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 1-3 of 3 results.

A374672 Numbers k such that k! has more infinitary divisors than (k+1)!.

Original entry on oeis.org

5, 9, 17, 27, 33, 34, 35, 43, 48, 51, 53, 59, 65, 68, 69, 75, 77, 87, 91, 97, 98, 99, 103, 115, 119, 125, 129, 134, 135, 139, 147, 149, 151, 155, 163, 164, 171, 179, 183, 189, 194, 195, 197, 199, 203, 211, 215, 221, 227, 229, 230, 231, 237, 245, 249, 257, 259
Offset: 1

Views

Author

Amiram Eldar, Jul 16 2024

Keywords

Comments

Numbers k such that k! has more Fermi-Dirac factors (A064547) than (k+1)!.
Numbers k such that A037445(k!) > A037445((k+1)!).
Numbers k such that A064547(k!) > A064547((k+1)!).
Numbers k such that A177329(k) > A177329(k+1).

Examples

			5 is a term since A037445(5!) = 16 > A037445(6!) = 8.

Links

Crossrefs

Cf. A037445, A064547, A177329, A343819, A374671, A374673, A374674.

Programs

  • Mathematica
    s[n_] := s[n] = Module[{e = FactorInteger[n!][[;; , 2]]}, Sum[DigitCount[e[[k]], 2, 1], {k, 1, Length[e]}]]; Select[Range[2, 300], s[#] > s[# + 1] &]
  • PARI
    s(n) = {my(e = factor(n!)[, 2]); sum(k=1, #e, hammingweight(e[k]));}
lista(kmax) = {my(s1 = s(1), s2); for(k = 2, kmax, s2 = s(k); if(s1 > s2, print1(k-1, ", ")); s1 = s2);}

A374673 a(n) is the start of the least run of exactly n consecutive positive numbers with an equal value of A177329, or -1 if no such run exists.

Original entry on oeis.org

2, 8, 44, 83, 4475, 75093, 164903, 59480, 1342805
Offset: 1

Views

Author

Amiram Eldar, Jul 16 2024

Keywords

Comments

For n > 1, a(n)! is the start of the least run of successive factorials of positive numbers (i.e., ignoring 0!) with an equal number of infinitary divisors (A037445).
a(9) > 320000, if it exists.

Examples

			  n |   a(n) | A177329(k), k = a(n), a(n)+1, ..., a(n)+n-1
  --|--------|------------------------------------------------
  1 |      2 | A177329(2) = 1
  2 |      8 | A177329(8) = A177329(9) = 6
  3 |     44 | A177329(44) = A177329(45) = A177329(46) = 21
  4 |     83 | A177329(83) = ... = A177329(86) = 35
  5 |   4475 | A177329(4475) = ... A177329(4479) = 923
  6 |  75093 | A177329(75093) = ... = A177329(75098) = 10857
  7 | 164903 | A177329(164903) = ... = A177329(164909) = 22038
  8 |  59480 | A177329(59480) = ... = A177329(59487) = 8814

Crossrefs

Cf. A177329, A318166, A374671, A374672, A374674.

Programs

  • Mathematica
    s[n_] := Module[{e = FactorInteger[n!][[;; , 2]]}, Sum[DigitCount[e[[k]], 2, 1], {k, 1, Length[e]}]]; seq[len_] := Module[{v = Table[0, {len}], w = {0}, c = 0, k = 3, m, s1}, While[c < len, s1 = s[k]; m = Length[w]; If[s1 == w[[m]], AppendTo[w, s1], If[m <= len && v[[m]] == 0, v[[m]] = k-m; c++]; w = {s1}]; k++]; v]; seq[5]
  • PARI
    s(n) = {my(e = factor(n!)[, 2]); sum(k=1, #e, hammingweight(e[k]));}
lista(len) = {my(v = vector(len), w = [0], c = 0, k = 3, m, s1); while(c < len, s1 = s(k); m = #w; if(s1 == w[m], w = concat(w, s1), if(m < = len && v[m] == 0, v[m] = k-m; c++); w = [s1]); k++); v;}
  • Python
    from itertools import count
from collections import Counter
from sympy import factorint
def A374673(n):
    if n==1: return 2
    c, a, l = Counter(), 0, 0
    for m in count(2):
        c += Counter(factorint(m))
        b = sum(map(int.bit_count,c.values()))
        if b==a:
            l += 1
        else:
            if l==n-1:
                return m-n
            l = 0
        a = b # Chai Wah Wu, Jul 18 2024

Extensions

a(9) from Chai Wah Wu, Jul 18 2024

A374674 a(n) is the start of the least run of exactly n consecutive positive numbers with strictly decreasing values of A177329, or -1 if no such run exists.

Original entry on oeis.org

2, 5, 68, 33, 709, 2313, 13251, 17961, 231881, 525323, 4172904, 7163595
Offset: 1

Views

Author

Amiram Eldar, Jul 16 2024

Keywords

Comments

For n > 1, a(n)! is the start the least run of successive factorials with strictly decreasing number of infinitary divisors (A037445).
a(9) > 170000, if it exists.

Examples

			  n |  a(n) | A177329(k), k = a(n), a(n)+1, ..., a(n)+n-1
  --|-------|------------------------------------------------------
  1 |     2 | 1
  2 |     5 | 4 > 3
  3 |    68 | 31 > 28 > 27
  4 |    33 | 21 > 17 > 16 > 15
  5 |   709 | 199 > 197 > 195 > 193 > 190
  6 |  2313 | 528 > 523 > 519 > 518 > 513 > 508
  7 | 13251 | 2355 > 2354 > 2353 > 2351 > 2350 > 2345 > 2343
  8 | 17961 | 3060 > 3056 > 3051 > 3049 > 3048 > 3047 > 3044 > 3041

Crossrefs

Cf. A037445, A177329, A374671, A374672, A374673.

Programs

  • Mathematica
    s[n_] := Module[{e = FactorInteger[n!][[;; , 2]]}, Sum[DigitCount[e[[k]], 2, 1], {k, 1, Length[e]}]]; seq[len_] := Module[{v = Table[0, {len}], w = {s[2]}, c = 0, k = 3, m, s1}, While[c < len, s1 = s[k]; m = Length[w]; If[s1 < w[[m]], AppendTo[w, s1], If[m <= len && v[[m]] == 0, v[[m]] = k-m; c++]; w = {s1}]; k++]; v]; seq[5]
  • PARI
    s(n) = {my(e = factor(n!)[, 2]); sum(k=1, #e, hammingweight(e[k]));}
lista(len) = {my(v = vector(len), w = [s(2)], c = 0, k = 3, m, s1); while(c < len, s1 = s(k); m = #w; if(s1 < w[m], w = concat(w, s1), if(m < = len && v[m] == 0, v[m] = k-m; c++); w = [s1]); k++); v;}
  • Python
    from itertools import count
from collections import Counter
from sympy import factorint
def A374674(n):
    if n==1: return 2
    c, a, l = Counter(), 0, 0
    for m in count(2):
        c += Counter(factorint(m))
        b = sum(map(int.bit_count,c.values()))
        if bChai Wah Wu, Jul 18 2024

Extensions

a(9)-a(12) from Chai Wah Wu, Jul 18 2024
Showing 1-3 of 3 results.

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