A191555 -id:A191555 - cp's OEIS frontend

A003961 Completely multiplicative with a(prime(k)) = prime(k+1).

1, 3, 5, 9, 7, 15, 11, 27, 25, 21, 13, 45, 17, 33, 35, 81, 19, 75, 23, 63, 55, 39, 29, 135, 49, 51, 125, 99, 31, 105, 37, 243, 65, 57, 77, 225, 41, 69, 85, 189, 43, 165, 47, 117, 175, 87, 53, 405, 121, 147, 95, 153, 59, 375, 91, 297, 115, 93, 61, 315, 67, 111, 275, 729, 119

Offset: 1

Marc LeBrun

Meyers (see Guy reference) conjectures that for all r >= 1, the least odd number not in the set {a(i): i < prime(r)} is prime(r+1). - N. J. A. Sloane, Jan 08 2021
Meyers' conjecture would be refuted if and only if for some r there were such a large gap between prime(r) and prime(r+1) that there existed a composite c for which prime(r) < c < a(c) < prime(r+1), in which case (by Bertrand's postulate) c would necessarily be a term of A246281. - Antti Karttunen, Mar 29 2021
a(n) is odd for all n and for each odd m there exists a k with a(k) = m (see A064216). a(n) > n for n > 1: bijection between the odd and all numbers. - Reinhard Zumkeller, Sep 26 2001
a(n) and n have the same number of distinct primes with (A001222) and without multiplicity (A001221). - Michel Marcus, Jun 13 2014
From Antti Karttunen, Nov 01 2019: (Start)
More generally, a(n) has the same prime signature as n, A046523(a(n)) = A046523(n). Also A246277(a(n)) = A246277(n) and A287170(a(n)) = A287170(n).
Many permutations and other sequences that employ prime factorization of n to encode either polynomials, partitions (via Heinz numbers) or multisets in general can be easily defined by using this sequence as one of their constituent functions. See the last line in the Crossrefs section for examples.
(End)

			a(12) = a(2^2 * 3) = a(prime(1)^2 * prime(2)) = prime(2)^2 * prime(3) = 3^2 * 5 = 45.
a(A002110(n)) = A002110(n + 1) / 2.

Richard K. Guy, editor, Problems From Western Number Theory Conferences, Labor Day, 1983, Problem 367 (Proposed by Leroy F. Meyers, The Ohio State U.).

Indranil Ghosh, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Index entries for sequences computed from indices in prime factorization.
Index entries for sequences related to Heinz numbers.

See A045965 for another version.
Row 1 of table A242378 (which gives the "k-th powers" of this sequence), row 3 of A297845 and of A306697. See also arrays A066117, A246278, A255483, A308503, A329050.
Cf. A064989 (a left inverse), A064216, A000040, A002110, A000265, A027746, A046523, A048673 (= (a(n)+1)/2), A108228 (= (a(n)-1)/2), A191002 (= a(n)*n), A252748 (= a(n)-2n), A286385 (= a(n)-sigma(n)), A283980 (= a(n)*A006519(n)), A341529 (= a(n)*sigma(n)), A326042, A049084, A001221, A001222, A122111, A225546, A260443, A245606, A244319, A246269 (= A065338(a(n))), A322361 (= gcd(n, a(n))), A305293.
Cf. A191555, A252738.
Cf. A249734, A249735 (bisections).
Cf. A246261 (a(n) is of the form 4k+1), A246263 (of the form 4k+3), A246271, A246272, A246259, A246281 (n such that a(n) < 2n), A246282 (n such that a(n) > 2n), A252742.
Cf. A275717 (a(n) > a(n-1)), A275718 (a(n) < a(n-1)).
Cf. A003972 (Möbius transform), A003973 (Inverse Möbius transform), A318321.
Cf. A300841, A305421, A322991, A250469, A269379 for analogous shift-operators in other factorization and quasi-factorization systems.
Cf. also following permutations and other sequences that can be defined with the help of this sequence: A005940, A163511, A122111, A260443, A206296, A265408, A265750, A275733, A275735, A297845, A091202 & A091203, A250245 & A250246, A302023 & A302024, A302025 & A302026.
A version for partition numbers is A003964, strict A357853.
A permutation of A005408.
Applying the same transformation again gives A357852.
Other multiplicative sequences: A064988, A357977, A357978, A357980, A357983.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000720, A076610, A296150.

a003961 1 = 1
a003961 n = product $ map (a000040 . (+ 1) . a049084) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012, Oct 09 2011
(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
(require 'factor)
(define (A003961 n) (apply * (map A000040 (map 1+ (map A049084 (factor n))))))
;; Antti Karttunen, May 20 2014

a:= n-> mul(nextprime(i[1])^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 13 2017

a[p_?PrimeQ] := a[p] = Prime[ PrimePi[p] + 1]; a[1] = 1; a[n_] := a[n] = Times @@ (a[#1]^#2& @@@ FactorInteger[n]); Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Dec 01 2011, updated Sep 20 2019 *)
Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[n == 1], {n, 65}] (* Michael De Vlieger, Mar 24 2017 *)

a(n)=local(f); if(n<1,0,f=factor(n); prod(k=1,matsize(f)[1],nextprime(1+f[k,1])^f[k,2]))

a(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Michel Marcus, May 17 2014

use ntheory ":all";  sub a003961 { vecprod(map { next_prime($) } factor(shift)); }  # _Dana Jacobsen, Mar 06 2016

from sympy import factorint, prime, primepi, prod
def a(n):
    f=factorint(n)
    return 1 if n==1 else prod(prime(primepi(i) + 1)**f[i] for i in f)
[a(n) for n in range(1, 11)] # Indranil Ghosh, May 13 2017

If n = Product p(k)^e(k) then a(n) = Product p(k+1)^e(k).
Multiplicative with a(p^e) = A000040(A000720(p)+1)^e. - David W. Wilson, Aug 01 2001
a(n) = Product_{k=1..A001221(n)} A000040(A049084(A027748(n,k))+1)^A124010(n,k). - Reinhard Zumkeller, Oct 09 2011 [Corrected by Peter Munn, Nov 11 2019]
A064989(a(n)) = n and a(A064989(n)) = A000265(n). - Antti Karttunen, May 20 2014 & Nov 01 2019
A001221(a(n)) = A001221(n) and A001222(a(n)) = A001222(n). - Michel Marcus, Jun 13 2014
From Peter Munn, Oct 31 2019: (Start)
a(n) = A225546((A225546(n))^2).
a(A225546(n)) = A225546(n^2).
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-nextprime(p))) = 2.06399637... . - Amiram Eldar, Nov 18 2022

A329050 Square array A(n,k) = prime(n+1)^(2^k), read by descending antidiagonals (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ...; Fermi-Dirac primes (A050376) in matrix form, sorted into rows by their prime divisor.

Original entry on oeis.org

2, 4, 3, 16, 9, 5, 256, 81, 25, 7, 65536, 6561, 625, 49, 11, 4294967296, 43046721, 390625, 2401, 121, 13, 18446744073709551616, 1853020188851841, 152587890625, 5764801, 14641, 169, 17, 340282366920938463463374607431768211456, 3433683820292512484657849089281, 23283064365386962890625, 33232930569601, 214358881, 28561, 289, 19

Offset: 0

Antti Karttunen and Peter Munn, Nov 02 2019

This sequence is a permutation of A050376, so every positive integer is the product of a unique subset, S_factors, of its terms. If we restrict S_factors to be chosen from a subset, S_0, consisting of numbers from specified rows and/or columns of this array, there are notable sequences among those that may be generated. See the examples. Other notable sequences can be generated if we restrict the intersection of S_factors with specific rows/columns to have even cardinality. In any of the foregoing cases, the numbers in the resulting sequence form a group under the binary operation A059897(.,.).
Shares with array A246278 the property that columns grow downward by iterating A003961, and indeed, this array can be obtained from A246278 by selecting its columns 1, 2, 8, 128, ..., 2^((2^k)-1), for k >= 0.
A(n,k) is the image of the lattice point with coordinates X=n and Y=k under the inverse of the bijection f defined in the first comment of A306697. This geometric relationship can be used to construct an isomorphism from the polynomial ring GF(2)[x,y] to a ring over the positive integers, using methods similar to those for constructing A297845 and A306697. See A329329, the ring's multiplicative operator, for details.

			The top left 5 X 5 corner of the array:
  n\k |   0     1       2           3                   4
  ----+-------------------------------------------------------
   0  |   2,    4,     16,        256,              65536, ...
   1  |   3,    9,     81,       6561,           43046721, ...
   2  |   5,   25,    625,     390625,       152587890625, ...
   3  |   7,   49,   2401,    5764801,     33232930569601, ...
   4  |  11,  121,  14641,  214358881,  45949729863572161, ...
Column 0 continues as a list of primes, column 1 as a list of their squares, column 2 as a list of their 4th powers, and so on.
Every nonnegative power of 2 (A000079) is a product of a unique subset of numbers from row 0; every squarefree number (A005117) is a product of a unique subset of numbers from column 0. Likewise other rows and columns generate the sets of numbers from sequences:
Row 1:                 A000244 Powers of 3.
Column 1:              A062503 Squares of squarefree numbers.
Row 2:                 A000351 Powers of 5.
Column 2:              A113849 4th powers of squarefree numbers.
Union of rows 0 and 1:     A003586 3-smooth numbers.
Union of columns 0 and 1:  A046100 Biquadratefree numbers.
Union of row 0 / column 0: A122132 Oddly squarefree numbers.
Row 0 excluding column 0:  A000302 Powers of 4.
Column 0 excluding row 0:  A056911 Squarefree odd numbers.
All rows except 0:         A005408 Odd numbers.
All columns except 0:      A000290\{0} Positive squares.
All rows except 1:         A001651 Numbers not divisible by 3.
All columns except 1:      A252895 (have odd number of square divisors).
If, instead of restrictions on choosing individual factors of the product, we restrict the product to be of an even number of terms from each row of the array, we get A262675. The equivalent restriction applied to columns gives us A268390; applied only to column 0, we get A028260 (product of an even number of primes).

Antti Karttunen, Table of n, a(n) for n = 0..77; the first 12 antidiagonals of array
Wikipedia, Polynomial ring

Transpose: A329049.
Permutation of A050376.
Rows 1-4: A001146, A011764, A176594, A165425 (after the two initial terms).
Columns 1-5: A000040, A001248, A030514, A179645, A030635.
Antidiagonal products: A191555.
Subtable of A182944, A242378, A246278, A329332.
A000290, A003961, A225546 are used to express relationship between terms of this sequence.
Related binary operations: A059897, A306697, A329329.
See also the table in the example section.

Table[Prime[#]^(2^k) &[m - k + 1], {m, 0, 7}, {k, m, 0, -1}] // Flatten (* Michael De Vlieger, Dec 28 2019 *)

up_to = 105;
A329050sq(n,k) = (prime(1+n)^(2^k));
A329050list(up_to) = { my(v = vector(up_to), i=0); for(a=0, oo, for(col=0, a, i++; if(i > up_to, return(v)); v[i] = A329050sq(col, a-col))); (v); };
v329050 = A329050list(up_to);
A329050(n) = v329050[1+n];
for(n=0,up_to-1,print1(A329050(n),", ")); \\ Antti Karttunen, Nov 06 2019

A(0,k) = 2^(2^k), and for n > 0, A(n,k) = A003961(A(n-1,k)).
A(n,k) = A182944(n+1,2^k).
A(n,k) = A329332(2^n,2^k).
A(k,n) = A225546(A(n,k)).
A(n,k+1) = A000290(A(n,k)) = A(n,k)^2.

Example annotated for clarity by Peter Munn, Feb 12 2020

A252738 Row products of irregular table A005940: a(0) = 1; a(1) = 2; for n > 1: 2^(2^(n-2)) * a(n-1) * A003961(a(n-1)); also row products of A163511, A253563, A253565, and A332977.

Original entry on oeis.org

1, 2, 12, 2160, 2449440000, 8488905214204800000000000, 3025568387202006082882734693673523654400000000000000000000000000

Offset: 0

Antti Karttunen, Dec 21 2014

nonn

			From _Michael De Vlieger_, Jul 21 2023: (Start)
a(0) = 1 = product of {1},
a(1) = 2^1 = product of {2},
a(2) = 2^2 * 3^1 = product of {3, 2^2},
a(3) = 2^4 * 3^3 * 5^1 = product of {5, 2^1*3^1, 3^2, 2^3},
a(4) = 2^8 * 3^7 * 5^4 * 7^1 = product of
  {7, 2^1*5^1, 3^1*5^1, 2^2*3^1, 5^2, 2^1*3^2, 3^3, 2^4},
...
Table of e(n,k) where a(n) = Product_{k=1..n+1} prime(k)^e(n,k):
prime(k)|    2    3    5   7  11  13  17  19 23 29 31 ...
   n\k  |    1    2    3   4   5   6   7   8  9 10 11 ...
   ----------------------------------------------------
    0   |    1
    1   |    2    1
    2   |    4    3    1
    3   |    8    7    4   1
    4   |   16   15   11   5   1
    5   |   32   31   26  16   6   1
    6   |   64   63   57  42  22   7   1
    7   |  128  127  120  99  64  29   8   1
    8   |  256  255  247 219 163  93  37   9  1
    9   |  512  511  502 466 382 256 130  46 10  1
   10   | 1024 1023 1013 968 848 638 386 176 56 11  1
  ... (End)

Alois P. Heinz, Table of n, a(n) for n = 0..9

These are row products of irregular tables A005940, A163511, A253563 and A253565, which all are shaped like a binary tree.
Partial products of A252740.
Cf. A252737 (row sums), A252739 (divided by n), A252741 (divided by n!).
Cf. A000079, A000225, A000290, A003961, A163511, A191555, A266639, A267096, A332977.

Table[Times @@ Array[Prime[# + 1]^Sum[Binomial[n, # + j], {j, 0, n}] &, n + 1, 0], {n, 0, 5}] (* Michael De Vlieger, Jul 21 2023 *)

allocatemem(234567890);
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A252738print(up_to_n) = { my(s, i=0, n=0); for(n=0, up_to_n, if(0 == n, s = 1, if(1 == n, s = 2; lev = vector(1); lev[1] = 2, oldlev = lev; lev = vector(2*length(oldlev)); s = 1; for(i = 0, (2^(n-1))-1, lev[i+1] = if((i%2),A003961(oldlev[(i\2)+1]),2*oldlev[(i\2)+1]); s *= lev[i+1]))); write("b252738.txt", n, " ", s)); }; \\ Counts them empirically.
A252738print(7);

(definec (A252738rec n) (if (<= n 1) (+ 1 n) (* (A000079 (A000079 (- n 2))) (A252738rec (- n 1)) (A003961 (A252738rec (- n 1)))))) ;; Implements the given recurrence; uses the memoizing definec-macro.
(define (A252738 n) (if (zero? n) 1 (mul A163511 (A000079 (- n 1)) (A000225 n))))
(define (mul intfun lowlim uplim) (let multloop ((i lowlim) (res 1)) (cond ((> i uplim) res) (else (multloop (+ 1 i) (* res (intfun i)))))))
;; Another alternative, implementing the new recurrence:
(definec (A252738 n) (if (<= n 1) (+ 1 n) (* (A267096 (- n 2)) (A000290 (A252738 (- n 1)))))) ;; Antti Karttunen, Feb 06 2016

a(0) = 1; a(1) = 2; for n > 1: a(n) = 2^(2^(n-2)) * a(n-1) * A003961(a(n-1)).
a(0) = 1; for n>=1: a(n) = Product_{k=A000079(n-1) .. A000225(n)} A163511(k) = Product_{k=2^(n-1) .. (2^n)-1} A163511(k).
a(0) = 1; a(1) = 2; for n > 1: a(n) = A267096(n-2) * a(n-1)^2. [Compare to the formulas of A191555] - Antti Karttunen, Feb 06 2016
From Michael De Vlieger, Jul 21 2023: (Start)
a(n) = Product_{k=1..n+1} prime(k)^e(n,k), where e(n,k) = k-th term in row n of A055248.
A067255(a(n)) = row n of A055248. (End)

Typos in the second formula corrected by Antti Karttunen, Feb 06 2016

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 *)

A225547 Fixed points of A225546.

Original entry on oeis.org

1, 2, 9, 12, 18, 24, 80, 108, 160, 216, 625, 720, 960, 1250, 1440, 1792, 1920, 2025, 3584, 4050, 5625, 7500, 8640, 11250, 15000, 16128, 17280, 18225, 21504, 24300, 32256, 36450, 43008, 48600, 50000, 67500, 100000, 135000, 143360, 162000, 193536, 218700, 286720, 321489, 324000, 387072, 437400, 450000, 600000

Offset: 1

Paul Tek, May 10 2013

nonn

Every number in this sequence is the product of a unique subset of A225548.
From Peter Munn, Feb 11 2020: (Start)
The terms are the numbers whose Fermi-Dirac factors (see A050376) occur symmetrically about the main diagonal of A329050.
Closed under the commutative binary operation A059897(.,.). As numbers are self-inverse under A059897, the sequence thereby forms a subgroup of the positive integers under A059897.
(End)

			The Fermi-Dirac factorization of 160 is 2 * 5 * 16. The factors 2, 5 and 16 are A329050(0,0), A329050(2,0) and A329050(0,2), having symmetry about the main diagonal of A329050. So 160 is in the sequence.

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A050376, A225546, A329050.
Closed under A059895, A059896, A059897, A306697, A329329.
Subsequences: A191554, A191555, A225548.
Cf. fixed points of the comparable A122111 involution: A088902.

A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
ff(fa) = {for (i=1, #fa~, my(p=fa[i, 1]); fa[i, 1] = A019565(fa[i, 2]); fa[i, 2] = 2^(primepi(p)-1); ); fa; } \\ A225546
pos(k, fs) = for (i=1, #fs, if (fs[i] == k, return(i)););
normalize(f) = {my(list = List()); for (k=1, #f~, my(fk = factor(f[k,1])); for (j=1, #fk~, listput(list, fk[j,1]));); my(fs = Set(list)); my(m = matrix(#fs, 2)); for (i=1, #m~, m[i,1] = fs[i]; for (k=1, #f~, m[i,2] += valuation(f[k,1], fs[i])*f[k,2];);); m;}
isok(n) = my(fa=factor(n), fb=ff(fa)); normalize(fb) == fa; \\ Michel Marcus, Aug 05 2022

A331592 a(n) is the smaller of the number of terms in the factorizations of n into (1) powers of distinct primes and (2) powers of squarefree numbers with distinct exponents that are powers of 2.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen and Peter Munn, Jan 21 2020

nonn

See A329332 for a description of the relationship between the two factorizations. From this relationship we get the formula a(n) = min(A001221(n), A001221(A225546(n))).
The result depends only on the prime signature of n.
k first appears at A191555(k).

			The factorization of 6 into powers of distinct primes is 6 = 2^1 * 3^1 = 2 * 3, which has 2 terms. Its factorization into powers of squarefree numbers with distinct exponents that are powers of 2 is 6 = 6^(2^0) = 6^1, which has 1 term. So a(6) is min(2,1) = 1.
The factorization of 40 into powers of distinct primes is 40 = 2^3 * 5^1 = 8 * 5, which has 2 terms. Its factorization into powers of squarefree numbers with distinct exponents that are powers of 2 is 40 = 10^(2^0) * 2^(2^1) = 10^1 * 2^2 = 10 * 4, which has 2 terms. So a(40) is min(2,2) = 2.

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

Cf. A000961, A001221, A191555, A293442, A329332.
Sequences with related definitions: A331308, A331591, A331593.
A003961, A225546 are used to express relationship between terms of this sequence.
Differs from = A071625 for the first time at n=216, where a(216) = 2, while A071625(216) = 1.

A331592(n) = min(omega(n), A331591(n)); \\ Uses also code from A331591.

a(n) = min(A001221(n), A331591(n)) = min(A001221(n), A001221(A293442(n))).
a(A225546(n)) = a(n).
a(A003961(n)) = a(n).
a(n^2) = a(n).

A337533 1 together with nonsquares whose square part's square root is in the sequence.

Original entry on oeis.org

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

Offset: 1

Peter Munn, Aug 31 2020

The appearance of a number is determined by its prime signature.
Every squarefree number is present, as the square root of the square part of a squarefree number is 1. Other 4th-power-free numbers are present if and only if they are nonsquare.
If the square part of nonsquarefree k is a 4th power, k does not appear.
Every positive integer k is the product of a unique subset S_k of the terms of A050376, which are arranged in array form in A329050 (primes in column 0, squares of primes in column 1, 4th powers of primes in column 2 and so on). k > 1 is in this sequence if and only if the members of S_k occur in consecutive columns of A329050, starting with column 0.
If the qualifying condition in the previous paragraph was based on the rows instead of the columns of A329050, we would get A055932. The self-inverse function defined by A225546 transposes A329050. A225546 also has multiplicative properties such that if we consider A055932 and this sequence as sets, A225546(.) maps the members of either set 1:1 onto the other set.

			4 is square and not 1, so 4 is not in the sequence.
12 = 3 * 2^2 is nonsquare, and has square part 4, whose square root (2) is in the sequence. So 12 is in the sequence.
32 = 2 * 4^2 is nonsquare, but has square part 16, whose square root (4) is not in the sequence. So 32 is not in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square part
Index to sequences related to prime signature

Complement of A337534.
Closed under A000188(.).
A209229, A267116 are used in a formula defining this sequence.
Subsequences: A005117, A036537, A179646, A179666, A179669, A189987, A191554, A191555, A298207, A317090.
Subsequence of A164514.
A007913, A008833, A008835, A335324 give the squarefree, square and comparably related parts of a number.
Cf. A050376, A329050.
Related to A055932 via A225546.

S:= {1}:
for n from 2 to 100 do
  if not issqr(n) then
    F:= ifactors(n)[2];
    s:= mul(t[1]^floor(t[2]/2),t=F);
    if member(s,S) then S:= S union {n} fi
  fi
od:
sort(convert(S,list)); # Robert Israel, Jan 07 2025

pow2Q[n_] := n == 2^IntegerExponent[n, 2]; Select[Range[100], # == 1 || pow2Q[1 + BitOr @@ (FactorInteger[#][[;; , 2]])] &] (* Amiram Eldar, Sep 18 2020 *)

Numbers m such that A209229(A267116(m) + 1) = 1.
If A008835(a(n)) > 1 then A335324(a(n)) > 1.
If A008833(a(n)) > 1 then A007913(a(n)) > 1.

A267117 Numbers m such that in their prime factorization m = p_1^e_1 * ... * p_k^e_k, there is no digit-position in the base-2 representation of the exponents e_1 .. e_k such that in that position all those exponents would have 1-bit.

Original entry on oeis.org

1, 12, 18, 20, 28, 44, 45, 48, 50, 52, 60, 63, 68, 75, 76, 80, 84, 90, 92, 98, 99, 112, 116, 117, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 162, 164, 171, 172, 175, 176, 180, 188, 192, 198, 204, 207, 208, 212, 220, 228, 234, 236, 240, 242, 244, 245, 252, 260, 261, 268, 272, 275, 276, 279, 284, 288, 292, 294, 300

Offset: 1

Antti Karttunen, Feb 03 2016

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 1, 21, 261, 2824, 29144, 294233, 2951313, 29542282, 295514868, 2955441810, 29555347819, ... . Apparently, the asymptotic density of this sequence exists and equals 0.2955... . - Amiram Eldar, Sep 09 2022

			60 = 2^2 * 3^1 * 5^1 is included, as bitwise-anding together the binary representations of the exponents, "10", "01" and "01" results "00", zero.

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

Indices of zeros in A267115.
Cf. A054753 (subsequence), A191555 (subsequence after the initial 2).
Cf. also A267114, A268376.

{1}~Join~Select[Range@ 300, BitAnd @@ Map[Last, FactorInteger@ #] == 0 &] (* Michael De Vlieger, Feb 07 2016 *)

Erroneous claim corrected by Antti Karttunen, Feb 07 2016

cp's OEIS Frontend

A266639 a(n) = A252738(n) / A191555(n).

Views

Author

Keywords

Crossrefs

Programs

Scheme

Formula

A003961 Completely multiplicative with a(prime(k)) = prime(k+1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Perl

Python

Formula

A329050 Square array A(n,k) = prime(n+1)^(2^k), read by descending antidiagonals (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ...; Fermi-Dirac primes (A050376) in matrix form, sorted into rows by their prime divisor.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A252738 Row products of irregular table A005940: a(0) = 1; a(1) = 2; for n > 1: 2^(2^(n-2)) * a(n-1) * A003961(a(n-1)); also row products of A163511, A253563, A253565, and A332977.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A225547 Fixed points of A225546.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A331592 a(n) is the smaller of the number of terms in the factorizations of n into (1) powers of distinct primes and (2) powers of squarefree numbers with distinct exponents that are powers of 2.

Views

Author

Keywords

Comments

Examples

Links