A240906 -id:A240906 - cp's OEIS frontend

A241123 Smallest k such that the factorization of k! over distinct terms of A050376 contains exactly n primes.

2, 3, 5, 13, 17, 21, 23, 37, 33, 42, 43, 56, 59, 57, 75, 84, 99, 101, 105, 109, 123, 119, 133, 139, 157, 162, 163, 182, 186, 183, 207, 208, 222, 219, 235, 220, 255, 257, 263, 268, 267, 303, 305, 307, 316, 315, 340, 344, 341, 343, 383, 385, 387, 397, 411, 425

Offset: 1

Vladimir Shevelev, Apr 16 2014

nonn

			Factorization of 5! over distinct terms of A050376 is 5! = 2*3*4*5. Thus 5 is the smallest k such that such a factorization contains 3 primes: 2,3,5. So a(3)=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 = 1..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. A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906.

f[p_, e_] := Mod[e, 2]; b[1] = 0; b[n_] := Plus @@ (f @@@ FactorInteger[n]); m = 56; v = Table[0, {m}]; c = 0; p = 1; n = 2; While[c < m, p *= n; i = b[p]; If[i <= m && v[[i]] == 0, c++; v[[i]] = n]; n++]; v (* Amiram Eldar, Sep 17 2019 *)

nbp(n) = {f = factor(n); sum (i=1, #f~, f[i,2] % 2);}
a(n) = {k = 1; while(nbp(k!) != n, k++); k;} \\ Michel Marcus, Apr 27 2014

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

A241124 Smallest k such that the factorization of k! over distinct terms of A050376 contains at least n nonprime terms of A050376.

Original entry on oeis.org

4, 6, 8, 12, 14, 15, 16, 24, 25, 26, 30, 32, 46, 46, 48, 48, 62, 63, 63, 64, 64, 87, 91, 95, 96, 96, 96, 114, 114, 122, 124, 125, 128, 129, 160, 161, 176, 177, 178, 178, 188, 189, 190, 192, 192, 192, 194, 225, 226, 226, 240, 252, 254, 255, 256, 288, 288, 289, 290, 320

Offset: 1

Vladimir Shevelev, Apr 16 2014

nonn

			For k=2,3,4,5,6, we have the following factorizations of k! over distinct terms of A050376: 2!=2, 3!=2*3, 4!=2*3*4, 5!=2*3*4*5, 6!=5*9*16.
Therefore, a(1)=4, a(2)=6.

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

f[n_] := DigitCount[n, 2, 1] - Mod[n, 2]; nb[n_] := Total@(f/@ FactorInteger[n][[;;,2]]); a[n_] := (k=1; While[nb[k!] < n, k++]; k); Array[a, 60] (* Amiram Eldar, Dec 16 2018 from the PARI code *)

nb(n) = {my(f = factor(n)); sum(k=1, #f~, hammingweight(f[k,2]) - (f[k,2] % 2));}
a(n) = {my(k=1); while (nb(k!) < n, k++); k;} \\ Michel Marcus, Dec 16 2018

More terms from Michel Marcus, Dec 16 2018

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

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

A235623 Numbers n for which in the prime power factorization of n!, the numbers of exponents 1 and >1 are equal.

Original entry on oeis.org

0, 1, 4, 7, 8, 9, 13, 19, 20, 21

Offset: 1

Vladimir Shevelev, Apr 20 2014

Number n is in the sequence, if and only if pi(n) = 2*pi(n/2), where pi(x) is the number of primes<=x. Indeed, all primes from interval (n/2, n] appear in prime power factorization of n! with exponent 1, while all primes from interval (0, n/2] appear in n! with exponents >1. However, it follows from Ehrhart's link that, for n>=22, pi(n) < 2*pi(n/2). Therefore, a(9)=21 is the last term of the sequence.

m is in this sequence if and only if the number of prime divisors of [m/2]! equals the number of unitary prime divisors of m! - Peter Luschny, Apr 29 2014

			21! = 2^20*3^9*5^4*7^3*11*13*17*19. Here 4 primes with exponent 1 and 4 primes with exponents >1, so 21 is in the sequence.

Eugene Ehrhart, On prime numbers, Fibonacci Quarterly 26:3 (1988), pp. 271-274.

Cf. A056171, A177329, A177333, A177334, A240537, A240588, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906, A241123, A241124, A241139, A241148, A241289.

with(numtheory): a := proc(n) factorset(n!); factorset(iquo(n,2)!);
`if`(nops(%% minus %) = nops(%), n, NULL) end: seq(a(n), n=0..30); # Peter Luschny, Apr 28 2014

isok(n) = {f = factor(n!); sum(i=1, #f~, f[i,2] == 1) == sum(i=1, #f~, f[i,2] > 1);} \\ Michel Marcus, Apr 20 2014

cp's OEIS Frontend

A241123 Smallest k such that the factorization of k! over distinct terms of A050376 contains exactly n primes.

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A241124 Smallest k such that the factorization of k! over distinct terms of A050376 contains at least n nonprime terms of A050376.

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

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

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

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A235623 Numbers n for which in the prime power factorization of n!, the numbers of exponents 1 and >1 are equal.

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