A177329 -id:A177329 - cp's OEIS frontend

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.

2, 8, 44, 83, 4475, 75093, 164903, 59480, 1342805

Offset: 1

Amiram Eldar, Jul 16 2024

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.

			  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

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

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]

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;}

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

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

Amiram Eldar, Jul 16 2024

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.

			  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

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

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]

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;}

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

a(9)-a(12) from Chai Wah Wu, Jul 18 2024

A177333 Smallest factor in the factorization of n! over distinct terms of A050376.

Original entry on oeis.org

2, 2, 2, 2, 5, 5, 2, 2, 7, 7, 3, 3, 2, 2, 2, 2, 5, 5, 4, 3, 2, 2, 4, 4, 2, 2, 2, 2, 4, 4, 2, 2, 3, 3, 3, 3, 2, 2, 4, 4, 2, 2, 2, 2, 3, 3, 4, 4, 2, 2, 2, 2, 4, 4, 2, 2, 3, 3, 7, 7, 2, 2, 2, 2, 3, 3, 3, 4, 2, 2, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 3, 4, 2, 2, 4, 4, 5, 3, 2, 2, 4, 4, 2, 2, 2, 2, 3, 3, 2, 2, 4

Offset: 2

Vladimir Shevelev, May 06 2010

nonn

			The factorization of 10! = 3628800 is 2^8*3^4*5^2*7^1, where 2^8 > 3^4 > 5^2 > 7, so a(10)=7 is the smallest of these 4 factors.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].

Amiram Eldar, Table of n, a(n) for n = 2..1000
S. Litsyn and V. S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Cf. A050376, A177329, A055460, A176525, A001358, A176472, A176509, A064380, A050292.

A177333 := proc(n) local a,p,pow2 ; a := n! ; for p in ifactors(n!)[2] do pow2 := convert( op(2,p),base,2) ; for j from 1 to nops(pow2) do if op(j,pow2) <> 0 then a := min(a,op(1,p)^(2^(j-1))) ; end if; end do: end do: return a ; end proc:
seq(A177333(n),n=2..120) ; # R. J. Mathar, Jun 16 2010

b[n_] :=2^(-1+Position[ Reverse@IntegerDigits[n, 2],?(#==1&)])//Flatten; a[n] := Module[{np = PrimePi[n]}, v=Table[0,{np}]; Do[p = Prime[k]; Do[v[[k]] += IntegerExponent[j, p], {j,2,n}],  {k,1,np}]; Min[(Prime/@Range[np])^(b/@v) // Flatten]]; Array[a, 105, 2] (* Amiram Eldar, Sep 17 2019 *)

Corrected from a(10) on and extended beyond a(30) by R. J. Mathar, Jun 16 2010

A177334 Largest factor in the factorization of n! over distinct terms of A050376.

Original entry on oeis.org

2, 3, 4, 5, 16, 16, 16, 81, 256, 256, 256, 256, 256, 256, 256, 256, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 4294967296, 4294967296, 4294967296, 4294967296, 4294967296, 4294967296

Offset: 2

Vladimir Shevelev, May 06 2010

nonn

Each number >=2 has a unique factorization over distinct terms of A050376.
This is obtained from the standard prime factor representation by splitting the exponents into a sum of powers of 2, and further factorization according to the nonzero term of this base-2 representation.
The largest factor of this representation of A000142(n) defines this sequence.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].

Amiram Eldar, Table of n, a(n) for n = 2..1000
S. Litsyn and V. S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Cf. A050376, A177329, A055460, A177333, A176525, A001358, A176509, A064380, A050292.

A177334 := proc(n) local a,p,pow2 ; a := 1 ; for p in ifactors(n!)[2] do pow2 := convert( op(2,p),base,2) ; for j from 1 to nops(pow2) do if op(j,pow2) <> 0 then a := max(a,op(1,p)^(2^(j-1))) ; end if; end do: end do: return a ; end proc:
seq(A177334(n),n=2..60) ; # R. J. Mathar, Jun 16 2010

b[n_] :=2^(-1+Position[ Reverse@IntegerDigits[n, 2],?(#==1&)])//Flatten; a[n] := Module[{np = PrimePi[n]}, v=Table[0,{np}]; Do[p = Prime[k]; Do[v[[k]] += IntegerExponent[j, p], {j,2,n}],  {k,1,np}]; Max[(Prime/@Range[np])^(b/@v) // Flatten]]; Array[a, 38, 2]  (* Amiram Eldar, Sep 17 2019 *)

a(18) and a(19) corrected and sequence extended by R. J. Mathar, Jun 16 2010

A241139 Number of nonprimes in factorization of n! over distinct terms of A050376.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6, 7, 7, 4, 4, 5, 5, 6, 6, 8, 9, 10, 10, 9, 9, 11, 11, 12, 12, 10, 9, 8, 8, 9, 10, 11, 11, 12, 12, 11, 12, 14, 14, 16, 15, 15, 15, 13, 13, 14, 14, 14, 14, 16, 16, 16, 16, 17, 19, 21, 21, 18, 18, 19, 16, 14, 14, 16, 16, 17

Offset: 2

Vladimir Shevelev, Apr 16 2014

nonn

			Factorization of 4! over distinct terms of A050376 is 4! = 2*3*4. This factorization contains only one A050376-nonprime. So a(4)=1.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].

Amiram Eldar, Table of n, a(n) for n = 2..1000
S. Litsyn and V. S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Cf. A177329, A177333, A177334, A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906, A241123, A241124.

b[n_] := 2^(-1 + Position[Reverse@IntegerDigits[n, 2], ?(# == 1 &)]) // Flatten; a[n] := Module[{np = PrimePi[n]}, v = Table[0, {np}]; Do[p = Prime[k]; Do[v[[k]] += IntegerExponent[j, p], {j, 2, n}], {k, 1, np}]; Length[Select[(b /@ v) // Flatten, # > 1 &]]]; Array[a, 73, 2]  (* Amiram Eldar, Sep 17 2019 *)

a(n)={my(f=factor(n!)[,2]); sum(i=1, #f~, hammingweight(f[i]>>1))} \\ Andrew Howroyd, Sep 17 2019

a(n) = A177329(n) - A055460(n).

More terms from Peter J. C. Moses, Apr 17 2014

A374671 Positive numbers k such that k! and (k+1)! have an equal number of infinitary divisors.

Original entry on oeis.org

8, 19, 23, 44, 45, 57, 67, 76, 80, 83, 84, 85, 105, 107, 116, 120, 123, 140, 141, 146, 158, 161, 165, 174, 177, 187, 201, 208, 214, 225, 235, 239, 241, 243, 244, 246, 247, 263, 269, 272, 277, 284, 297, 309, 315, 321, 322, 325, 337, 339, 341, 342, 344, 360, 363

Offset: 1

Amiram Eldar, Jul 16 2024

nonn

Positive numbers such that k! and (k+1)! have an equal number of Fermi-Dirac factors (A064547).
Positive numbers k such that A037445(k!) = A037445((k+1)!).
Positive numbers k such that A064547(k!) = A064547((k+1)!).
Positive numbers k such that A177329(k) = A177329(k+1).

			8 is a term since A037445(8!) = A037445(9!) = 64.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A037445, A064547, A177329, A343819, A374672, A374673, A374674.

s[n_] := s[n] = Module[{e = FactorInteger[n!][[;; , 2]]}, Sum[DigitCount[e[[k]], 2, 1], {k, 1, Length[e]}]]; Select[Range[2, 400], s[#] == s[# + 1] &]

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);}

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

Amiram Eldar, Jul 16 2024

nonn

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).

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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] &]

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);}

A375635 The number of infinitary divisors of n!.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 8, 16, 64, 64, 16, 32, 128, 256, 512, 1024, 2048, 4096, 256, 512, 512, 2048, 4096, 8192, 8192, 16384, 32768, 65536, 16384, 32768, 65536, 131072, 524288, 2097152, 131072, 65536, 32768, 65536, 131072, 262144, 524288, 1048576, 4194304, 8388608

Offset: 0

Amiram Eldar, Aug 22 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 0..3000

Cf. A027423, A037445, A048656, A177329.

f[p_, e_] := 2^DigitCount[e, 2, 1]; a[0] = a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 100, 0]

a(n) = vecprod(apply(x -> 2^hammingweight(x), factor(n!)[,2]));

from collections import Counter
from sympy import factorint
def A375635(n): return 1<Chai Wah Wu, Aug 22 2024

a(n) = A037445(n!).
a(n) = 2^A177329(n).
A048656(n) <= a(n) <= A027423(n).

A241148 Number of factorials k!, 0<=k<=n, relatively prime to n! in Fermi-Dirac arithmetic.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 5, 5, 2, 2, 7, 7, 4, 4, 4, 2, 2, 2, 5, 5, 7, 4, 3, 3, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 3, 3, 4, 4, 3, 2, 4, 4, 3, 3, 2, 4, 5, 5, 4, 4, 2, 2, 2, 2, 6, 5, 2, 2, 3, 3, 7, 7, 3, 2, 2, 2, 3, 3, 3, 4, 3, 3, 4, 4, 2, 2, 2, 2, 6, 6, 4, 4, 2, 2, 2, 3, 4

Offset: 0

Vladimir Shevelev, Apr 16 2014

nonn

Or, equivalently, the number of factorials k!, 0<=k<=n, for which k! and n! have no common A050376-factors in their factorizations over distinct terms of A050376.

Note that 1 (=0!=1!) corresponds to an empty subset of A050376.

			0!=1, 1!=1; further we have the following factorizations of k! over distinct terms of A050376 for k = 2,3,4,5,6:
2!=2, 3!=2*3, 4!=2*3*4, 5!=2*3*4*5, 6!=5*9*16. Thus, in the sense of the factorizations being considered, 6! is relatively prime to 0!,1!,2!,3!, and 4!. So a(6)=5.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 (Russian; MR 2000f: 11097, pp. 3912-3913).

Amiram Eldar, Table of n, a(n) for n = 0..500
S. Litsyn and V. S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Cf. A177329, A177333, A177334, A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906, A241123, A241124, A241139.

b[n_] := 2^(-1 + Position[Reverse@IntegerDigits[n, 2], ?(# == 1 &)]) // Flatten; infp[n] := Module[{np = PrimePi[n]}, v = Table[0, {np}]; Do[p = Prime[k]; Do[v[[k]] += IntegerExponent[j, p], {j, 2, n}], {k, 1, np}]; (Prime /@ Range[np])^(b /@ v) // Flatten]; infCoprimeQ[x_, y_] := Intersection[infp[x], infp[y]] == {}; a[n_] := Length @ Select[Range[0, n], infCoprimeQ[n, #] & ]; Array[a, 87, 0] (* Amiram Eldar, Sep 17 2019 *)

More terms from Peter J. C. Moses, Apr 18 2014

A241289 Numbers n for which in the factorization of n! over distinct terms of A050376, the numbers of prime and nonprime terms are equal.

Original entry on oeis.org

7, 8, 9, 13, 18, 22, 37, 57, 71

Offset: 1

Vladimir Shevelev, Apr 18 2014

a(10), if it exists, should be more than 5000. Is a(9)=71 the last term of sequence? - Peter J. C. Moses, Apr 19 2014

One can prove that a(9)=71 indeed is the last term of this sequence. - Vladimir Shevelev, Apr 19 2014.

			7 is in the sequence, since 7! in the considered factorization is 5*7*9*16, and here we have 2 primes and 2 nonprimes.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].

S. Litsyn and V. S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Cf. A177329, A177333, A177334, A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906, A241123, A241124, A241139, A241148.

Terms a(7) - a(9) from Peter J. C. Moses, Apr 19 2014

cp's OEIS Frontend

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.

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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.

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A177333 Smallest factor in the factorization of n! over distinct terms of A050376.

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A177334 Largest factor in the factorization of n! over distinct terms of A050376.

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A241139 Number of nonprimes in factorization of n! over distinct terms of A050376.

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A374671 Positive numbers k such that k! and (k+1)! have an equal number of infinitary divisors.

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

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