] <= nn, Sow[m]; k++]; k = 0; i++; p *= Prime[i] ] ][[-1, 1]] ] (* generate A070175 *);
r = 0; Reap[Do[If[# > r, r = #; Sow[n]] &@ Binomial[#2 + #1 - 1, #1] & @@ {PrimeNu[n], PrimeOmega[n]}, {n, f[10]}] ][[-1, 1]]
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