A050376 -id:A050376 - cp's OEIS frontend

A334868 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j) for all i, j >= 1, where f(1) = 0 and for n > 1, f(n) = -1 if n is in A050376, and f(n) = A334870(n) otherwise.

1, 2, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 8, 2, 2, 9, 2, 10, 11, 12, 2, 13, 2, 14, 15, 16, 2, 17, 2, 18, 19, 20, 21, 22, 2, 23, 24, 25, 2, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 2, 40, 41, 6, 42, 43, 2, 44, 45, 46, 2, 47, 2, 48, 49, 50, 51, 52, 2, 53, 2, 54, 2, 55, 56, 57, 58, 59, 2, 60, 61, 62, 63, 64, 65, 66, 2, 67, 68, 8, 2, 69, 2, 70, 71

Offset: 1

Antti Karttunen, Jun 08 2020

nonn

For all i, j: A305979(i) = A305979(j) => a(i) = a(j) => A334872(i) = A334872(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A050376 (positions of 2's), A305979, A334869, A334870, A334872.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A334870(n) = if(issquare(n),sqrtint(n),my(c=core(n), m=n); forprime(p=2, , if(!(c % p), m/=p; break, m*=p)); (m));
A334868aux(n) = if(1==n,0,if(A302777(n),-1,A334870(n)));
v334868 = rgs_transform(vector(up_to,n,A334868aux(n)));
A334868(n) = v334868[n];

A335428 Prime exponent of the first Fermi-Dirac factor (number of form p^(2^k), A050376) reached, when starting from n and iterating with A334870, with a(1) = 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 1, 2, 2, 1, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Jun 26 2020

nonn

			For n=27, when iterating with A334870, we obtain the path 27 -> 18 -> 9, with that 9 being the first prime power encountered that has an exponent of the form 2^k, as 9 = 3^2, thus a(27) = 2. See the binary tree A334860 or A334866 for how such paths go.
For n=900, when iterating with A334870 we obtain the path 900 -> 30 -> 15 -> 10 -> 5, and 5^1 is finally a prime power with an exponent that is two's power, thus a(900) = 1. Note that 900 is the first such position of 1 in this sequence that is not listed in A333634.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A050376, A209229, A302777, A334860, A334866, A334870, A334872.

A209229(n) = (n && !bitand(n,n-1));
A302777(n) = A209229(isprimepower(n));
A334870(n) = if(issquare(n),sqrtint(n),my(c=core(n), m=n); forprime(p=2, , if(!(c % p), m/=p; break, m*=p)); (m));
A335428(n) = { while(n>1 && !A302777(n), n = A334870(n)); isprimepower(n); };

\\ Faster, A209229 and A302777 like in above:
A335428(n) = if(1==n,0, while(!A302777(n), if(issquarefree(n), return(1)); if(issquare(n), n = sqrtint(n), n /= core(n))); isprimepower(n));

A050379 Number of ordered factorizations of n into members of A050376.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 5, 1, 2, 2, 6, 1, 5, 1, 5, 2, 2, 1, 10, 2, 2, 3, 5, 1, 6, 1, 10, 2, 2, 2, 14, 1, 2, 2, 10, 1, 6, 1, 5, 5, 2, 1, 22, 2, 5, 2, 5, 1, 10, 2, 10, 2, 2, 1, 18, 1, 2, 5, 18, 2, 6, 1, 5, 2, 6, 1, 32, 1, 2, 5, 5, 2, 6, 1, 22, 6, 2, 1, 18, 2, 2, 2, 10, 1, 18, 2, 5, 2, 2, 2

Offset: 1

Christian G. Bower, Nov 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3 * 3 and 375 = 3 * 5^3 both have prime signature (3,1).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A002033, A050376, A050377, A050378, A050380.

Cf. also A000142, A002110, A023359.

read(transforms) :
L := [1] :
for n from 2 to 100  do
    if isA050376(n) then
        L := [op(L),-1] ;
    else
        L := [op(L),0] ;
    end if;
end do :
a050379 := DIRICHLETi(L) ; # R. J. Mathar, May 26 2017

A064547(n) = {my(f = factor(n)[, 2]); sum(k=1, #f, hammingweight(f[k])); } \\ Michel Marcus, Feb 10 2016
isA050376(n) = ((1==omega(n)) && (1==A064547(n))); \\ Checking that omega(n) is 1 is just an optimization here.
A050379(n) = if(1==n,n,sumdiv(n,d,if(dA050376(n/d)*A050379(d),0))); \\ Antti Karttunen, Oct 20 2017

Dirichlet g.f.: 1/(1-B(s)) where B(s) is D.g.f. of characteristic function of A050376.
a(p^k) = A023359(k), for any prime p.
a(A002110(n)) = A000142(n) = n!.
a(n) = A050380(A101296(n)). - R. J. Mathar, May 26 2017

A176620 Primes p for which the factorization of p! over distinct terms of A050376 does not contain 2.

Original entry on oeis.org

7, 11, 13, 19, 31, 37, 41, 47, 59, 61, 67, 73, 79, 97, 103, 107, 109, 127, 131, 137, 151, 157, 167, 173, 179, 181, 191, 193, 199, 211, 223, 227, 229, 233, 239, 241, 251, 271, 283, 307, 313, 331, 367, 379, 397, 409, 419, 421, 431, 433, 439, 443, 457, 463, 487

Offset: 1

Vladimir Shevelev, Apr 22 2010

nonn

Equivalent definition: primes p for which A007814(p!) is even. Apparently, the sequence is A027697 without the 2 (see A014499). [R. J. Mathar, Oct 26 2010]

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

Cf. A050376, A000040, A000142.

Select[Range[500], PrimeQ[#] && EvenQ @ IntegerExponent[#!, 2] &]  (* Amiram Eldar, Sep 13 2019 *)

Corrected (37 added, 41 added, 43 removed...) and extended by R. J. Mathar, Oct 26 2010

A181970 Places of nonprimes in A050376.

Original entry on oeis.org

3, 6, 9, 13, 20, 28, 37, 47, 63, 71, 83, 111, 127, 160, 177, 235, 280, 301, 348, 377, 430, 509, 542, 633, 700, 731, 838, 875, 915, 1030, 1194, 1284, 1327, 1415, 1458, 1559, 1752, 1915, 2015, 2181, 2231, 2531, 2590, 2773, 2960, 3089, 3154, 3289, 3485, 3562, 3919, 3997, 4142

Offset: 1

Vladimir Shevelev, Apr 06 2012

nonn

Or numbers not expressed in the form pi(p) + pi(sqrt(p)) + pi((sqrt(sqrt(p)))) +... with prime p.

			Show that 28 not expressed in form pi(p) + pi(sqrt(p)) + pi((sqrt(sqrt(p)))) +... with prime p. Indeed, for p=79, this sum is 22+4+1=27, while for p=83, it is 23+4+2=29.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A050376, A000040, A181962.

first_few(lim)=my(v=List(apply(n->n^2, primes(primepi(sqrtint(lim))))),u,t); forprime(p=2,(lim+.5)^(1/4),t=p^2;while((t=t^2)<=lim,listput(v,t)));listput(v,1);v=vecsort(Vec(v));u=vector(#v-1,i,sum(j=v[i]+1,v[i+1]-1,isprime(j)));u[1]++;for(i=2, #u, u[i]+=u[i-1]+1);u \\ Charles R Greathouse IV, Apr 10 2012

from sympy import primepi, integer_nthroot
def A181970(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(integer_nthroot(x,1<Chai Wah Wu, Feb 18-19 2025

a(n) ~ 0.5 n^2 log n. - Charles R Greathouse IV, Apr 11 2012

a(30)-a(53) from Charles R Greathouse IV, Apr 10 2012

A228518 a(n) is the least r > 1 for which the interval (rn, r(n+1)) contains no terms of A050376 or a(n) = 0 if no such r exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 4, 0, 2, 3, 0, 2, 0, 6, 2, 2, 3, 2, 7, 3, 2, 4, 0, 2, 7, 2, 2, 4, 3, 2, 2, 4, 2, 4, 4, 2, 2, 3, 5, 3, 2, 2, 3, 2, 2, 2, 3, 2, 4, 3, 2, 3, 4, 2, 0, 2, 2, 2, 5, 2, 2, 4, 2, 5, 2, 2, 3, 3, 2, 2, 2, 2, 4, 4, 2, 2, 3, 2, 2, 3, 2, 4, 3, 2, 3, 2

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Aug 24 2013

nonn

It is an a Fermi-Dirac analog of A218831, since terms of A050376 play a role of primes in Fermi-Dirac arithmetic (see comments in A050376).
Conjecture: a(n) = 0 iff n = 1, 2, 3, 4, 5, 6, 9, 12, 14, 24, 56.
A proof that the interval(r*n, r*(n+1)) for r > 1 always contains a term from A050376 for n = 1, 2, 3, 4, 5, 6, 9, 12, 14, 24, 56 uses similar methods of analog of Ramanujan numbers (cf. A228520) and their generalization.

Peter J. C. Moses, Table of n, a(n) for n = 1..10000

Cf. A218831, A228520, A188672.

If a(n)*A218831(n) is not 0, then a(n) >= A218831(n).

If a(n)*A188672(n) is not 0, then a(n) <= A188672(n).

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

A242165 Smallest k>=0, such that n+/-k are both Fermi-Dirac primes (A050376).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 3, 2, 0, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 3, 0, 1, 0, 9, 4, 3, 6, 5, 0, 9, 2, 3, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 9, 0, 5, 6, 3, 4, 9, 0, 1, 0, 9, 4, 3, 6, 5, 0, 15, 2, 3, 0, 1, 0, 7, 4, 3, 4, 5, 0, 1, 0, 1, 0, 5, 4, 3, 14, 9, 0, 7, 10, 9, 4, 13, 6, 7, 0

Offset: 2

Vladimir Shevelev, May 05 2014

nonn

The existence of a(n)>=0 for all n >= 2 is equivalent to the Goldbach conjecture in Fermi-Dirac arithmetic (cf. comment in A241927) that every even number >= 4 is a sum of two terms of A050376 (it is slightly weaker than Goldbach conjecture for primes).

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

Peter J. C. Moses, Table of n, a(n) for n = 2..10001

Cf. A082467, A241922, A241927, A241947.

a(A050376(n)) = 0.

A244343 Least even k such that sfdf(k-3) > sfdf(k-1) >= A050376(n), where sfdf(n) is the smallest Fermi-Dirac factor of n (A223490).

Original entry on oeis.org

16, 46, 46, 64, 100, 254, 326, 392, 392, 590, 776, 776, 1190, 1520, 1814, 2420, 2624, 3764, 3764, 3764, 4454, 4454, 4892, 5752, 6400, 6400, 7210, 9380, 9524, 11414, 11414, 13190, 13190, 13190, 18272, 18272, 19940, 20414, 20414, 21824, 24614, 24614, 25592

Offset: 2

Vladimir Shevelev and Peter J. C. Moses, Jun 26 2014

nonn

a(n) is a Fermi-Dirac analog of A242719.

			If k is even such that k-1 is either 1 or in A050376, then k cannot be required. Thus, if n=2, then k=2,4,6,8,10,12,14 are not required, while for k=16 we have sfdf(16-3) = 13 > sfdf(16-1) = 3 = A050376(2). So a(2)=16.

Cf. A050376, A223490, A242719.

A248713 a(1)=1; starting with n>1, concatenate distinct divisors which are in A050376 in increasing order and repeat until a term of A050376 is reached (a(n)=0 if no term of A050376 is ever reached).

Original entry on oeis.org

1, 2, 3, 4, 5, 23, 7, 731173, 9, 25, 11, 31397, 13, 313, 1129, 16, 17, 29, 19, 59, 37, 211, 23, 731173, 25, 3251, 313, 47, 29, 547, 31, 313289, 311, 31397, 1129, 49, 37, 373, 313, 961, 41, 379, 43, 3137, 59, 223, 47, 479, 49, 71443, 317, 31123, 53, 239, 773

Offset: 1

Vladimir Shevelev, Oct 12 2014

Fermi-Dirac analog of A037274 (terms of A050376 are Fermi-Dirac primes).

			We have 40 = 2*4*5 -> 245 = 5*49 -> 549 = 9*61 -> 961 is in A050376. So a(40) = 961.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..100

Cf. A037274, A050376, A084318.

a(8) and a(34) corrected by Hiroaki Yamanouchi, Oct 13 2014

cp's OEIS Frontend

A334868 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j) for all i, j >= 1, where f(1) = 0 and for n > 1, f(n) = -1 if n is in A050376, and f(n) = A334870(n) otherwise.

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A335428 Prime exponent of the first Fermi-Dirac factor (number of form p^(2^k), A050376) reached, when starting from n and iterating with A334870, with a(1) = 0.

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A050379 Number of ordered factorizations of n into members of A050376.

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A176620 Primes p for which the factorization of p! over distinct terms of A050376 does not contain 2.

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Mathematica

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A181970 Places of nonprimes in A050376.

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A228518 a(n) is the least r > 1 for which the interval (r*n, r*(n+1)) contains no terms of A050376 or a(n) = 0 if no such r exists.

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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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A242165 Smallest k>=0, such that n+/-k are both Fermi-Dirac primes (A050376).

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A228518 a(n) is the least r > 1 for which the interval (rn, r(n+1)) contains no terms of A050376 or a(n) = 0 if no such r exists.