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.

A098565 Numbers that appear as binomial coefficients exactly 6 times.

Original entry on oeis.org

120, 210, 1540, 7140, 11628, 24310, 61218182743304701891431482520
Offset: 1

Views

Author

Paul D. Hanna, Oct 27 2004

Keywords

Links

Crossrefs

See A098564 for more information.
Cf. A185024, A182237. Subsequence of A003015.
Cf. A059233.

Programs

  • Haskell
    import Data.List (elemIndices)
a098565 n = a098565_list !! (n-1)
a098565_list = map (+ 2 ) $ elemIndices 3 a059233_list
-- Reinhard Zumkeller, Dec 24 2012

Formula

A059233(a(n)) = 3. - Reinhard Zumkeller, Dec 24 2012

Extensions

a(7) from T. D. Noe, Jul 13 2005

A137905 Numbers that appear as binomial coefficients exactly twice.

Original entry on oeis.org

3, 4, 5, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85, 86, 87, 88
Offset: 1

Views

Author

David Wasserman, Feb 21 2008

Keywords

Comments

Complement of A006987; a(n) = A058084(a(n)). - Reinhard Zumkeller, Mar 20 2009

Examples

			7 is a member because 7 = binomial(7, 1) = binomial(7, 6) and no other binomial coefficient equals 7. [clarified by _Jonathan Sondow_, Jan 12 2018]

Crossrefs

Cf. A006987, A058084, A062527, A098564, A098566.

Programs

  • PARI
    isok(n) = (sum(i=0, n, sum(j=0, i, binomial(i,j)==n)) == 2) \\ Michel Marcus, Jun 16 2013

Formula

a(n) = A185024(n+1). - Elijah Beregovsky, May 14 2019

A180058 Smallest number occurring in exactly n rows of Pascal's triangle.

Original entry on oeis.org

2, 6, 120, 3003
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 24 2012

Keywords

Comments

A059233(a(n)) = n and A059233(m) < n for m < a(n).

Examples

			.  n  A180058  referred equal binomial coefficients (A007318)  A059233
.  -  -------  ----------------------------------------------  -------
.  1        2   C (2, 1)                                             1
.  2        6   C (4, 2)   C (6, 1)                                  2
.  3      120   C (10, 3)  C (16, 2)  C (120, 1)                     3
.  4     3003   C (14, 6)  C (15, 5)  C (78, 2)   C (3003, 1)        4 .

Links

Crossrefs

Cf. A185024, A182237, A098565, A003015, A003016.

Programs

  • Haskell
    import Data.List (elemIndex)
import Data.Maybe (fromJust)
a180058 = (+ 2) . fromJust . (`elemIndex` a059233_list)

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
Showing 1-4 of 4 results.

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