A050376 -id:A050376 - cp's OEIS frontend

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

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

A366247 The number of infinitary divisors of n that are "Fermi-Dirac primes" (A050376) and terms of A366243.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 05 2023

First differs from A101436 at n = 32.

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

Cf. A050376, A064547, A101436, A139352, A366242, A366243, A366246.

s[0] = 0; s[n_] := s[n] = s[Floor[n/4]] + If[Mod[n, 4] > 1, 1, 0]; f[p_, e_] := s[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

s(e) = if(e>3, s(e\4)) + e%4\2 \\ after Charles R Greathouse IV at A139352
a(n) = vecsum(apply(s, factor(n)[, 2]));

Additive with a(p^e) = A139352(e).
a(n) = A064547(n) - A366246(n).
a(n) = A064547(A366245(n)).
a(n) >= 0, with equality if and only if n is in A366242.
a(n) <= A064547(n), with equality if and only if n is in A366243.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.39310573826635831710..., where f(x) = Sum_{k>=0} (x^(2*4^k)/(1+x^(2*4^k))).

A186945 The smallest integer x > 0 such that the number of terms of A050376 in (x/2,x] equals n.

Original entry on oeis.org

2, 3, 5, 13, 17, 25, 31, 49, 61, 71, 73, 81, 103, 109, 113, 131, 139, 157, 173, 181, 191, 193, 199, 239, 241, 257, 269, 271, 283, 289, 293, 313, 353, 361, 373, 379, 409, 419, 421, 433, 439, 443, 463, 491, 499, 509, 571, 577, 599, 601, 607, 613, 619, 643, 647

Offset: 1

Vladimir Shevelev, Aug 30 2013

nonn

The sequence is an analog of Labos primes (A080359) in Fermi-Dirac arithmetic, since in this arithmetic terms of A050376 play role of primes (see comments in A050376).

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

Cf. A080359, A228520, A228592.

a(n) <= A228520(n).

More terms from Peter J. C. Moses

A187042 Numbers the expansion of which over distinct terms of A050376 contains the same number of primes and perfect squares.

Original entry on oeis.org

8, 12, 18, 20, 27, 28, 32, 44, 45, 48, 50, 52, 63, 68, 75, 76, 80, 92, 98, 99, 112, 116, 117, 124, 125, 147, 148, 153, 162, 164, 171, 172, 175, 176, 188, 207, 208, 212, 216, 236, 242, 243, 244, 245, 261, 268, 272, 275, 279, 284, 292, 304, 316, 325, 332, 333, 338

Offset: 1

Vladimir Shevelev, Mar 02 2011

nonn

The sequence does not contain squarefree numbers or perfect squares.

Initially the sequence matches A378494 (the intersection of A000379 and A026424). The first differences are the absence here of 120 and 168 and the inclusion here of 216. - Peter Munn, Jul 13 2024 (edited by Paolo Xausa, Nov 29 2024).

			147 and 216 are in the sequence, since their expansions over distinct terms of A050376 are 3*49 and 2*3*4*9 respectively. Thus the expansion of 147 contains one prime and one perfect square, while the expansion of 216 contains two primes and two perfect squares.

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

Cf. A050376, A187039, A378494.

aQ[n_] := Total @ (d = IntegerDigits[Last /@ FactorInteger[n], 2])[[;; , -1]] == Total @ Flatten @d / 2; Select[Range[350], aQ] (* Amiram Eldar, Oct 01 2019 *)

a(28)=153 inserted and more terms added by Amiram Eldar, Oct 01 2019

A229064 Lesser of Fermi-Dirac twin primes: both a(n)(>=5) and a(n)+2 are in A050376.

Original entry on oeis.org

5, 7, 9, 11, 17, 23, 29, 41, 47, 59, 71, 79, 81, 101, 107, 137, 149, 167, 179, 191, 197, 227, 239, 269, 281, 311, 347, 359, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 839, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1367

Offset: 1

Vladimir Shevelev, Sep 17 2013

nonn

Terms of A050376 play the role of primes in Fermi-Dirac arithmetic. Therefore, if q and q+2 are consecutive terms of A050376, then we call them twin primes in Fermi-Dirac arithmetic. The sequence lists lessers of them.
There exist conjecturally only 5 Fermat primes F, such that both F-1 and F are in A050376. If we add pair (3,4), then we obtain exactly 6 such pairs as an analog of the unique pair (2,3) in usual arithmetic, which is not considered as a pair of twin primes.
For n>4, numbers n such that n and n+2 are of the form p^(2^k), where p is prime and k >= 0. - Ralf Stephan, Sep 23 2013
If a(n) is not the lesser of twin primes (A001359), then either a(n) or a(n)+2 is a perfect square. For example, a(4)=9 and a(7)=23. Note that the first case is possible only if a(n) = 3^(2^m), m>=1. - Vladimir Shevelev, Jun 27 2014

			2, 3 are not in the sequence, although pairs (2,4) and (3,5) are in A050376. Indeed, 2 and 4 as well as 3 and 5 are not consecutive terms of A050376.

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..10000
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. A001359.

inA050376Q[1]:=False; inA050376Q[n_] := Length[#] == 1 && (Union[Rest[IntegerDigits[#[[1]][[2]], 2]]] == {0} || #[[1]][[2]] == 1)&[FactorInteger[n]]; nextA050376[n_] := NestWhile[#+1&, n+1, !inA050376Q[#] == True&]; Select[Range[1500], inA050376Q[#] && (nextA050376[#]-#) == 2&] (* Peter J. C. Moses, Sep 19 2013 *)

isok(n)={my(e1=isprimepower(n), e2=isprimepower(n+2)); n >= 5 && e1 && e2 && !bitand(e1,e1-1) && !bitand(e2,e2-1)} \\ Andrew Howroyd, Oct 16 2024

A268392 a(n) = A268385(A050376(n)).

Original entry on oeis.org

2, 3, 8, 5, 7, 27, 11, 13, 32, 17, 19, 23, 125, 29, 31, 37, 41, 43, 47, 343, 53, 59, 61, 67, 71, 73, 79, 243, 83, 89, 97, 101, 103, 107, 109, 113, 1331, 127, 131, 137, 139, 149, 151, 157, 163, 167, 2197, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 32768, 257, 263, 269, 271, 277, 281, 283

Offset: 1

Antti Karttunen, Feb 10 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A050376, A268385.

Cf. A268391 (same sequence sorted into ascending order).

(define (A268392 n) (A268385 (A050376 n)))

a(n) = A268385(A050376(n)).

A322823 a(n) = 0 if n is 1 or a Fermi-Dirac prime (A050376), otherwise a(n) = 1 + a(A300840(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 2, 0, 0, 1, 0, 3, 2, 1, 0, 2, 0, 1, 2, 3, 0, 2, 0, 1, 2, 1, 4, 3, 0, 1, 2, 3, 0, 2, 0, 3, 4, 1, 0, 2, 0, 1, 2, 3, 0, 2, 4, 3, 2, 1, 0, 3, 0, 1, 5, 3, 4, 2, 0, 3, 2, 4, 0, 3, 0, 1, 2, 3, 5, 2, 0, 4, 0, 1, 0, 3, 4, 1, 2, 3, 0, 4, 5, 3, 2, 1, 4, 2, 0, 1, 6, 3, 0, 2, 0, 3, 4

Offset: 1

Antti Karttunen, Dec 29 2018

nonn

For n > 1, a(n) gives the number of edges needed to traverse from n to reach the leftmost branch (where the terms of A050376 are located) in the binary tree illustrated in A052330.

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

Cf. A050376, A052330, A052331, A302777, A300840, A322822.

up_to = 10000;
ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A050376list(up_to) = { my(v=vector(up_to), i=0); for(n=1,oo,if(A302777(n), i++; v[i] = n); if(i == up_to,return(v))); };
v050376 = A050376list(up_to);
A050376(n) = v050376[n];
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A300840(n) = A052330(A052331(n)>>1);
A322823(n) = if((1==n)||(1==A302777(n)),0,1+A322823(A300840(n)));

a(1) = 0; for n > 1, if A302777(n) == 1, a(n) = 0, otherwise a(n) = 1 + a(A300840(n)).

A330688 Record values in A050377, number of ways to factor n into "Fermi-Dirac primes" (A050376).

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 14, 16, 20, 24, 28, 40, 56, 60, 80, 84, 104, 112, 120, 144, 160, 168, 184, 200, 208, 216, 224, 240, 260, 288, 320, 360, 368, 400, 416, 432, 460, 480, 520, 576, 600, 624, 640, 720, 736, 800, 864, 920, 960, 1040, 1104, 1120, 1152, 1200, 1440, 1456, 1472, 1480, 1600, 1840, 2016, 2080, 2400, 2576, 2880, 2960, 3360

Offset: 1

Antti Karttunen, Dec 28 2019

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1004 (calculated using the b-file at A330687; terms 1..285 from Antti Karttunen)

Cf. A050376, A050377, A330687.

upto_e = 101; \\ 101 --> 211 terms
A018819(n) = if( n<1, n==0, if( n%2, A018819(n-1), A018819(n/2)+A018819(n-1))); \\ From A018819
v018819 = vector(upto_e,n,A018819(n));
A050377(n) = factorback(apply(e -> v018819[e], factor(n)[, 2]));
A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
A330688list(e) = { my(lista = List([1, 2]), i=2, u = 2^e, t, m=0, v025487); while(lista[i] != u, if(2*lista[i] <= u, listput(lista,2*lista[i]); t = A283980(lista[i]); if(t <= u, listput(lista,t))); i++); v025487 = vecsort(Vec(lista)); lista = List([]); for(i=1,#v025487,if((t=A050377(v025487[i]))>m, listput(lista,t); m=t)); Vec(lista); };
v330688 = A330688list(upto_e);
A330688(n) = v330688[n];

a(n) = A050377(A330687(n)).

A330690 Number of ways to factor A108951(n) into "Fermi-Dirac primes" (A050376), where A108951 is fully multiplicative with a(prime(k)) = k-th primorial.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 4, 2, 1, 2, 1, 2, 4, 4, 1, 4, 1, 2, 4, 2, 1, 4, 8, 2, 4, 2, 1, 4, 1, 4, 4, 2, 8, 8, 1, 2, 4, 4, 1, 4, 1, 2, 4, 2, 1, 4, 16, 8, 4, 2, 1, 8, 8, 4, 4, 2, 1, 8, 1, 2, 4, 6, 8, 4, 1, 2, 4, 8, 1, 8, 1, 2, 8, 2, 16, 4, 1, 4, 16, 2, 1, 8, 8, 2, 4, 4, 1, 8, 16, 2, 4, 2, 8, 6, 1, 16, 4, 16, 1, 4, 1, 4, 8

Offset: 1

Antti Karttunen, Dec 28 2019

nonn

a(64) = 6 is the first term which is not a power of 2.

Antti Karttunen, Table of n, a(n) for n = 1..10201
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A034386, A018819, A050376, A050377, A050378, A108951, A329901.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A018819(n) = if( n<1, n==0, if( n%2, A018819(n-1), A018819(n/2)+A018819(n-1))); \\ From A018819
A050377(n) = factorback(apply(e -> A018819(e), factor(n)[, 2]));
A330690(n) = A050377(A108951(n));

a(n) = A050377(A108951(n)).

a(n) = A050378(A329901(n)).

cp's OEIS Frontend

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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A366247 The number of infinitary divisors of n that are "Fermi-Dirac primes" (A050376) and terms of A366243.

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A186945 The smallest integer x > 0 such that the number of terms of A050376 in (x/2,x] equals n.

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A187042 Numbers the expansion of which over distinct terms of A050376 contains the same number of primes and perfect squares.

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Mathematica

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A229064 Lesser of Fermi-Dirac twin primes: both a(n)(>=5) and a(n)+2 are in A050376.

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A268392 a(n) = A268385(A050376(n)).

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A322823 a(n) = 0 if n is 1 or a Fermi-Dirac prime (A050376), otherwise a(n) = 1 + a(A300840(n)).

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