A268388 -id:A268388 - cp's OEIS frontend

A050376 "Fermi-Dirac primes": numbers of the form p^(2^k) where p is prime and k >= 0.

2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241

Offset: 1

Christian G. Bower, Nov 15 1999

Every number n is a product of a unique subset of these numbers. This is sometimes called the Fermi-Dirac factorization of n (see A182979). Proof: In the prime factorization n = Product_{j>=1} p(j)^e(j) expand every exponent e(j) as binary number and pick the terms of this sequence corresponding to the positions of the ones in binary (it is clear that both n and n^2 have the same number of factors in this sequence, and that each factor appears with exponent 1 or 0).
Or, a(1) = 2; for n>1, a(n) = smallest number which cannot be obtained as the product of previous terms. This is evident from the unique factorization theorem and the fact that every number can be expressed as the sum of powers of 2. - Amarnath Murthy, Jan 09 2002
Except for the first term, same as A084400. - David Wasserman, Dec 22 2004
The least number having 2^n divisors (=A037992(n)) is the product of the first n terms of this sequence according to Ramanujan.
According to the Bose-Einstein distribution of particles, an unlimited number of particles may occupy the same state. On the other hand, according to the Fermi-Dirac distribution, no two particles can occupy the same state (by the Pauli exclusion principle). Unique factorizations of the positive integers by primes (A000040) and over terms of A050376 one can compare with two these distributions in physics of particles. In the correspondence with this, the factorizations over primes one can call "Bose-Einstein factorizations", while the factorizations over distinct terms of A050376 one can call "Fermi-Dirac factorizations". - Vladimir Shevelev, Apr 16 2010
The numbers of the form p^(2^k), where p is prime and k >= 0, might thus be called the "Fermi-Dirac primes", while the classic primes might be called the "Bose-Einstein primes". - Daniel Forgues, Feb 11 2011
In the theory of infinitary divisors, the most natural name of the terms is "infinitary primes" or "i-primes". Indeed, n is in the sequence, if and only if it has only two infinitary divisors. Since 1 and n are always infinitary divisors of n>1, an i-prime has no other infinitary divisors. - Vladimir Shevelev, Feb 28 2011
{a(n)} is the minimal set including all primes and closed with respect to squaring. In connection with this, note that n and n^2 have the same number of factors in their Fermi-Dirac representations. - Vladimir Shevelev, Mar 16 2012
In connection with this sequence, call an integer compact if the factors in its Fermi-Dirac factorization are pairwise coprime. The density of such integers equals (6/Pi^2)*Product_{prime p} (1+(Sum_{i>=1} p^(-(2^i-1))/(p+1))) = 0.872497... It is interesting that there exist only 7 compact factorials listed in A169661. - Vladimir Shevelev, Mar 17 2012
The first k terms of the sequence solve the following optimization problem:
Let x_1, x_2,..., x_k be integers with the restrictions: 2<=x_1A064547(Product{i=1..k} x_i) >= k.  Let the goal function be Product_{i=1..k} x_i. Then the minimal value of the goal function is Product_{i=1..k} a(i). - Vladimir Shevelev, Apr 01 2012
From Joerg Arndt, Mar 11 2013: (Start)
Similarly to the first comment, for the sequence "Numbers of the form p^(3^k) or p^(2*3^k) where p is prime and k >= 0" one obtains a factorization into distinct factors by using the ternary expansion of the exponents (here n and n^3 have the same number of such factors).
The generalization to base r would use "Numbers of the form p^(r^k), p^(2*r^k), p^(3*r^k), ..., p^((r-1)*r^k) where p is prime and k >= 0" (here n and n^r have the same number of (distinct) factors).  (End)
The first appearance of this sequence as a multiplicative basis in number theory with some new notions, formulas and theorems may have been in my 1981 paper (see the Abramovich reference). - Vladimir Shevelev, Apr 27 2014
Numbers n for which A064547(n) = 1. - Antti Karttunen, Feb 10 2016
Lexicographically earliest sequence of distinct nonnegative integers such that no term is a product of 2 or more distinct terms. Removing the distinctness requirement, the sequence becomes A000040 (the prime numbers); and the equivalent sequence where the product is of 2 distinct terms is A026416 (without its initial term, 1). - Peter Munn, Mar 05 2019
The sequence was independently developed as a multiplicative number system in 1985-1986 (and first published in 1995, see the Uhlmann reference) using a proof method involving representations of positive integers as sums of powers of 2. This approach offers an arguably simpler and more flexible means for analyzing the sequence. - Jeffrey K. Uhlmann, Nov 09 2022

			Prime powers which are not terms of this sequence:
  8 = 2^3 = 2^(1+2), 27 = 3^3 = 3^(1+2), 32 = 2^5 = 2^(1+4),
  64 = 2^6 = 2^(2+4), 125 = 5^3 = 5^(1+2), 128 = 2^7 = 2^(1+2+4)
"Fermi-Dirac factorizations":
  6 = 2*3, 8 = 2*4, 24 = 2*3*4, 27 = 3*9, 32 = 2*16, 64 = 4*16,
  108 = 3*4*9, 120 = 2*3*4*5, 121 = 121, 125 = 5*25, 128 = 2*4*16.

V. S. Abramovich, On an analog of the Euler function, Proceeding of the North-Caucasus Center of the Academy of Sciences of the USSR (Rostov na Donu) (1981) No. 2, 13-17 (Russian; MR0632989(83a:10003)).
S. Ramanujan, Highly Composite Numbers, Collected Papers of Srinivasa Ramanujan, p. 125, Ed. G. H. Hardy et al., AMS Chelsea 2000.
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).
J. K. Uhlmann, Dynamic map building and localization: new theoretical foundations, Doctoral Dissertation, University of Oxford, Appendix 16, 1995.

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Simon Litsyn and Vladimir Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.
OEIS Wiki, "Fermi-Dirac representation" of n.
Vladimir Shevelev, Compact integers and factorials, Acta Arith. 126 (2007), no.3, 195-236.
J. K. Uhlmann, Appendix 16, Doctoral Dissertation, University of Oxford, page 243, 1995.

Cf. A000040 (primes, is a subsequence), A026416, A026477, A037992 (partial products), A050377-A050380, A052330, A064547, A066724, A084400, A176699, A182979.
Cf. A268388 (complement without 1).
Cf. A124010, subsequence of A000028, A000961, A213925, A223490.
Cf. A228520, A186945 (Fermi-Dirac analog of Ramanujan primes, A104272, and Labos primes, A080359).
Cf. also A268385, A268391, A268392.

a050376 n = a050376_list !! (n-1)
a050376_list = filter ((== 1) . a209229 . a100995) [1..]
-- Reinhard Zumkeller, Mar 19 2013

isA050376 := proc(n)
    local f,e;
    f := ifactors(n)[2] ;
    if nops(f) = 1 then
        e := op(2,op(1,f)) ;
        if isA000079(e) then
            true;
        else
            false;
        end if;
    else
        false;
    end if;
end proc:
A050376 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        2 ;
    else
        for a from procname(n-1)+1 do
            if isA050376(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, May 26 2017

nn = 300; t = {}; k = 1; While[lim = nn^(1/k); lim > 2,  t = Join[t, Prime[Range[PrimePi[lim]]]^k]; k = 2 k]; t = Union[t] (* T. D. Noe, Apr 05 2012 *)

{a(n)= local(m, c, k, p); if(n<=1, 2*(n==1), n--; c=0; m=2; while( cMichael Somos, Apr 15 2005; edited by Michel Marcus, Aug 07 2021

lst(lim)=my(v=primes(primepi(lim)),t); forprime(p=2,sqrt(lim),t=p; while((t=t^2)<=lim,v=concat(v,t))); vecsort(v) \\ Charles R Greathouse IV, Apr 10 2012

is_A050376(n)=2^#binary(n=isprimepower(n))==n*2 \\ M. F. Hasler, Apr 08 2015

ispow2(n)=n && n>>valuation(n,2)==1
is(n)=ispow2(isprimepower(n)) \\ Charles R Greathouse IV, Sep 18 2015

isok(n)={my(e=isprimepower(n)); e && !bitand(e,e-1)} \\ Andrew Howroyd, Oct 16 2024

from sympy import isprime, perfect_power
def ok(n):
  if isprime(n): return True
  answer = perfect_power(n)
  if not answer: return False
  b, e = answer
  if not isprime(b): return False
  while e%2 == 0: e //= 2
  return e == 1
def aupto(limit):
  alst, m = [], 1
  for m in range(1, limit+1):
    if ok(m): alst.append(m)
  return alst
print(aupto(241)) # Michael S. Branicky, Feb 03 2021

from sympy import primepi, integer_nthroot
def A050376(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

(define A050376 (MATCHING-POS 1 1 (lambda (n) (= 1 (A064547 n)))))
;; Requires also my IntSeq-library. - Antti Karttunen, Feb 09 2016

From Vladimir Shevelev, Mar 16 2012: (Start)
Product_{i>=1} a(i)^k_i = n!, where k_i = floor(n/a(i)) - floor(n/a(i)^2) + floor(n/a(i)^3) - floor(n/a(i)^4) + ...
Denote by A(x) the number of terms not exceeding x.
Then A(x) = pi(x) + pi(x^(1/2)) + pi(x^(1/4)) + pi(x^(1/8)) + ...
Conversely, pi(x) = A(x) - A(sqrt(x)). For example, pi(37) = A(37) - A(6) = 16-4 = 12. (End)
A209229(A100995(a(n))) = 1. - Reinhard Zumkeller, Mar 19 2013
From Vladimir Shevelev, Aug 31 2013: (Start)
A Fermi-Dirac analog of Euler product: Zeta(s) = Product_{k>=1} (1+a(k)^(-s)), for s > 1.
In particular, Product_{k>=1} (1+a(k)^(-2)) = Pi^2/6. (End)
a(n) = A268385(A268392(n)). [By their definitions.] - Antti Karttunen, Feb 10 2016
A000040 union A001248 union A030514 union A179645 union A030635 union .... - R. J. Mathar, May 26 2017

Edited by Charles R Greathouse IV, Mar 17 2010

More examples from Daniel Forgues, Feb 09 2011

A225546 Tek's flip: Write n as the product of distinct factors of the form prime(i)^(2^(j-1)) with i and j integers, and replace each such factor with prime(j)^(2^(i-1)).

Original entry on oeis.org

1, 2, 4, 3, 16, 8, 256, 6, 9, 32, 65536, 12, 4294967296, 512, 64, 5, 18446744073709551616, 18, 340282366920938463463374607431768211456, 48, 1024, 131072, 115792089237316195423570985008687907853269984665640564039457584007913129639936, 24, 81, 8589934592, 36, 768

Offset: 1

Paul Tek, May 10 2013

This is a multiplicative self-inverse permutation of the integers.
A225547 gives the fixed points.
From Antti Karttunen and Peter Munn, Feb 02 2020: (Start)
This sequence operates on the Fermi-Dirac factors of a number. As arranged in array form, in A329050, this sequence reflects these factors about the main diagonal of the array, substituting A329050[j,i] for A329050[i,j], and this results in many relationships including significant homomorphisms.
This sequence provides a relationship between the operations of squaring and prime shift (A003961) because each successive column of the A329050 array is the square of the previous column, and each successive row is the prime shift of the previous row.
A329050 gives examples of how significant sets of numbers can be formed by choosing their factors in relation to rows and/or columns. This sequence therefore maps equivalent derived sets by exchanging rows and columns. Thus odd numbers are exchanged for squares, squarefree numbers for powers of 2 etc.
Alternative construction: For n > 1, form a vector v of length A299090(n), where each element v[i] for i=1..A299090(n) is a product of those distinct prime factors p(i) of n whose exponent e(i) has the bit (i-1) "on", or 1 (as an empty product) if no such exponents are present. a(n) is then Product_{i=1..A299090(n)} A000040(i)^A048675(v[i]). Note that because each element of vector v is squarefree, it means that each exponent A048675(v[i]) present in the product is a "submask" (not all necessarily proper) of the binary string A087207(n).
This permutation effects the following mappings:
A000035(a(n)) = A010052(n), A010052(a(n)) = A000035(n). [Odd numbers <-> Squares]
A008966(a(n)) = A209229(n), A209229(a(n)) = A008966(n). [Squarefree numbers <-> Powers of 2]
(End)
From Antti Karttunen, Jul 08 2020: (Start)
Moreover, we see also that this sequence maps between A016825 (Numbers of the form 4k+2) and A001105 (2*squares) as well as between A008586 (Multiples of 4) and A028983 (Numbers with even sum of the divisors).
(End)

			  7744  = prime(1)^2^(2-1)*prime(1)^2^(3-1)*prime(5)^2^(2-1).
a(7744) = prime(2)^2^(1-1)*prime(3)^2^(1-1)*prime(2)^2^(5-1) = 645700815.

Cf. A225547 (fixed points) and the subsequences listed there.
Transposes A329050, A329332.
An automorphism of positive integers under the binary operations A059895, A059896, A059897, A306697, A329329.
An automorphism of A059897 subgroups: A000379, A003159, A016754, A122132.
Permutes lists where membership is determined by number of Fermi-Dirac factors: A000028, A050376, A176525, A268388.
Also permutes: A036554, A059404, A066427, A066428, A072774, A331593.
Sequences f that satisfy f(a(n)) = f(n): A048675, A064179, A064547, A097248, A302777, A331592.
Pairs of sequences (f,g) that satisfy a(f(n)) = g(a(n)): (A000265,A008833), (A000290,A003961), (A005843,A334747), (A006519,A007913), (A008586,A334748).
Pairs of sequences (f,g) that satisfy a(f(n)) = g(n), possibly with offset change: (A000040,A001146), (A000079,A019565).
Pairs of sequences (f,g) that satisfy f(a(n)) = g(n), possibly with offset change: (A000035, A010052), (A008966, A209229), (A007814, A248663), (A061395, A299090), (A087207, A267116), (A225569, A227291).
Cf. A331287 [= gcd(a(n),n)].
Cf. A331288 [= min(a(n),n)], see also A331301.
Cf. A331309 [= A000005(a(n)), number of divisors].
Cf. A331590 [= a(a(n)*a(n))].
Cf. A331591 [= A001221(a(n)), number of distinct prime factors], see also A331593.
Cf. A331740 [= A001222(a(n)), number of prime factors with multiplicity].
Cf. A331733 [= A000203(a(n)), sum of divisors].
Cf. A331734 [= A033879(a(n)), deficiency].
Cf. A331735 [= A009194(a(n))].
Cf. A331736 [= A000265(a(n)) = a(A008833(n)), largest odd divisor].
Cf. A335914 [= A038040(a(n))].
A self-inverse isomorphism between pairs of A059897 subgroups: (A000079,A005117), (A000244,A062503), (A000290\{0},A005408), (A000302,A056911), (A000351,A113849 U {1}), (A000400,A062838), (A001651,A252895), (A003586,A046100), (A007310,A000583), (A011557,A113850 U {1}), (A028982,A042968), (A053165,A065331), (A262675,A268390).
A bijection between pairs of sets: (A001248,A011764), (A007283,A133466), (A016825, A001105), (A008586, A028983).
Cf. also A336321, A336322 (compositions with another involution, A122111).

Array[If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 28] (* Michael De Vlieger, Jan 21 2020 *)

A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
a(n) = {my(f=factor(n)); for (i=1, #f~, my(p=f[i,1]); f[i,1] = A019565(f[i,2]); f[i,2] = 2^(primepi(p)-1);); factorback(f);} \\ Michel Marcus, Nov 29 2019

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A225546(n) = if(1==n,1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); prod(i=1,u,prime(i)^A048675(prods[i]))); \\ Antti Karttunen, Feb 02 2020

from math import prod
from sympy import prime, primepi, factorint
def A225546(n): return prod(prod(prime(i) for i, v in enumerate(bin(e)[:1:-1],1) if v == '1')**(1<Chai Wah Wu, Mar 17 2023

Multiplicative, with a(prime(i)^j) = A019565(j)^A000079(i-1).
a(prime(i)) = 2^(2^(i-1)).
From Antti Karttunen and Peter Munn, Feb 06 2020: (Start)
a(A329050(n,k)) = A329050(k,n).
a(A329332(n,k)) = A329332(k,n).
Equivalently, a(A019565(n)^k) = A019565(k)^n. If n = 1, this gives a(2^k) = A019565(k).
a(A059897(n,k)) = A059897(a(n), a(k)).
The previous formula implies a(n*k) = a(n) * a(k) if A059895(n,k) = 1.
a(A000040(n)) = A001146(n-1); a(A001146(n)) = A000040(n+1).
a(A000290(a(n))) = A003961(n); a(A003961(a(n))) = A000290(n) = n^2.
a(A000265(a(n))) = A008833(n); a(A008833(a(n))) = A000265(n).
a(A006519(a(n))) = A007913(n); a(A007913(a(n))) = A006519(n).
A007814(a(n)) = A248663(n); A248663(a(n)) = A007814(n).
A048675(a(n)) = A048675(n) and A048675(a(2^k * n)) = A048675(2^k * a(n)) = k + A048675(a(n)).
(End)
From Antti Karttunen and Peter Munn, Jul 08 2020: (Start)
For all n >= 1, a(2n) = A334747(a(n)).
In particular, for n = A003159(m), m >= 1, a(2n) = 2*a(n). [Note that A003159 includes all odd numbers]
(End)

Name edited by Peter Munn, Feb 14 2020

"Tek's flip" prepended to the name by Antti Karttunen, Jul 08 2020

A064547 Sum of binary digits (or count of 1-bits) in the exponents of the prime factorization of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 3, 3

Offset: 1

Wouter Meeussen, Oct 09 2001

This sequence is different from A058061 for n containing 6th, 8th, ..., k-th powers in its prime decomposition, where k runs through the integers missing from A064548.
For n > 1, n is a product of a(n) distinct members of A050376. - Matthew Vandermast, Jul 13 2004
For n > 1: a(n) = length of n-th row in A213925. - Reinhard Zumkeller, Mar 20 2013
Number of Fermi-Dirac factors of n. - Peter Munn, Dec 27 2019

			For n = 54, n = 2^1 * 3^3 with exponents (1) and (11) in binary, so a(54) = A000120(1) + A000120(3) = 1 + 2 = 3.

Antti Karttunen, Table of n, a(n) for n = 1..32768 (terms 1..2000 from Harry J. Smith)
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences related to binary expansion of n.

Cf. A000028 (positions of odd terms), A000379 (of even terms).
Cf. A050376 (positions of ones), A268388 (terms larger than ones).
Row lengths of A213925.
A000120, A007814, A028234, A037445, A052331, A064989, A067029, A156552, A223491, A286574 are used in formulas defining this sequence.
Cf. A005117, A058061 (to which A064548 relates), A138302.
Cf. other sequences counting factors of n: A001221, A001222.
Cf. other sequences where a(n) depends only on the prime signature of n: A181819, A267116, A268387.
A003961, A007913, A008833, A059895, A059896, A059897, A225546 are used to express relationship between terms of this sequence.
Cf. A077761, A176699, A037992, A382294.

a064547 1 = 0
a064547 n = length $ a213925_row n  -- Reinhard Zumkeller, Mar 20 2013

expts:=proc(n) local t1,t2,t3,t4,i; if n=1 then RETURN([0]); fi; if isprime(n) then RETURN([1]); fi; t1:=ifactor(n); if nops(factorset(n))=1 then RETURN([op(2,t1)]); fi; t2:=nops(t1); t3:=[]; for i from 1 to t2 do t4:=op(i,t1); if nops(t4) = 1 then t3:=[op(t3),1]; else t3:=[op(t3),op(2,t4)]; fi; od; RETURN(t3); end;
A000120 := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end:
LamMos:= proc(n) local t1,t2,t3,i; t1:=expts(n); add( A000120(t1[i]),i=1..nops(t1)); end; # N. J. A. Sloane, Dec 20 2007
# alternative Maple program:
A064547:= proc(n) local F;
F:= ifactors(n)[2];
add(convert(convert(f[2],base,2),`+`),f=F)
end proc:
map(A064547,[$1..100]); # Robert Israel, May 17 2016

Table[Plus@@(DigitCount[Last/@FactorInteger[k], 2, 1]), {k, 105}]

a(n) = {my(f = factor(n)[,2]); sum(k=1, #f, hammingweight(f[k]));} \\ Michel Marcus, Feb 10 2016

from sympy import factorint
def wt(n): return bin(n).count("1")
def a(n):
    f=factorint(n)
    return sum([wt(f[i]) for i in f]) # Indranil Ghosh, May 30 2017

;; uses memoizing-macro definec
(definec (A064547 n) (cond ((= 1 n) 0) (else (+ (A000120 (A067029 n)) (A064547 (A028234 n))))))
;; Antti Karttunen, Feb 09 2016

;; uses memoizing-macro definec
(definec (A064547 n) (if (= 1 n) 0 (+ (A000120 (A007814 n)) (A064547 (A064989 n)))))
;; Antti Karttunen, Feb 09 2016

a(m*n) <= a(m)*a(n). - Reinhard Zumkeller, Mar 20 2013
From Antti Karttunen, Feb 09 2016: (Start)
a(1) = 0, and for n > 1, a(n) = A000120(A067029(n)) + a(A028234(n)).
a(1) = 0, and for n > 1, a(n) = A000120(A007814(n)) + a(A064989(n)).
(End)
a(n) = log_2(A037445(n)). - Vladimir Shevelev, May 13 2016
a(n) = A286574(A156552(n)). - Antti Karttunen, May 28 2017
Additive with a(p^e) = A000120(e). - Jianing Song, Jul 28 2018
a(n) = A000120(A052331(n)). - Peter Munn, Aug 26 2019
From Peter Munn, Dec 18 2019: (Start)
a(A000379(n)) mod 2 = 0.
a(A000028(n)) mod 2 = 1.
A001221(n) <= a(n) <= A001222(n).
A001221(n) < a(n) => a(n) < A001222(n).
a(n) = A001222(n) if and only if n is in A005117.
a(n) = A001221(n) if and only if n is in A138302.
a(n^2) = a(n).
a(A003961(n)) = a(n).
a(A225546(n)) = a(n).
a(n) = a(A007913(n)) + a(A008833(n)).
a(A050376(n)) = 1.
a(A059897(n,k)) + 2 * a(A059895(n,k)) = a(n) + a(k).
a(A059896(n,k)) + a(A059895(n,k)) = a(n) + a(k).
Alternative definition: a(1) = 0; a(n * m) = a(n) + 1 for m = A050376(k) > A223491(n).
(End)
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 0.13605447049622836522... (A382294), where f(x) = -x + Sum_{k>=0} x^(2^k)/(1+x^(2^k)). - Amiram Eldar, Sep 28 2023
a(n) << log n/log log n. - Charles R Greathouse IV, Nov 29 2024

A046644 From square root of Riemann zeta function: form Dirichlet series Sum b_n/n^s whose square is zeta function; sequence gives denominator of b_n.

Original entry on oeis.org

1, 2, 2, 8, 2, 4, 2, 16, 8, 4, 2, 16, 2, 4, 4, 128, 2, 16, 2, 16, 4, 4, 2, 32, 8, 4, 16, 16, 2, 8, 2, 256, 4, 4, 4, 64, 2, 4, 4, 32, 2, 8, 2, 16, 16, 4, 2, 256, 8, 16, 4, 16, 2, 32, 4, 32, 4, 4, 2, 32, 2, 4, 16, 1024, 4, 8, 2, 16, 4, 8, 2, 128, 2, 4, 16, 16, 4, 8

Offset: 1

N. J. A. Sloane

From Antti Karttunen, Aug 21 2018: (Start)

a(n) is the denominator of any rational-valued sequence f(n) which has been defined as f(n) = (1/2) * (b(n) - Sum_{d|n, d>1, d

Proof:

Proof is by induction. We assume as our induction hypothesis that the given multiplicative formula for A046644 (resp. additive formula for A046645) holds for all proper divisors d|n, dA046645(p) = 1. [Remark: for squares of primes, f(p^2) = (4*b(p^2) - 1)/8, thus a(p^2) = 8.]

First we note that A005187(x+y) <= A005187(x) + A005187(y), with equivalence attained only when A004198(x,y) = 0, that is, when x and y do not have any 1-bits in the shared positions. Let m = Sum_{e} A005187(e), with e ranging over the exponents in prime factorization of n.

For [case A] any n in A268388 it happens that only when d (and thus also n/d) are infinitary divisors of n will Sum_{e} A005187(e) [where e now ranges over the union of multisets of exponents in the prime factorizations of d and n/d] attain value m, which is the maximum possible for such sums computed for all divisor pairs d and n/d. For any n in A268388, A037445(n) = 2^k, k >= 2, thus A037445(n) - 2 = 2 mod 4 (excluding 1 and n from the count, thus -2). Thus, in the recursive formula above, the maximal denominator that occurs in the sum is 2^m which occurs k times, with k being an even number, but not a multiple of 4, thus the factor (1/2) in the front of the whole sum will ensure that the denominator of the whole expression is 2^m [which thus is equal to 2^A046645(n) = a(n)].

On the other hand [case B], for squares in A050376 (A082522, numbers of the form p^(2^k) with p prime and k>0), all the sums A005187(x)+A005187(y), where x+y = 2^k, 0 < x <= y < 2^k are less than A005187(2^k), thus it is the lonely "middle pair" f(p^(2^(k-1))) * f(p^(2^(k-1))) among all the pairs f(d)*f(n/d), 1 < d < n = p^(2^k) which yields the maximal denominator. Furthermore, as it occurs an odd number of times (only once), the common factor (1/2) for the whole sum will increase the exponent of 2 in denominator by one, which will be (2*A005187(2^(k-1))) + 1 = A005187(2^k) = A046645(p^(2^k)).

(End)

From Antti Karttunen, Aug 21 2018: (Start)

The following list gives a few such pairs num(n), b(n) for which b(n) is Dirichlet convolution of num(n)/a(n). Here ε stands for sequence A063524 (1, 0, 0, ...).

Numerators Dirichlet convolution of numerator(n)/a(n) yields

------- -----------

A046643 A000012

A257098 A008683

A317935 A003557

A318321 A003961

A317830 A175851

A317833 A078898

A317834 A078899

A317847 A303757

A317835 ε + A003415

A317845 ε + A001065

A317846 ε + A051953

A317936 ε + A100995

A317937 ε + A001221

A317938 ε + A001222

A317939 ε + A010051 (= A080339)

(End)

This sequence gives an upper bound for the denominators of any rational-valued sequence obtained as the "Dirichlet Square Root" of any integer-valued sequence. - Andrew Howroyd, Aug 23 2018

Antti Karttunen, Table of n, a(n) for n = 1..8192
Wikipedia, Dirichlet convolution
Index entries for sequences computed from exponents in factorization of n

Cf. A000079, A005187, A028234, A037445, A050376, A067029, A082522, A268388.
See A046643 for more details. See also A046645, A317940.
Cf. A299150, A299152, A317832, A317926, A317932, A317934 (for denominator sequences of other similar constructions).

b[1] = 1; b[n_] := b[n] = (dn = Divisors[n]; c = 1;
Do[c = c - b[dn[[i]]]*b[n/dn[[i]]], {i, 2, Length[dn] - 1}]; c/2); a[n_] := Denominator[b[n]]; a /@ Range[78] (* Jean-François Alcover, Apr 04 2011, after Maple code in A046643 *)
a18804[n_] := Sum[n EulerPhi[d]/d, {d, Divisors[n]}];
f[1] = 1; f[n_] := f[n] = 1/2 (a18804[n] - Sum[f[d] f[n/d], {d, Divisors[ n][[2 ;; -2]]}]);
a[n_] := f[n] // Denominator;
Array[a, 78] (* Jean-François Alcover, Sep 13 2018, after A318443 *)

A046643perA046644(n) = { my(c=1); if(1==n,c,fordiv(n,d, if((d>1)&&(dA046643perA046644(d)*A046643perA046644(n/d)))); (c/2)); }
A046644(n) = denominator(A046643perA046644(n)); \\ After the Maple-program given in A046643, Antti Karttunen, Jul 08 2017

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A046644(n) = factorback(apply(e -> 2^A005187(e),factor(n)[,2])); \\ Antti Karttunen, Aug 12 2018

for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-X)^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 04 2025

;; With memoization-macro definec.
(definec (A046644 n) (if (= 1 n) n (* (A000079 (A005187 (A067029 n))) (A046644 (A028234 n))))) ;; Antti Karttunen, Jul 08 2017

From Antti Karttunen, Jul 08 2017: (Start)
Multiplicative with a(p^n) = 2^A005187(n).
a(1) = 1; for n > 1, a(n) = A000079(A005187(A067029(n))) * a(A028234(n)).
a(n) = A000079(A046645(n)).
(End)
Sum_{j=1..n} A046643(j)/A046644(j) ~ n / sqrt(Pi*log(n)) * (1 + (1 - gamma/2)/(2*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 04 2025

A000379 Numbers where total number of 1-bits in the exponents of their prime factorization is even; a 2-way classification of integers: complement of A000028.

Original entry on oeis.org

1, 6, 8, 10, 12, 14, 15, 18, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, 36, 38, 39, 44, 45, 46, 48, 50, 51, 52, 55, 57, 58, 62, 63, 64, 65, 68, 69, 74, 75, 76, 77, 80, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 100, 106, 111, 112, 115, 116, 117, 118, 119, 120, 122, 123, 124, 125, 129

Offset: 1

N. J. A. Sloane

This sequence and A000028 (its complement) give the unique solution to the problem of splitting the positive integers into two classes in such a way that products of pairs of distinct elements from either class occur with the same multiplicities [Lambek and Moser]. Cf. A000069, A001969.
See A000028 for precise definition, Maple program, etc.
The sequence contains products of even number of distinct terms of A050376. - Vladimir Shevelev, May 04 2010
From Vladimir Shevelev, Oct 28 2013: (Start)
Numbers m such that the infinitary Möbius function (A064179) of m equals 1. (This follows from the definition of A064179.)
A number m is in the sequence iff the number k = k(m) of terms of A050376 that divide m with odd maximal exponent is even (see example).
(End)
Numbers k for which A064547(k) [or equally, A268386(k)] is even. Numbers k for which A010060(A268387(k)) = 0. - Antti Karttunen, Feb 09 2016
The sequence is closed under the commutative binary operation A059897(.,.). As integers are self-inverse under A059897(.,.), it therefore forms a subgroup of the positive integers considered as a group under A059897(.,.). Specifically (expanding on the comment above dated May 04 2010) it is the subgroup of even length words in A050376, which is the group's lexicographically earliest ordered minimal set of generators. A000028, the set of odd length words in A050376, is its complementary coset. - Peter Munn, Nov 01 2019
From Amiram Eldar, Oct 02 2024: (Start)
Numbers whose number of infinitary divisors (A037445) is a square.
Numbers whose exponentially odious part (A367514) has an even number of distinct prime factors, i.e., numbers k such that A092248(A367514(k)) = 0. (End)

			If m = 120, then the maximal exponent of 2 that divides 120 is 3, for 3 it is 1, for 4 it is 1, for 5 it is 1. Thus k(120) = 4 and 120 is a term. - _Vladimir Shevelev_, Oct 28 2013

Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 22.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
J. Lambek and L. Moser, On some two way classifications of integers, Canad. Math. Bull. 2 (1959), 85-89.
Eric Weisstein's World of Mathematics, Group.
Wikipedia, Generating set of a group.

Subsequences: A030229, A238748, A262675, A268390.
Subsequence of A268388 (apart from the initial 1).
Complement: A000028.
Sequences used in definitions of this sequence: A133008, A050376, A059897, A064179, A064547, A124010 (prime exponents), A268386, A268387, A010060.
Other 2-way classifications: A000069/A001969 (to which A000120 and A010060 are relevant), A000201/A001950.
This is different from A123240 (e.g., does not contain 180). The first difference occurs already at n=31, where A123240(31) = 60, a value which does not occur here, as a(31+1) = 62. The same is true with respect to A131181, as A131181(31) = 60.
Cf. A030231, A037445, A092248, A367514.

a000379 n = a000379_list !! (n-1)
a000379_list = filter (even . sum . map a000120 . a124010_row) [1..]
-- Reinhard Zumkeller, Oct 05 2011

Select[ Range[130], EvenQ[ Count[ Flatten[ IntegerDigits[#, 2]& /@ Transpose[ FactorInteger[#]][[2]]], 1]]&] // Prepend[#, 1]& (* Jean-François Alcover, Apr 11 2013, after Harvey P. Dale *)

is(n)=my(f=factor(n)[,2]); sum(i=1,#f,hammingweight(f[i]))%2==0 \\ Charles R Greathouse IV, Aug 31 2013
(Scheme, two variants)
(define A000379 (MATCHING-POS 1 1 (COMPOSE even? A064547)))
(define A000379 (MATCHING-POS 1 1 (lambda (n) (even? (A000120 (A268387 n))))))
;; Both require also my IntSeq-library. - Antti Karttunen, Feb 09 2016

Edited by N. J. A. Sloane, Dec 20 2007, to restore the original definition.

A332820 Integers in the multiplicative subgroup of positive rationals generated by the products of two consecutive primes and the cubes of primes. Numbers k for which A048675(k) is a multiple of three.

Original entry on oeis.org

1, 6, 8, 14, 15, 20, 26, 27, 33, 35, 36, 38, 44, 48, 50, 51, 58, 63, 64, 65, 68, 69, 74, 77, 84, 86, 90, 92, 93, 95, 106, 110, 112, 117, 119, 120, 122, 123, 124, 125, 141, 142, 143, 145, 147, 156, 158, 160, 161, 162, 164, 170, 171, 177, 178, 185, 188, 196, 198, 201, 202, 208, 209, 210, 214, 215, 216, 217, 219, 221, 225

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

nonn

The positive integers are partitioned between this sequence, A332821 and A332822, which list the integers in respective cosets of the subgroup.
As the sequence lists the integers in a multiplicative subgroup of the positive rationals, the sequence is closed under multiplication and, provided the result is an integer, under division.
It follows that for any n in this sequence, all powers n^k are present (k >= 0), as are all cubes.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, the resulting numbers are a permutation of the full sequence; and if we take the square root of each square term we get the full sequence.
There are no primes in the sequence, therefore if k is present and p is a prime, k*p and k/p are absent (noting that k/p might not be an integer). This property extends from primes to all terms of A050376 (often called Fermi-Dirac primes), therefore to squares of primes, 4th powers of primes etc.
The terms are the even numbers in A332821 halved. The terms are also the numbers m such that 5m is in A332821, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332822, and so on for alternate primes: 7, 13, 19 etc.
The numbers that are half of the even terms of this sequence are in A332822, which consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332821, which consists exactly of those numbers. These properties extend in a pattern of alternating primes as described in the previous paragraph.
If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.
If m and n are in this sequence then so is m*n (the definition of "multiplicative semigroup"), while if n is in this sequence, and x is in the complement A359830, then n*x is in A359830. This essentially follows from the fact that A048675 is totally additive sequence. Compare to A329609. - Antti Karttunen, Jan 17 2023

Positions of zeros in A332823; equivalently, numbers in row 3k of A277905 for some k >= 0.
Cf. A048675, A195017, A332821, A332822, A353350 (characteristic function), A353348 (its Dirichlet inverse), A359830 (complement).
Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
Subsequences: A000578\{0}, A006094, A090090, A099788, A245630 (A191002 in ascending order), A244726\{0}, A325698, A338471, A338556, A338907.
Subsequence of {1} U A268388.

Select[Range@ 225, Or[Mod[Total@ #, 3] == 0 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]], # == 1] &] (* Michael De Vlieger, Mar 15 2020 *)

isA332820(n) =  { my(f = factor(n)); !((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); };

{a(n) : n >= 1} = {1} U {2 * A332822(k) : k >= 1} U {A003961(a(k)) : k >= 1}.
{a(n) : n >= 1} = {1} U {a(k)^2 : k >= 1} U {A331590(2, A332822(k)) : k >= 1}.
From Peter Munn, Mar 17 2021: (Start)
{a(n) : n >= 1} = {k : k >= 1, 3|A048675(k)}.
{a(n) : n >= 1} = {k : k >= 1, 3|A195017(k)}.
{a(n) : n >= 1} = {A332821(k)/2 : k >= 1, 2|A332821(k)}.
{a(n) : n >= 1} = {A332822(k)/3 : k >= 1, 3|A332822(k)}.
(End)

New name from Peter Munn, Mar 08 2021

Showing 1-6 of 6 results.

cp's OEIS Frontend

A050376 "Fermi-Dirac primes": numbers of the form p^(2^k) where p is prime and k >= 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

PARI

PARI

Python

Python

Scheme

Formula

Extensions

A225546 Tek's flip: Write n as the product of distinct factors of the form prime(i)^(2^(j-1)) with i and j integers, and replace each such factor with prime(j)^(2^(i-1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

Extensions

A064547 Sum of binary digits (or count of 1-bits) in the exponents of the prime factorization of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Scheme

Scheme

Formula

A046644 From square root of Riemann zeta function: form Dirichlet series Sum b_n/n^s whose square is zeta function; sequence gives denominator of b_n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Scheme

Formula

A000379 Numbers where total number of 1-bits in the exponents of their prime factorization is even; a 2-way classification of integers: complement of A000028.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs