A026477 -id:A026477 - cp's OEIS frontend

A276630 a(n) = number of prime signature permutations which can prohibit the appearance of terms in A026477 that are members of the same signature set (see explanation in "Comments" and "Examples").

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

Offset: 1

Bob Selcoe, Sep 07 2016

nonn

Explanation:
Prime signatures are multisets of prime exponents in descending order, representing the prime factors of k. For example, k = 200 has prime signature {3,2} because 2^3*5^2 = 200. Prime signature sets contain all numbers that have the same signature; so {72, 108, 200, 392, 500, 675, 968, 1125, 1323..} are members of signature set {3,2}. All signature sets are infinite except {} == {0} = 1 (because p^0 = 1).
A026477 begins a(1)=1, a(2)=2 and a(3)=3; a(n) is smallest positive integer > a(n-1) not a product of any 3 distinct prior terms. Consequently, either all or no terms appear in A026477 that are members of a given three-exponent signature set Z. Call "IN" those Z's whose members are terms which appear, and call "OUT" those Z's whose members are terms which do not appear.
Determining which Z's are IN or OUT involves taking three IN's and summing various permutations of exponents, such that no two are identical in both order and placement (because all terms in A026477 are unique). If no permutation exists which generates a specific Z, then that Z is IN, otherwise OUT. Evaluating Z in graded reverse lexicographic order (see A080577), the first few IN are: {}, {1}, {2}, {4}, {1,1,1,1}, {3,1,1}, {3,3}, {2,2,2,1}, {1,1,1,1,1,1,1}. All other Z whose signature sums < 8 are OUT.
Only one permutation of exponents summing to Z is required for Z to be OUT. However, more than one permutation may exist; a(n) therefore represents the number of three IN's whose sums permute to Z, where n reflects the position of Z in the above order of partitions. Thus a(n)=0 are IN, all others are OUT. See "Examples" for additional discussion.
It appears that a(n) tends to increase as n increases.

			NOTE: Partitions are in graded reverse lexicographic order.
a(8) = 0; the 8th partition is {4}, therefore members of {4} (i.e., primes p^4) appear in A026477.
a(14) = 1; the 14th partition is {4,1}. The one group of three previously-appearing members of signature sets in A026477 whose sums permute to {4,1}  is  {4} + {1} + {}, so members of {4,1} do not appear in A026477.  Note that while {2} (p^2) also appears and {2} + {2} + {1} also permutes to {4,1}, it does not pertain here because {2} + {2} = {4} iff p^2*p^2 = p^4, violating the condition that all terms are products of distinct prior terms. So a(14) = 1, rather than 2.
a(59) = 3; the 59th partition is {3,3,1,1}. The three applicable signature sets which permute to {3,3,1,1} are A: {1,1,1,1} + {2} + {2}; B: {3,1,1} + {2} + {1} and C: {3,3} + {1} + {1}; so members of {3,3,1,1} do not appear in A026477. Note that A and C pertain here with repeated signature sets ({2} for A and {1} for C) because, unlike a(14), their placement in the permutation is different. So for instance, {1,1,1,1} in A = primes p*q*r*s, so one {2} may = p^2 while the other {2} may = q^2.

Cf. A026477, A080577.

Cf. A276631 (signature sets whose members appear in A026477; i.e, partitions corresponding with a(n)=0 in this sequence).

A276631 Expansion of signature sets in graded reverse lexicographic order whose members appear as terms in A026477.

Original entry on oeis.org

0, 1, 2, 4, 1, 1, 1, 1, 3, 1, 1, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 8, 7, 1, 3, 1, 1, 1, 1, 1, 6, 2, 1, 5, 2, 2, 3, 3, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 6, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 10, 3, 4, 4, 4, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Bob Selcoe, Sep 07 2016

nonn

See A276630 for detailed explanation about which signature sets appear here.

It appears that the number of signature sets with the same sum tend to increase as the sums increase.

			Signature sets start:
{}
{1}
{2}
{4}
{1, 1, 1, 1}
{3, 1, 1}
{3, 3}
{2, 2, 2, 1}
{1, 1, 1, 1, 1, 1, 1}
{8}
{7, 1}
{3, 1, 1, 1, 1, 1}
{6, 2, 1}
{5, 2, 2}
{3, 3, 1, 1, 1}
{3, 3, 3, 1}
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1}
{8, 1, 1}
{6, 5}
{3, 1, 1, 1, 1, 1, 1, 1, 1}
{10, 1, 1}
{10, 3}
{4, 4, 4, 1}
{2, 2, 2, 2, 2, 2, 1}
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}

Cf. A026477, A276630.

A278266 a(n) = A046523(A026477(n)).

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 4, 2, 2, 16, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 16, 2, 2, 2, 2, 2, 2, 2, 2, 120, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 120, 4, 2, 2, 2, 2, 2, 2, 2, 210, 2, 216, 2, 2, 2, 2, 2, 2, 2, 256, 2, 2, 120, 2, 120, 2, 2, 120, 2, 2, 4, 2, 2, 2, 120, 2, 2, 210, 2, 2, 2, 2, 2, 2, 4, 2, 2, 120, 2, 2, 384, 2, 210, 2, 2, 120, 2, 2, 2, 2, 2, 2

Offset: 1

Antti Karttunen, Nov 16 2016

nonn

Smallest number with the same prime signature as A026477(n). See further comments there.

Terms and b-file computed from b-file of A026477 provided by Charles R Greathouse IV.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A026477, A046523.

(define (A278266 n) (A046523 (A026477 n)))

a(n) = A046523(A026477(n)).

A275911 Intersection of A025487 and A026477.

Original entry on oeis.org

1, 2, 4, 16, 120, 210, 216, 256, 384, 2880, 6300, 7200, 15360, 15552, 26880, 27648, 32768, 49152, 73728, 83160, 120120, 189000, 510510, 921600, 1399680, 1966080, 2365440, 2822400, 2985984, 3440640, 4194304, 4860000, 5670000, 6291456, 6912000, 9437184

Offset: 1

N. J. A. Sloane, Aug 26 2016

nonn

Don Reble, Posting to Sequence Fans Mailing List, Aug 26 2016

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

Original entry on oeis.org

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

A001146 a(n) = 2^(2^n).

Original entry on oeis.org

2, 4, 16, 256, 65536, 4294967296, 18446744073709551616, 340282366920938463463374607431768211456, 115792089237316195423570985008687907853269984665640564039457584007913129639936

Offset: 0

N. J. A. Sloane

Or, write previous term in base 2, read in base 4.
a(1) = 2, a(n) = smallest power of 2 which does not divide the product of all previous terms.
Number of truth tables generated by Boolean expressions of n variables. - C. Bradford Barber (bradb(AT)shore.net), Dec 27 2005
From Ross Drewe, Feb 13 2008: (Start)
Or, number of distinct n-ary operators in a binary logic. The total number of n-ary operators in a k-valued logic is T = k^(k^n), i.e., if S is a set of k elements, there are T ways of mapping an ordered subset of n elements from S to an element of S. Some operators are "degenerate": the operator has arity p, if only p of the n input values influence the output. Therefore the set of operators can be partitioned into n+1 disjoint subsets representing arities from 0 to n.
For n = 2, k = 2 gives the familiar Boolean operators or functions, C = F(A,B). There are 2^2^2 = 16 operators, composed of: arity 0: 2 operators (C = 0 or 1), arity 1: 4 operators (C = A, B, not(A), not(B)), arity 2: 10 operators (including well-known pairs AND/NAND, OR/NOR, XOR/EQ). (End)
From José María Grau Ribas, Jan 19 2012: (Start)
  Or, numbers that can be formed using the number 2, the power operator (^), and parenthesis. (End) [The paper by Guy and Selfridge (see also A003018) shows that this is the same as the current sequence. - N. J. A. Sloane, Jan 21 2012]
a(n) is the highest value k such that A173419(k) = n+1. - Charles R Greathouse IV, Oct 03 2012
Let b(0) = 8 and b(n+1) = the smallest number not in the sequence such that b(n+1) - Product_{i=0..n} b(i) divides b(n+1)*Product_{i=0..n} b(i). Then b(n) = a(n) for n > 0. - Derek Orr, Jan 15 2015
Twice the number of distinct minimal toss sequences of a coin to obtain all sequences of length n, which is 2^(2^n-1). This derives from the 2^n ways to cut each of the de Bruijn sequences B(2,n). - Maurizio De Leo, Feb 28 2015
I conjecture that { a(n) ; n>1 } are the numbers such that n^4-1 divides 2^n-1, intersection of A247219 and A247165. - M. F. Hasler, Jul 25 2015
Erdős has shown that it is an irrationality sequence (see Guy reference). - Stefano Spezia, Oct 13 2024

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section E24.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.
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).

Vincenzo Librandi, Table of n, a(n) for n = 0..11
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Jan Brandts and Apo Cihangir, Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group, Special Matrices, Vol. 5, No. 1 (2017), pp. 158-201, arXiv preprint, arXiv:1512.03044 [math.CO], 2015.
John H. Conway, Sphere packings, lattices, codes and greed, pp. 45-55 of Proc. Intern. Congr. Math., Vol. 2, 1994, alternative link.
Jose María Grau and A. M. Oller-Marcén On the last digit and the last non-zero digit of n^n in base b, arXiv:1203.4066 [math.NT], 2012.
Richard K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
Rudolf Ondrejka, Exact values of 2^n, n=1(1)4000, Math. Comp., 23 (1969), 456.
Rudolf Ondrejka, Letter to N. J. A. Sloane, May 15 1976
Eric Weisstein's World of Mathematics, Irrationality Sequence, Quadratic Recurrence Equation, Coin Tossing.
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A000079, A000215, A007404, A026477, A062090, A062091, A112535, A155538, A215016.

Cf. also A003018, A051179, A173419, A165420, A247165, A247219.

a001146 = (2 ^) . (2 ^)
a001146_list = iterate (^ 2) 2  -- Reinhard Zumkeller, Jun 04 2012

[2^(2^n): n in [0..8]]; // Vincenzo Librandi, Jun 20 2011

A001146:=n->2^(2^n): seq(A001146(n), n=0..9); # Wesley Ivan Hurt, Sep 19 2014

2^2^Range[0,10] (* Harvey P. Dale, Jul 20 2011 *)

a(n)=1<<2^n \\ Charles R Greathouse IV, Jul 25 2011

a(n)=2^2^n \\ Charles R Greathouse IV, Oct 03 2012

def A001146(n): return 1<<(1<Chai Wah Wu, Mar 14 2023

a(n+1) = (a(n))^2.
1 = Sum_{n>=0} a(n)/A051179(n+1) = 2/3 + 4/15 + 16/255 + 256/65535, ..., with partial sums: 2/3, 14/15, 254/255, 65534/65535, ... - Gary W. Adamson, Jun 15 2003
a(n) = A000079(A000079(n)). - Robert Israel, Jan 15 2015
Sum_{n>=0} 1/a(n) = A007404. - Amiram Eldar, Oct 14 2020
From Amiram Eldar, Jan 28 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = 2.
Product_{n>=0} (1 - 1/a(n)) = A215016. (End)

A000028 Let k = p_1^e_1 p_2^e_2 p_3^e_3 ... be the prime factorization of n. Sequence gives k such that the sum of the numbers of 1's in the binary expansions of e_1, e_2, e_3, ... is odd.

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 24, 25, 29, 30, 31, 37, 40, 41, 42, 43, 47, 49, 53, 54, 56, 59, 60, 61, 66, 67, 70, 71, 72, 73, 78, 79, 81, 83, 84, 88, 89, 90, 96, 97, 101, 102, 103, 104, 105, 107, 108, 109, 110, 113, 114, 121, 126, 127, 128, 130, 131, 132, 135, 136, 137

Offset: 1

N. J. A. Sloane, Simon Plouffe

This sequence and A000379 (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.
Contains (for example) 180, so is different from A123193. - Max Alekseyev, Sep 20 2007
The sequence contains products of odd number of distinct terms of A050376. - Vladimir Shevelev, May 04 2010
From Vladimir Shevelev, Oct 28 2013: (Start)
Numbers m such that infinitary Moebius function of m (A064179) equals -1. This follows from the definition of A064179.
Number m is in the sequence if and only if the number k = k(m) of terms of A050376 which divide m with odd maximal exponent is odd.
For example, if m = 96, then the maximal exponent of 2 that divides 96 is 5, for 3 it is 1, for 4 it is 2, for 16 it is 1. Thus k(96) = 3 and 96 is a term.
(End)
Positions of odd terms in A064547, A268386 and A293439. - Antti Karttunen, Nov 09 2017
Lexicographically earliest sequence of distinct nonnegative integers such that no term is the A059897 product of 2 terms. (A059897 can be considered as a multiplicative operator related to the Fermi-Dirac factorization of numbers described in A050376.) Specifying that the A059897 product be of 2 distinct terms leaves the sequence unchanged. The equivalent sequences using standard integer multiplication are A026416 (with the 2 terms specified as distinct) and A026424 (otherwise). - Peter Munn, Mar 16 2019
From Amiram Eldar, Oct 02 2024: (Start)
Numbers whose number of infinitary divisors (A037445) is not a square.
Numbers whose exponentially odious part (A367514) has an odd number of distinct prime factors, i.e., numbers k such that A092248(A367514(k)) = 1. (End)

			If k = 96 then the maximal exponent of 2 that divides 96 is 5, for 3 it is 1. 5 in binary is 101_2 and has so has a sum of binary digits of 1 + 0 + 1 = 2. 1 in binary is 1_2 and so has a sum of binary digits of 1. Thus the sum of digits of binary exponents is 2 + 1 = 3 which is odd and so 96 is a term. - _Vladimir Shevelev_, Oct 28 2013, edited by _David A. Corneth_, Mar 20 2019

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.
Index entries for sequences computed from exponents in factorization of n.

Cf. A133008, A000379 (complement), A000120 (binary weight function), A064547; also A066724, A026477, A050376, A084400, A268386, A293439.
Note that A000069 and A001969, also A000201 and A001950 give other decompositions of the integers into two classes.
Cf. A124010 (prime exponents).
Cf. A026416, A026424, A059897.
Cf. A030230, A037445, A092248, A367514.

a000028 n = a000028_list !! (n-1)
a000028_list = filter (odd . sum . map a000120 . a124010_row) [1..]
-- Reinhard Zumkeller, Oct 05 2011

(Maple program from N. J. A. Sloane, Dec 20 2007) 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; # returns a list of the exponents e_1, e_2, ...
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: # returns weight of binary expansion
LamMos:= proc(n) local t1,t2,t3,i; t1:=expts(n); add( A000120(t1[i]),i=1..nops(t1)); end; # returns sum of weights of exponents
M:=400; t0:=[]; t1:=[]; for n from 1 to M do if LamMos(n) mod 2 = 0 then t0:=[op(t0),n] else t1:=[op(t1),n]; fi; od: t0; t1; # t0 is A000379, t1 is the present sequence

iMoebiusMu[ n_ ] := Switch[ MoebiusMu[ n ], 1, 1, -1, -1, 0, If[ OddQ[ Plus@@ (DigitCount[ Last[ Transpose[ FactorInteger[ n ] ] ], 2, 1 ]) ], -1, 1 ] ]; q=Select[ Range[ 20000 ],iMoebiusMu[ # ]===-1& ] (* Wouter Meeussen, Dec 21 2007 *)
Rest[Select[Range[150],OddQ[Count[Flatten[IntegerDigits[#,2]&/@ Transpose[ FactorInteger[#]][[2]]],1]]&]] (* Harvey P. Dale, Feb 25 2012 *)

is(n)=my(f=factor(n)[,2]); sum(i=1,#f,hammingweight(f[i]))%2 \\ Charles R Greathouse IV, Aug 31 2013

Entry revised by N. J. A. Sloane, Dec 20 2007, restoring the original definition, correcting the entries and adding a new b-file.

A026416 A 2-way classification of integers: a(1) = 1, a(2) = 2 and for n > 2, a(n) is the smallest number not of the form a(i)*a(j) for 1 <= i < j < n.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 24, 25, 29, 30, 31, 37, 40, 41, 42, 43, 47, 49, 53, 54, 56, 59, 61, 66, 67, 70, 71, 73, 78, 79, 81, 83, 88, 89, 97, 101, 102, 103, 104, 105, 107, 109, 110, 113, 114, 121, 127, 128, 130, 131, 135, 136, 137, 138, 139

Offset: 1

Clark Kimberling

An equivalent definition is: a(1) = 1, a(2) = 2; and for n > 2, a(n) = least positive integer > a(n-1) and not of the form a(i)*a(j) for 1 <= i < j < n.
a(2) to a(29) match the initial terms of A000028. [corrected by Peter Munn, Mar 15 2019]
This has a simpler definition than A000028, but the resulting pair lacks the crucial property of the A000028/A000379 pair (see the comment in A000028). - N. J. A. Sloane, Sep 28 2007
Contains (for example) 180, so is different from A123193. - Max Alekseyev, Sep 20 2007
From Vladimir Shevelev, Apr 05 2013: (Start)
1) The sequence does not contain (for example) 140, so is different from A000028.
2) Representation of numbers which are absent in the sequence as a product of two different terms of the sequence is, generally speaking, not unique. For example, 210 = 2*105 = 3*70 = 5*42 = 7*30.
(End)
Excluding a(1) = 1, the lexicographically earliest sequence of distinct nonnegative integers such that no term is a product of 2 distinct terms. Removing the latter distinctness requirement, the sequence becomes A026424; and the equivalent sequence where the product is of 2 or more distinct terms is A050376. A000028 is similarly the equivalent sequence when A059897 is used as multiplicative operator in place of standard integer multiplication. - Peter Munn, Mar 15 2019

			a(8) is not 10 because we already have 10 = 2*5. Of course all primes appear. 16 appears because 16 is not a product of earlier terms.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..400 from Vincenzo Librandi)

Complement of A131181. Cf. A000028, A059897.
Cf. A066724, A026477, A050376, A084400.
Similar sequences with different starting conditions: A026417 (1,3), A026419 (1,4), A026420 (2,4), A026421 (3,4).
Related sequences with definition using any products (not necessarily distinct) and with various starting conditions: A026422 (1,2),A026423 (1,3), A026424 (2,3), A026425 (1,4), A026426 (2,4), A026427 (3,4).
See also families of related sequences: A026431 (excluding product-1), A026443 (excluding product+2), A026453 (excluding product-2) and references therein.

a[1]=1; a[2]=2; a[n_] := a[n] = For[k = a[n-1] + 1, True, k++, If[ FreeQ[ Table[ a[i]*a[j], {i, 1, n-2}, {j, i+1, n-1}], k], Return[k]]]; Table[a[n], {n, 1, 101}] (* Jean-François Alcover, May 16 2013 *)

from itertools import count, islice
def agen(): # generator of terms
    a, products = [1, 2], {2}
    yield from a
    for k in count(3):
        if k not in products:
            yield k
            products.update(k*a[i] for i in range(len(a)))
            a.append(k)
        products.discard(k)
print(list(islice(agen(), 62))) # Michael S. Branicky, Jun 09 2025

More terms from Max Alekseyev, Sep 23 2007

Edited by N. J. A. Sloane, Jul 13 2008 at the suggestion of R. J. Mathar and Max Alekseyev

cp's OEIS Frontend

A276630 a(n) = number of prime signature permutations which can prohibit the appearance of terms in A026477 that are members of the same signature set (see explanation in "Comments" and "Examples").

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A276631 Expansion of signature sets in graded reverse lexicographic order whose members appear as terms in A026477.

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A278266 a(n) = A046523(A026477(n)).

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A275911 Intersection of A025487 and A026477.

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

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A001146 a(n) = 2^(2^n).

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A000028 Let k = p_1^e_1 p_2^e_2 p_3^e_3 ... be the prime factorization of n. Sequence gives k such that the sum of the numbers of 1's in the binary expansions of e_1, e_2, e_3, ... is odd.

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A026478 a(n) = least positive integer > a(n-1) and not of form a(i)a(j)a(k) for 1<=i<=j<=k