A177334 -id:A177334 - cp's OEIS frontend

A240906 Maximal A050376-factors in factorization of factorials over distinct terms of A050376.

2, 3, 4, 5, 16, 81, 256, 65536, 4294967296, 18446744073709551616, 340282366920938463463374607431768211456, 115792089237316195423570985008687907853269984665640564039457584007913129639936

Offset: 2

Vladimir Shevelev, Apr 14 2014

nonn

In contrast to prime power factorization of n! where maximal prime does not exceed n, in the considered factorization of n!, the maximal A050376-factor could be much greater than n. For example, for n=18, it is 65536.

Essentially, a(n) is A177334 with duplicates removed. - Ivan Neretin, May 31 2016

			81 is in the sequence, since 9! = 2*4*5*7*16*81.

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

a(9) from A177334 by Robert Price, Aug 31 2019

a(10)-a(12) from Amiram Eldar, Sep 17 2019

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

A241270 Numbers with the property that in their factorization over distinct terms of A050376, the sums of prime and nonprime terms of A050376 are equal.

Original entry on oeis.org

126, 468, 624, 792, 880, 1056, 1150, 2900, 3264, 4606, 5824, 6375, 6624, 8320, 9856, 10388, 11375, 12798, 13650, 16400, 16704, 19250, 20925, 30135, 32625, 36720, 39150, 39900, 53784, 56446, 56925, 57000, 59500, 63455, 65520, 71400, 71500, 72471

Offset: 1

Vladimir Shevelev, Apr 18 2014

nonn

The corresponding sequence of the sum over the primes, which equals the sum over the nonprimes, is 9, 13, 16, 13, 16, 16, 25, 29, 20, 49, 20, 25, 25, 20, 20, 53, 25, 81, 25, 41, 29, 25, 34, 49, 34, 25, 34, 29, 85, 169, 34, 29, 29, 49, 25, 29, 29, 49, ... - Wolfdieter Lang, Apr 25 2014

			126 and 468 are in the sequence since the factorizations are 2*7*9 and 4*9*13 respectively, and 2+7=9, 4+9=13.

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

Peter J. C. Moses, Table of n, a(n) for n = 1..2000
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. A187039, A187042, A177329, A177333, A177334.

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

New extension from Wolfdieter Lang, Apr 25 2014

cp's OEIS Frontend

A240906 Maximal A050376-factors in factorization of factorials 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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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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A241270 Numbers with the property that in their factorization over distinct terms of A050376, the sums of prime and nonprime terms of A050376 are equal.

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