A138302 -id:A138302 - cp's OEIS frontend

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

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

A209061 Exponentially squarefree numbers.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Mar 13 2012

nonn

Numbers having only squarefree exponents in their canonical prime factorization.
According to the formula of Theorem 3 [Toth], the density of the exponentially squarefree numbers is 0.9559230158619... (A262276). - Peter J. C. Moses and Vladimir Shevelev, Sep 10 2015
From Vladimir Shevelev, Sep 24 2015: (Start)
A generalization. Let S be a finite or infinite increasing integer sequence s=s(n), s(0)=0.
Let us call a positive number N an exponentially S-number, if all exponents in its prime power factorization are in the sequence S.
Let {u(n)} be the characteristic function of S. Then, for the density h=h(S) of the exponentially S-numbers, we have the representations
h(S) = Product_{prime p} Sum_{j in S} (p-1)/p^(j+1) = Product_{p} (1 + Sum_{j>=1} (u(j) - u(j-1))/p^j). In particular, if S = {0,1}, then the exponentially S-numbers are squarefree numbers; if S consists of 0 and {2^k}A138302%20(see%20%5BShevelev%5D,%202007);%20if%20S%20consists%20of%200%20and%20squarefree%20numbers,%20then%20u(n)=%7Cmu(n)%7C,%20where%20mu(n)%20is%20the%20M%C3%B6bius%20function%20(A008683),%20we%20obtain%20the%20density%20h%20of%20the%20exponentially%20squarefree%20numbers%20(cf.%20Toth's%20link,%20Theorem%203);%20the%20calculation%20of%20h%20with%20a%20very%20high%20degree%20of%20accuracy%20belongs%20to%20_Juan%20Arias-de-Reyna">{k>=0}, then the exponentially S-numbers form A138302 (see [Shevelev], 2007); if S consists of 0 and squarefree numbers, then u(n)=|mu(n)|, where mu(n) is the Möbius function (A008683), we obtain the density h of the exponentially squarefree numbers (cf. Toth's link, Theorem 3); the calculation of h with a very high degree of accuracy belongs to _Juan Arias-de-Reyna (A262276). Note that if S contains 1, then h(S) >= 1/zeta(2) = 6/Pi^2; otherwise h(S) = 0. Indeed, in the latter case, the density of the sequence of exponentially S-numbers does not exceed the density of A001694, which equals 0. (End)
The term "exponentially squarefree number" was apparently coined by Subbarao (1972). - Amiram Eldar, May 28 2025

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Xiaodong Cao and Wenguang Zhai, Some arithmetic functions involving exponential divisors, JIS 13 (2010), Article 10.3.7.
Y.-F. S. Petermann, Arithmetical functions involving exponential divisors: note on two papers by L. Toth, Ann. Univ. Sci. Budapest, Sect. Comp. 32 (2010) 143-149.
Vladimir Shevelev, Compact integers and factorials, Acta Arithmetica 126:3 (2007), pp. 195-236.
Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015.
Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv:1511.03860 [math.NT], 2015.
Vladimir Shevelev, A fast computation of density of exponentially S-numbers, arXiv:1602.04244 [math.NT], 2016.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395.
H. M. Stark, On the asymptotic density of the k-free integers, Proc. Amer. Soc. 17 (1966), 1211-1214.
M. V. Subbarao, On some arithmetic convolutions, in: A. A. Gioia and D. L. Goldsmith (eds.), The Theory of Arithmetic Functions: Proceedings of the Conference at Western Michigan University, April 29 - May 1, 1971, Berlin, Heidelberg: Springer Berlin Heidelberg, 2006, pp. 247-271; alternative link.
László Tóth, On certain arithmetic functions involving exponential divisors, II., Annales Univ. Sci. Budapest., Sect. Comp., 27 (2007), 155-166 and arXiv:0708.3557 [math.NT], 2007-2009.

Complement of A130897.
A005117, A004709, and A046100 are subsequences.
Cf. A001221, A001694, A008683, A008966, A036537, A115063, A138302, A197680, A262276, A262675, A268335, A270428.

a209061 n = a209061_list !! (n-1)
a209061_list = filter
   (all (== 1) . map (a008966 . fromIntegral) . a124010_row) [1..]

Select[Range@ 69, Times @@ Boole@ Map[SquareFreeQ, Last /@ FactorInteger@ #] > 0 &] (* Michael De Vlieger, Sep 07 2015 *)

is(n)=my(f=factor(n)[,2]); for(i=1,#f,if(!issquarefree(f[i]), return(0))); 1 \\ Charles R Greathouse IV, Sep 02 2015

A166234(a(n)) <> 0.
Product_{k=1..A001221(n)} A008966(A124010(n,k)) = 1.
One can prove that the principal term of Toth's asymptotics for the density of this sequence (cf. Toth's link, Theorem 3) equals also Product_{prime p}(Sum_{j in S}(p-1)/p^{j+1})*x, where S is the set of 0 and squarefree numbers. The remainder term O(x^(0.2+t)), where t>0 is arbitrarily small, was obtained by L. Toth while assuming the Riemann Hypothesis. - Vladimir Shevelev, Sep 12 2015

A270428 Exponentially odious numbers: 1 together with positive integers n such that all exponents in prime factorization of n are odious numbers (A000069).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84

Offset: 1

Antti Karttunen, May 26 2016

A268385 maps each term of this sequence to a unique term of A268335, and vice versa.

The asymptotic density of this sequence is Product_{p prime} f(1/p) = 0.87686263163054480657..., where f(x) = 1 - x + (1 - (1-x) * Product_{k>=0} (1-x^(2^k)))/2. - Amiram Eldar, Oct 27 2023

Antti Karttunen, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175 (2016), 385-395.

Apart from 1, a subsequence of A270420.
Indices of ones in A270419.
Sequence A270436 sorted into ascending order.
Cf. A010060, A028234, A067029, A355825 (characteristic function).
Cf. also A262675, A268335, A268385.
Differs from its subsequence A138302 for the first time at n=113, where a(113) = 128 = 2^7, a value which does not occur in A138302.

odiousQ[n_] := OddQ[DigitCount[n, 2, 1]]; Select[Range[100], AllTrue[FactorInteger[#][[;;, 2]], odiousQ] &] (* Amiram Eldar, May 18 2023 *)

A355825(n) = factorback(apply(e->(hammingweight(e)%2), factor(n)[, 2]));
isA270428(n) = A355825(n); \\ Antti Karttunen, Jul 21 2022

;; Requires Antti Karttunen's IntSeq-library.
(define A270428 (ZERO-POS 1 1 (COMPOSE sub1 A270419)))

;; Requires Antti Karttunen's IntSeq-library.
(definec (chA270428 n) (cond ((= 1 n) 1) (else (* (A010060 (A067029 n)) (chA270428 (A028234 n))))))
(define A270428 (NONZERO-POS 1 1 chA270428))

A262675 Exponentially evil numbers.

Original entry on oeis.org

1, 8, 27, 32, 64, 125, 216, 243, 343, 512, 729, 864, 1000, 1024, 1331, 1728, 1944, 2197, 2744, 3125, 3375, 4000, 4096, 4913, 5832, 6859, 7776, 8000, 9261, 10648, 10976, 12167, 13824, 15552, 15625, 16807, 17576, 19683, 21952, 23328, 24389, 25000, 27000, 27648, 29791

Offset: 1

Vladimir Shevelev, Sep 27 2015

Or the numbers whose prime power factorization contains primes only in evil exponents (A001969): 0, 3, 5, 6, 9, 10, 12, ...
If n is in the sequence, then n^2 is also in the sequence.
A268385 maps each term of this sequence to a unique nonzero square (A000290), and vice versa. - Antti Karttunen, May 26 2016

			864 = 2^5*3^3; since 5 and 3 are evil numbers, 864 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

Subsequence of A036966.
Apart from 1, a subsequence of A270421.
Indices of ones in A270418.
Sequence A270437 sorted into ascending order.
Cf. A001969, A209061, A138302, A197680, A000578, A000584, A001014, A001017, A008456, A010803, A010805, A010806, A010808, A010811, A010812, A001221, A010059, A124010, A268385, A270428.

a262675 n = a262675_list !! (n-1)
a262675_list = filter
   (all (== 1) . map (a010059 . fromIntegral) . a124010_row) [1..]
-- Reinhard Zumkeller, Oct 25 2015

{1}~Join~Select[Range@ 30000, AllTrue[Last /@ FactorInteger[#], EvenQ@ First@ DigitCount[#, 2] &] &] (* Michael De Vlieger, Sep 27 2015, Version 10 *)
expEvilQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], EvenQ[DigitCount[#, 2, 1]] &]; With[{max = 30000}, Select[Union[Flatten[Table[i^2*j^3, {j, Surd[max, 3]}, {i, Sqrt[max/j^3]}]]], expEvilQ]] (* Amiram Eldar, Dec 01 2023 *)

isok(n) = {my(f = factor(n)); for (i=1, #f~, if (hammingweight(f[i,2]) % 2, return (0));); return (1);} \\ Michel Marcus, Sep 27 2015

use ntheory ":all"; sub isok { my @f = factor_exp($[0]); return scalar(grep { !(hammingweight($->[1]) % 2) } @f) == @f; } # Dana Jacobsen, Oct 26 2015

Product_{k=1..A001221(n)} A010059(A124010(n,k)) = 1. - Reinhard Zumkeller, Oct 25 2015

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=2} 1/p^A001969(k)) = Product_{p prime} f(1/p) = 1.2413599378..., where f(x) = (1/(1-x) + Product_{k>=0} (1 - x^(2^k)))/2. - Amiram Eldar, May 18 2023, Dec 01 2023

More terms from Michel Marcus, Sep 27 2015

A335275 Numbers k such that the largest square dividing k is a unitary divisor of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76

Offset: 1

Amiram Eldar, Jul 06 2020

nonn

Numbers k such that gcd(A008833(k), k/A008833(k)) = 1.
Numbers whose prime factorization contains exponents that are either 1 or even.
Numbers whose powerful part (A057521) is a square.
First differs from A220218 at n = 227: a(227) = 256 is not a term of A220218.
The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.881513... (A065465).
Complement of A295661. - Vaclav Kotesovec, Jul 07 2020
Differs from A096432 in having or not having 1, 256, 432, 648, 768, 1280, 1728, 1792, 2000, 2160, 2304,... - R. J. Mathar, Jul 22 2020
Equivalently, numbers k whose squarefree part (A007913) is a unitary divisor, or gcd(A007913(k), A008833(k)) = 1. - Amiram Eldar, Oct 09 2022

			12 is a term since the largest square dividing 12 is 4, and 4 and 12/4 = 3 are coprime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Trans. Amer. Math. Soc., Vol. 112, No. 2 (1964), pp. 214-227. See corollary 3.1.2, p. 222.

A000290, A138302 and A220218 are subsequences.

Cf. A007913, A008833, A057521, A065465, A295661.

seqQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # == 1 || EvenQ[#] &];  Select[Range[100], seqQ]

isok(k) = my(d=k/core(k)); gcd(d, k/d) == 1; \\ Michel Marcus, Jul 07 2020

A268375 Numbers k for which A001222(k) = A267116(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 73, 75, 76, 79, 80, 81, 83, 89, 92, 97, 98, 99, 101, 103, 107, 109, 112, 113, 116, 117, 121, 124, 125, 127, 128, 131, 137, 139, 144, 147, 148, 149, 151, 153

Offset: 1

Antti Karttunen, Feb 03 2016

Numbers k whose prime factorization k = p_1^e_1 * ... * p_m^e_m contains no pair of exponents e_i and e_j (i and j distinct) whose base-2 representations have at least one shared digit-position in which both exponents have a 1-bit.
Equivalently, numbers k such that the factors in the (unique) factorization of k into powers of squarefree numbers with distinct exponents that are powers of two, are prime powers. For example, this factorization of 90 is 10^1 * 3^2, so 90 is not included, as 10 is not prime; whereas this factorization of 320 is 5^1 * 2^2 * 2^4, so 320 is included as 5 and 2 are both prime. - Peter Munn, Jan 16 2020
A225546 maps the set of terms 1:1 onto A138302. - Peter Munn, Jan 26 2020
Equivalently, numbers k for which A064547(k) = A331591(k). - Amiram Eldar, Dec 23 2023

			12 = 2^2 * 3^1 is included in the sequence as the exponents 2 ("10" in binary) and 1 ("01" in binary) have no 1-bits in the same position, and 18 = 2^1 * 3^2 is included for the same reason.
On the other hand, 24 = 2^3 * 3^1 is NOT included in the sequence as the exponents 3 ("11" in binary) and 1 ("01" in binary) have 1-bit in the same position 0.
720 = 2^4 * 3^2 * 5^1 is included as the exponents 1, 2 and 4 ("001", "010" and "100" in binary) have no 1-bits in shared positions.
Likewise, 10! = 3628800 = 2^8 * 3^4 * 5^2 * 7^1 is included as the exponents 1, 2, 4 and 8 ("0001", "0010", "0100" and "1000" in binary) have no 1-bits in shared positions. And similarly for any term of A191555.

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

Indices of zeros in A268374, also in A289618.
Cf. A001222, A064547, A267116, A046645, A289617, A331591.
Cf. A091862 (characteristic function), A268376 (complement).
Cf. A000961, A054753, A191555 (subsequences).
Related to A138302 via A225546.
Cf. also A318363 (a permutation).

{1}~Join~Select[Range@ 160, PrimeOmega@ # == BitOr @@ Map[Last, FactorInteger@ #] &] (* Michael De Vlieger, Feb 04 2016 *)

A384247 The number of integers from 1 to n whose largest divisor that is an infinitary divisor of n is 1.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 4, 8, 4, 10, 6, 12, 6, 8, 15, 16, 8, 18, 12, 12, 10, 22, 8, 24, 12, 18, 18, 28, 8, 30, 16, 20, 16, 24, 24, 36, 18, 24, 16, 40, 12, 42, 30, 32, 22, 46, 30, 48, 24, 32, 36, 52, 18, 40, 24, 36, 28, 58, 24, 60, 30, 48, 48, 48, 20, 66, 48, 44, 24

Offset: 1

Amiram Eldar, May 23 2025

Analogous to A047994, as A064380 is analogous to A116550.

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

Row lengths of A384246.

Cf. A000010, A001146, A006519, A047994, A064380, A091732, A116550, A138302, A268335, A384248.

f[p_, e_] := p^e*(1 - 1/p^(2^(IntegerExponent[e, 2]))); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); n * prod(i = 1, #f~, (1 - 1/f[i,1]^(1 << valuation(f[i,2], 2))));}

Multiplicative with a(p^e) = p^e * (1 - 1/p^A006519(e)).
a(n) >= A091732(n), with equality if and only if n is in A138302.
a(n) <= A047994(n), with equality if and only if n is in A138302.
a(n) >= A000010(n), with equality if and only if n is an exponentially odd number (A268335).
a(n) is odd if and only if n = 1 or 2^(2^k) for k >= 0 (A001146). a(2^(2^k)) = 2^(2^k)-1.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 0.66718130416373472394..., and f(x) = 1 - (1-x)*Sum_{k>=1} x^(2^k)/(1-x^(2^k)).

A191555 a(n) = Product_{k=1..n} prime(k)^(2^(n-k)).

Original entry on oeis.org

1, 2, 12, 720, 3628800, 144850083840000, 272760108249915378892800000000, 1264767303092594444142256488682840323816161280000000000000000

Offset: 0

Rick L. Shepherd, Jun 06 2011

x^(2^n) - a(n) is the minimal polynomial over Q for the algebraic number sqrt(p(1)*sqrt(p(2)*...*sqrt(p(n-1)*sqrt(p(n)))...)), where p(k) is the k-th prime.  Each such monic polynomial is irreducible by Eisenstein's Criterion (using p = p(n)).
A prime version of Somos's quadratic recurrence sequence A052129(n) = A052129(n-1)^2 * n = Product_{k=1..n} k^(2^(n-k)). - Jonathan Sondow, Mar 29 2014
All positive integers have unique factorizations into powers of distinct primes, and into powers of squarefree numbers with distinct exponents that are powers of 2. (See A329332 for a description of the relationship between the two.) a(n) is the least number such that both factorizations have n factors. - Peter Munn, Dec 15 2019
From Peter Munn, Jan 24 2020 to Feb 06 2020: (Start)
For n >= 0, a(n+1) is the n-th power of 12 in the monoid defined by A306697.
a(n) is the least positive integer that cannot be expressed as the product of fewer than n terms of A072774 (powers of squarefree numbers).
All terms that are less than the order of the Monster simple group (A003131) are divisors of the group's order, with a(6) exceeding its square root.
(End)
It is remarkable that 4 of the first 5 terms are factorials. - Hal M. Switkay, Jan 21 2025

			a(1) = 2^1 = 2 and x^2 - 2 is the minimal polynomial for the algebraic number sqrt(2).
a(4) = 2^8*3^4*5^2*7^1 = 3628800 and x^16 - 3628800 is the minimal polynomial for the algebraic number sqrt(2*sqrt(3*sqrt(5*sqrt(7)))).

Alois P. Heinz, Table of n, a(n) for n = 0..11

Sequences with related definitions: A006939, A052129, A191554, A239350 (and thence A239349), A252738, A266639.
Cf. also A000040, A000079, A003131, A072774, A329332, A331592.
A000290, A003961, A059896, A306697 are used to express relationship between terms of this sequence.
Subsequence of A025487, A138302, A225547, A267117 (apart from a(1) = 2), A268375, A331593.
Antidiagonal products of A329050.

[n le 1 select 2 else Self(n-1)^2*NthPrime(n): n in [1..10]]; // Vincenzo Librandi, Feb 06 2016

a:= proc(n) option remember;
      `if`(n=0, 1, a(n-1)^2*ithprime(n))
    end:
seq(a(n), n=0..8);  # Alois P. Heinz, Mar 05 2020

RecurrenceTable[{a[1] == 2, a[n] == a[n-1]^2 Prime[n]}, a, {n, 10}] (* Vincenzo Librandi, Feb 06 2016 *)
Table[Product[Prime[k]^2^(n-k),{k,n}],{n,0,10}] (* or *) nxt[{n_,a_}]:={n+1,a^2 Prime[n+1]}; NestList[nxt,{0,1},10][[All,2]] (* Harvey P. Dale, Jan 07 2022 *)

a(n) = prod(k=1, n, prime(k)^(2^(n-k)))

;; Two variants, both with memoization-macro definec.
(definec (A191555 n) (if (= 1 n) 2 (* (A000040 n) (A000290 (A191555 (- n 1)))))) ;; After the original recurrence.
(definec (A191555 n) (if (= 1 n) 2 (* (A000079 (A000079 (- n 1))) (A003961 (A191555 (- n 1)))))) ;; After the alternative recurrence - Antti Karttunen, Feb 06 2016

For n > 0, a(n) = a(n-1)^2 * prime(n); a(0) = 1. [edited to extend to a(0) by Peter Munn, Feb 13 2020]
a(0) = 1; for n > 0, a(n) = 2^(2^(n-1)) * A003961(a(n-1)). - Antti Karttunen, Feb 06 2016, edited Feb 13 2020 because of the new prepended starting term.
For n > 1, a(n) = A306697(a(n-1),12) = A059896(a(n-1)^2, A003961(a(n-1))). - Peter Munn, Jan 24 2020

a(0) added by Peter Munn, Feb 13 2020

A359411 a(n) is the number of divisors of n that are both infinitary and exponential.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 30 2022

First differs from A318672 and A325989 at n = 32.
If e > 0 is the exponent of the highest power of p dividing n (where p is a prime), then for each divisor d of n that is both an infinitary and an exponential divisor, the exponent of the highest power of p dividing d is a number k such that k | e and the bitwise AND of e and k is equal to k.
The least term that is higher than 2 is a(216) = 4.
The position of the first appearance of a prime p in this sequence is 2^A359081(p), if A359081(p) > -1. E.g., 2^39 = 549755813888 for p = 3, 2^175 = 4.789...*10^52 for p = 5, and 2^1275 = 6.504...*10^383 for p = 7.
This sequence is unbounded since A246600 is unbounded (see A359082).

			a(8) = 2 since 8 has 2 divisors that are both infinitary and exponential: 2 and 8.

Cf. A037445, A049419, A077609, A080948, A138302, A246600, A322791, A359081, A359082, A359412.

Cf. A318672, A325989.

s[n_] := DivisorSum[n, 1 &, BitAnd[n, #] == # &]; f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = sumdiv(n, d, bitand(d, n)==d);
a(n) = {my(f = factor(n)); prod(i = 1, #f~, s(f[i,2]));}

from math import prod
from sympy import divisors, factorint
def A359411(n): return prod(sum(1 for d in divisors(e,generator=True) if e|d == e) for e in factorint(n).values()) # Chai Wah Wu, Sep 01 2023

Multiplicative with a(p^e) = A246600(e).
a(n) = 1 if and only if n is in A138302.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} A246600(k)/p^k) = 1.135514937... .

A356194 a(n) is the smallest multiple of n whose prime factorization exponents are all powers of 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 16, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 48, 25, 26, 81, 28, 29, 30, 31, 256, 33, 34, 35, 36, 37, 38, 39, 80, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 162, 55, 112, 57, 58, 59, 60, 61, 62, 63, 256, 65, 66, 67

Offset: 1

Amiram Eldar, Jul 29 2022

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

Cf. A138302, A353897.

Similar sequences: A053143, A053149, A053167, A066638, A087320, A087321, A197863, A356191, A356192, A356193.

f[p_, e_] := p^(2^Ceiling[Log2[e]]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = {my(e=logint(n,2)); if(n == 2^e, n, 2^(e+1))};
a(n) = {my(f=factor(n)); prod(i=1, #f~, f[i,1]^s(f[i,2]))};

Multiplicative with a(p^e) = p^(2^ceiling(log_2(e))).

a(n) = n iff n is in A138302.

cp's OEIS Frontend

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

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A209061 Exponentially squarefree numbers.

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A270428 Exponentially odious numbers: 1 together with positive integers n such that all exponents in prime factorization of n are odious numbers (A000069).

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A262675 Exponentially evil numbers.

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A335275 Numbers k such that the largest square dividing k is a unitary divisor of k.

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A268375 Numbers k for which A001222(k) = A267116(k).

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A384247 The number of integers from 1 to n whose largest divisor that is an infinitary divisor of n is 1.

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