A225546 -id:A225546 - cp's OEIS frontend

A331590 Square array A(n,k) = A225546(A225546(n) * A225546(k)), n >= 1, k >= 1, read by descending antidiagonals.

1, 2, 2, 3, 3, 3, 4, 6, 6, 4, 5, 8, 5, 8, 5, 6, 10, 12, 12, 10, 6, 7, 5, 15, 9, 15, 5, 7, 8, 14, 10, 20, 20, 10, 14, 8, 9, 12, 21, 24, 7, 24, 21, 12, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 15, 27, 18, 35, 15, 35, 18, 27, 15, 11, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12, 13, 24, 33, 40, 45, 20, 11, 20, 45, 40, 33, 24, 13

Offset: 1

Peter Munn, Jan 21 2020

As a binary operation, this sequence defines a commutative monoid over the positive integers that is isomorphic to multiplication. The self-inverse permutation A225546(.) provides an isomorphism. This monoid therefore has unique factorization. Its primes are the even terms of A050376: 2, 4, 16, 256, ..., which in standard integer multiplication are the powers of 2 with powers of 2 as exponents.
In this monoid, in contrast, the powers of 2 run through the squarefree numbers, the k-th power of 2 being A019565(k). 4 is irreducible and its powers are the squares of the squarefree numbers, the k-th power of 4 being A019565(k)^2 (where "^2" denotes standard integer squaring); and so on with powers of 16, 256, ...
In many cases the product of two numbers is the same here as in standard integer multiplication. See the formula section for details.

			From _Antti Karttunen_, Feb 02 2020: (Start)
The top left 16 X 16 corner of the array:
   1,  2,  3,  4,  5,  6,   7,   8,   9,  10,  11,  12,  13,  14,  15,  16, ...
   2,  3,  6,  8, 10,  5,  14,  12,  18,  15,  22,  24,  26,  21,  30,  32, ...
   3,  6,  5, 12, 15, 10,  21,  24,  27,  30,  33,  20,  39,  42,   7,  48, ...
   4,  8, 12,  9, 20, 24,  28,  18,  36,  40,  44,  27,  52,  56,  60,  64, ...
   5, 10, 15, 20,  7, 30,  35,  40,  45,  14,  55,  60,  65,  70,  21,  80, ...
   6,  5, 10, 24, 30, 15,  42,  20,  54,   7,  66,  40,  78,  35,  14,  96, ...
   7, 14, 21, 28, 35, 42,  11,  56,  63,  70,  77,  84,  91,  22, 105, 112, ...
   8, 12, 24, 18, 40, 20,  56,  27,  72,  60,  88,  54, 104,  84, 120, 128, ...
   9, 18, 27, 36, 45, 54,  63,  72,  25,  90,  99, 108, 117, 126, 135, 144, ...
  10, 15, 30, 40, 14,  7,  70,  60,  90,  21, 110, 120, 130, 105,  42, 160, ...
  11, 22, 33, 44, 55, 66,  77,  88,  99, 110,  13, 132, 143, 154, 165, 176, ...
  12, 24, 20, 27, 60, 40,  84,  54, 108, 120, 132,  45, 156, 168,  28, 192, ...
  13, 26, 39, 52, 65, 78,  91, 104, 117, 130, 143, 156,  17, 182, 195, 208, ...
  14, 21, 42, 56, 70, 35,  22,  84, 126, 105, 154, 168, 182,  33, 210, 224, ...
  15, 30,  7, 60, 21, 14, 105, 120, 135,  42, 165,  28, 195, 210,  35, 240, ...
  16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240,  81, ...
(End)

Antti Karttunen, Antidiagonals n = 1..144, flattened
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..80200; (antidiagonals n = 1..400)
Eric Weisstein's World of Mathematics, Monoid

Isomorphic to A003991 with A225546 as isomorphism.
Cf. A003961(main diagonal), A048675, A059895, A059896, A059897.
Rows/columns, sorted in ascending order: 2: A000037, 3: A028983, 4: A252849.
A019565 lists powers of 2 in order of increasing exponent.
Powers of k, sorted in ascending order: k=2: A005117, k=3: A056911, k=4: A062503, k=5: A276378, k=6: intersection of A325698 and A005117, k=7: intersection of A007775 and A005117, k=8: A062838.
Irreducibles are A001146 (even terms of A050376).

up_to = 1275;
A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331590sq(x,y) = if(1==x,y,if(1==y,x, my(fx=factor(x),fy=factor(y),u=max(#binary(vecmax(fx[, 2])),#binary(vecmax(fy[, 2]))),prodsx=vector(u,x,1),m=1); for(i=1,u,for(k=1,#fx~, if(bitand(fx[k,2],m),prodsx[i] *= fx[k,1])); for(k=1,#fy~, if(bitand(fy[k,2],m),prodsx[i] *= fy[k,1])); m<<=1); prod(i=1,u,A019565(A048675(prodsx[i]))^(1<<(i-1)))));
A331590list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A331590sq(col,(a-(col-1))))); (v); };
v331590 = A331590list(up_to);
A331590(n) = v331590[n]; \\ Antti Karttunen, Feb 02 2020

Alternative definition: A(n,1) = n; A(n,k) = A(A059897(n,k), A003961(A059895(n,k))).
Main derived identities: (Start)
A(n,k) = A(k,n).
A(1,n) = n.
A(n, A(m,k)) = A(A(n,m), k).
A(m,m) = A003961(m).
A(n^2, k^2) = A(n,k)^2.
A(A003961(n), A003961(k)) = A003961(A(n,k)).
A(A019565(n), A019565(k)) = A019565(n+k).
(End)
Characterization of conditions for A(n,k) = n * k: (Start)
The following 4 conditions are equivalent:
(1) A(n,k) = n * k;
(2) A(n,k) = A059897(n,k);
(3) A(n,k) = A059896(n,k);
(4) A059895(n,k) = 1.
If gcd(n,k) = 1, A(n,k) = n * k.
If gcd(n,k) = 1, A(A225546(n), A225546(k)) = A225546(n) * A225546(k).
The previous formula implies A(n,k) = n * k in the following cases:
(1) for n = A005117(m), k = j^2;
(2) more generally for n = A005117(m_1)^(2^i_1), k = A005117(m_2)^(2^i_2), with A004198(i_1, i_2) = 0.
(End)

A331591 a(n) is the number of distinct prime factors of A225546(n), or equally, number of distinct prime factors of A293442(n).

Original entry on oeis.org

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

a(n) is the number of terms in the unique factorization of n into powers of squarefree numbers with distinct exponents that are powers of 2. See A329332 for a description of the relationship between this factorization, canonical (prime power) factorization and A225546.
The result depends only on the prime signature of n.
a(n) is the number of distinct bit-positions where there is a 1-bit in the binary representation of an exponent in the prime factorization of n. - Antti Karttunen, Feb 05 2020
The first 3 is a(128) = a(2^1 * 2^2 * 2^4) = 3 and in general each m occurs first at position 2^(2^m-1) = A058891(m+1). - Peter Munn, Mar 07 2022

			From _Peter Munn_, Jan 28 2020: (Start)
The factorization of 6 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) = 1.
Similarly, 40 = 10^(2^0) * 2^(2^1) = 10^1 * 2^2 = 10 * 4, which has 2 terms. So a(40) = 2.
Similarly, 320 = 5^(2^0) * 2^(2^1) * 2^(2^2) = 5^1 * 2^2 * 2^4 = 5 * 4 * 16, which has 3 terms. So a(320) = 3.
10^100 (a googol) factorizes in this way as 10^4 * 10^32 * 10^64. So a(10^100) = 3.
(End)

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

Cf. A000120, A005117, A048675, A225546, A267116, A293442, A293447, A299090, A329332, A337533, A337534.
Sequences with related definitions: A001221, A331309, A331592, A331593, A331740.
Positions of records: A058891.
Positions of 1's: A340682.
Sequences used to express relationships between the terms: A007913, A008833, A059796, A331590.

Array[PrimeNu@ If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 105] (* Michael De Vlieger, Jan 24 2020 *)
f[e_] := Position[Reverse[IntegerDigits[e, 2]], 1] // Flatten; a[n_] := CountDistinct[Flatten[f /@ FactorInteger[n][[;; , 2]]]]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Dec 23 2023 *)

A331591(n) = if(1==n,0,my(f=factor(n),u=#binary(vecmax(f[, 2])),xs=vector(u),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),xs[i]++)); m<<=1); #select(x -> (x>0),xs));

A331591(n) = if(1==n, 0, hammingweight(fold(bitor, factor(n)[, 2]))); \\ Antti Karttunen, Feb 05 2020

A331591(n) = if(n==1, 0, (core(n)>1) + A331591(core(n,1)[2])) \\ Peter Munn, Mar 08 2022

a(n) = A001221(A293442(n)) = A001221(A225546(n)).
From Peter Munn, Jan 28 2020: (Start)
a(n) = A000120(A267116(n)).
a(n) = a(A007913(n)) + a(A008833(n)).
For m >= 2, a(A005117(m)) = 1.
a(n^2) = a(n).
(End)
a(n) <= A331740(n) <= A048675(n) <= A293447(n). - Antti Karttunen, Feb 05 2020
From Peter Munn, Mar 07 2022: (Start)
a(n) <= A299090(n).
a(A337533(n)) = A299090(A337533(n)).
a(A337534(n)) < A299090(A337534(n)).
max(a(n), a(k)) <= a(A059796(n, k)) = a(A331590(n, k)) <= a(n) + a(k).
(End)

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

A331733 a(n) = sigma(A225546(n)), where sigma is the sum of divisors.

Original entry on oeis.org

1, 3, 7, 4, 31, 15, 511, 12, 13, 63, 131071, 28, 8589934591, 1023, 127, 6, 36893488147419103231, 39, 680564733841876926926749214863536422911, 124, 2047, 262143, 231584178474632390847141970017375815706539969331281128078915168015826259279871, 60, 121, 17179869183, 91, 2044

Offset: 1

Antti Karttunen, Feb 02 2020

nonn

Cf. A000203, A048675, A225546, A331309, A331734, A331735, A331741.

Cf. A323243, A323173, A324054, A324184, A324545 for other permutations of sigma, and also A324573, A324653.

Array[If[# == 1, 1, DivisorSigma[1, #] &@ Apply[Times, Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]]] &, 28] (* Michael De Vlieger, Feb 08 2020 *)

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331733(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)^(1+A048675(prods[i]))-1)/(prime(i)-1)));

a(n) = A000203(A225546(n)).

For all n >= 1, A000035(a(A016754(n))) = 1. [Result is odd for all odd squares]

A331287 a(n) = gcd(n, A225546(n)).

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 2, 9, 2, 1, 12, 1, 2, 1, 1, 1, 18, 1, 4, 1, 2, 1, 24, 1, 2, 9, 4, 1, 2, 1, 2, 1, 2, 1, 9, 1, 2, 1, 8, 1, 2, 1, 4, 9, 2, 1, 4, 1, 2, 1, 4, 1, 18, 1, 8, 1, 2, 1, 12, 1, 2, 9, 1, 1, 2, 1, 4, 1, 2, 1, 18, 1, 2, 3, 4, 1, 2, 1, 80, 1, 2, 1, 12, 1, 2, 1, 8, 1, 18, 1, 4, 1, 2, 1, 8, 1, 2, 9, 1, 1, 2, 1, 8, 1

Offset: 1

Antti Karttunen, Jan 20 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..11250
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A225546, A225547 (fixed points), A297845, A331288, A331310, A331311.

PARI
```
A331287(n) = gcd(n, A225546(n));
```

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331287(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,if(!(n%prime(i)),for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1]))); m<<=1); prod(i=1,u,prime(i)^min(valuation(n,prime(i)),A048675(prods[i]))));

a(n) = gcd(n, A225546(n)).
a(n) = A331310(n) * A331311(n).
a(A297845(n,9)) = A297845(a(n),9). - Peter Munn, Jan 24 2020

A331309 a(n) = A000005(A225546(n)), where A000005 gives the number of divisors of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 4, 9, 4, 3, 6, 17, 6, 33, 10, 7, 2, 65, 6, 129, 10, 11, 18, 257, 8, 5, 34, 9, 18, 513, 8, 1025, 4, 19, 66, 13, 4, 2049, 130, 35, 12, 4097, 12, 8193, 34, 15, 258, 16385, 6, 9, 10, 67, 66, 32769, 12, 21, 20, 131, 514, 65537, 14, 131073, 1026, 27, 4, 37, 20, 262145, 130, 259, 14, 524289, 8, 1048577, 2050, 15

Offset: 1

Antti Karttunen, Jan 21 2020

nonn

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

Cf. A000005, A048675, A225546, A331287, A331308, A331591.

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331309(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,1+A048675(prods[i])));

a(n) = A000005(A225546(n)).

A340675 Exponential of Mangoldt function conjugated by Tek's flip: a(n) = A225546(A014963(A225546(n))).

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 2, 1, 4, 2, 2, 1, 2, 2, 2, 16, 2, 1, 2, 1, 2, 2, 2, 1, 4, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 4, 2, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 16, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 4, 2, 2, 2, 1, 2

Offset: 1

Antti Karttunen and Peter Munn, Feb 01 2021

nonn

Nonunit squarefree numbers take the value 2, other nonsquares take the value 1, and squares take the square of the value taken by their square root.

Sequences used in a definition of this sequence: A014963, A048298, A225546, A267116, A297108, A340676.
Cf. A051903, A051144, A331591, A340673.
Positions of 1's: {1} U A340681, 2's: A005117 \ {1}, of 4's: A062503 \ {1}, of 16's: A113849.
Positions of terms > 1: A340682, of terms > 2: A340674.
Sequences used to express relationship between terms of this sequence: A003961, A331590.

A340675(n) = if(1==n,n,if(issquarefree(n), 2, if(!issquare(n), 1, A340675(sqrtint(n))^2)));

a(n) = A225546(A340673(n)) = A225546(A014963(A225546(n))).
a(n) = 2^A048298(A267116(n)).
If A340673(n) = 1, then a(n) = 1, otherwise a(n) = 2^A297108(A340673(n)).
If A340676(n) = 0, then a(n) = 1, otherwise a(n) = 2^(2^(A340676(n)-1)).
If n = s^(2^k), s squarefree >= 2, k >= 0, then a(n) = 2^(2^k), otherwise a(n) = 1.
For n, k > 1, if a(n) = a(k) then a(A331590(n, k)) = a(n), otherwise a(A331590(n, k)) = 1.
a(n^2) = a(n)^2.
a(A003961(n)) = a(n).
a(A051144(n)) = 1.
a(n) = 1 if and only if A331591(n) <> 1, otherwise a(n) = 2^A051903(n).

A334109 a(n) = A329697(A225546(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 4, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 8, 4, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 4, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 8, 2, 5, 0, 0, 0, 1, 0

Offset: 1

Antti Karttunen, Apr 29 2020

nonn

Conjecture: Each k >= 0 occurs for the first time at A334110(k) = A019565(k)^2. Note that each k must occur first time on square n, because of the identity a(n) = a(A008833(n)). However, is there any reason to exclude squares with prime exponents > 2 from the candidates? See also comments in A334204.

Antti Karttunen, Table of n, a(n) for n = 1..10200
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001248, A005117 (positions of zeros), A334110.
Cf. A003961, A008833, A019565, A059895, A225546, A329697, A331590, A331736, A334204, A334860.
Cf. also A334107, A334108.

Map[-1 + Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] &, Array[If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 105] ] (* Michael De Vlieger, May 26 2020 *)

A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
A334109(n) = { my(f=factor(n),pis=apply(primepi,f[,1]),es=f[,2]); sum(k=1,#f~,(2^(pis[k]-1))*A329697(A019565(es[k]))); };

Additive with a(prime(i)^j) = A000079(i-1) * A329697(A019565(j)), a(m*n) = a(m)+a(n) if gcd(m,n) = 1.
Alternatively, additive with a(prime(i)^(2^k)) = 2^(i-1) * A329697(prime(k+1)), a(m*n) = a(m)+a(n) if A059895(m,n) = 1. - Peter Munn, May 04 2020
a(n) = A329697(A225546(n)) = A329697(A331736(n)).
a(n) = a(A008833(n)).
For all n >= 0, a(A334110(n)) = n, a(A334860(n)) = A334204(n).
a(A331590(m,k)) = a(m) + a(k); a(A003961(n)) = 2*a(n). - Peter Munn, Apr 30 2020

A336321 a(n) = A122111(A225546(n)).

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 19, 6, 9, 11, 53, 10, 131, 23, 13, 8, 311, 15, 719, 22, 29, 59, 1619, 14, 49, 137, 21, 46, 3671, 17, 8161, 12, 61, 313, 37, 25, 17863, 727, 139, 26, 38873, 31, 84017, 118, 39, 1621, 180503, 20, 361, 77, 317, 274, 386093, 33, 71, 58, 733, 3673, 821641, 34, 1742537, 8167, 87, 18, 151, 67, 3681131, 626, 1627, 41, 7754077, 35, 16290047

Offset: 1

Antti Karttunen and Peter Munn, Jul 17 2020

nonn

A122111 and A225546 are both self-inverse permutations of the positive integers based on prime factorizations, and they share further common properties. For instance, they map the prime numbers to powers of 2: A122111 maps the k-th prime to 2^k, whereas A225546 maps it to 2^2^(k-1).

In composing these permutations, this sequence maps the squarefree numbers, as listed in A019565, to the prime numbers in increasing order; and the list of powers of 2 to the "normal" numbers (A055932), as listed in A057335.

			From _Peter Munn_, Jan 04 2021: (Start)
In this set of examples we consider [a(n)] as a function a(.) with an inverse, a^-1(.).
First, a table showing mapping of the powers of 2:
  n     a^-1(2^n) =    2^n =        a(2^n) =
        A001146(n-1)   A000079(n)   A057335(n)
  0             (1)         1            1
  1               2         2            2
  2               4         4            4
  3              16         8            6
  4             256        16            8
  5           65536        32           12
  6      4294967296        64           18
  ...
Next, a table showing mapping of the squarefree numbers, as listed in A019565 (a lexicographic ordering by prime factors):
  n   a^-1(A019565(n))   A019565(n)      a(A019565(n))   a^2(A019565(n))
      Cf. {A337533}      Cf. {A005117}   = prime(n)      = A033844(n-1)
  0              1               1             (1)               (1)
  1              2               2               2                 2
  2              3               3               3                 3
  3              8               6               5                 7
  4              6               5               7                19
  5             12              10              11                53
  6             18              15              13               131
  7            128              30              17               311
  8              5               7              19               719
  9             24              14              23              1619
  ...
As sets, the above columns are A337533, A005117, A008578, {1} U A033844.
Similarly, we get bijections between sets A000290\{0} -> {1} U A070003; and {1} U A335740 -> A005408 -> A066207.
(End)

A122111 composed with A225546.
Cf. A336322 (inverse permutation).
Other sequences used in a definition of this sequence: A000040, A000188, A019565, A248663, A253550, A253560.
Sequences used to express relationship between terms of this sequence: A003159, A003961, A297002, A334747.
Cf. A057335.
A mapping between the binary tree sequences A334866 and A253563.
Lists of sets (S_1, S_2, ... S_j) related by the bijection defined by the sequence: (A000290\{0}, {1} U A070003), ({1} U A001146, A000079, A055932), ({1} U A335740, A005408, A066207), (A337533, A005117, A008578, {1} U A033844).

a(n) = A122111(A225546(n)).
Alternative definition: (Start)
Write n = m^2 * A019565(j), where m = A000188(n), j = A248663(n).
a(1) = 1; otherwise for m = 1, a(n) = A000040(j), for m > 1, a(n) = A253550^j(A253560(a(m))).
(End)
a(A000040(m)) = A033844(m-1).
a(A001146(m)) = 2^(m+1).
a(2^n) = A057335(n).
a(n^2) = A253560(a(n)).
For n in A003159, a(2n) = b(a(n)), where b(1) = 2, b(n) = A253550(n), n >= 2.
More generally, a(A334747(n)) = b(a(n)).
a(A003961(n)) = A297002(a(n)).
a(A334866(m)) = A253563(m).

A336322 a(n) = A225546(A122111(n)).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 16, 9, 12, 10, 32, 15, 24, 18, 256, 30, 64, 7, 48, 27, 20, 14, 512, 36, 40, 81, 96, 21, 128, 42, 65536, 54, 60, 72, 1024, 35, 120, 45, 768, 70, 192, 105, 80, 162, 28, 210, 131072, 25, 144, 90, 160, 11, 4096, 108, 1536, 135, 56, 22, 2048, 33, 84, 243, 4294967296, 216, 384, 66, 240, 270, 288, 55, 262144, 110, 168, 324, 480, 50

Offset: 1

Antti Karttunen and Peter Munn, Jul 17 2020

nonn

A225546 and A122111 are both self-inverse permutations of the positive integers based on prime factorizations, and they share further common properties. For instance, they map the prime numbers to powers of 2: A225546 maps the k-th prime to 2^2^(k-1), whereas A122111 maps it to 2^k.

In composing these permutations, this sequence maps the list of prime numbers to the squarefree numbers, as listed in A019565; and the "normal" numbers (A055932), as listed in A057335, to ascending powers of 2.

Index entries for sequences that are permutations of the natural numbers

A225546 composed with A122111.
Sorted even bisection: A335738.
Sorted odd bisection (excluding 1): A335740.
Sequences used to express relationship between terms of this sequence: A001222, A003961, A253560, A331590, A350066.
Sequences of sequences (S_1, S_2, ... S_j) with the property a(S_i) = S_{i+1}, or essentially so: (A033844, A000040, A019565), (A057335, A000079, A001146), (A000244, A011764), (A001248, A334110), (A253563, A334866).
The inverse permutation, A336321, lists sequences where the property is weaker (between the sets of terms).

a(A033844(m)) = A000040(m+1). [Offset corrected Peter Munn, Feb 14 2022]
a(A000040(m)) = A019565(m).
a(A057335(m)) = 2^m.
For m >= 1, a(2^m) = A001146(m-1).
a(A253563(m)) = A334866(m).
From Peter Munn, Feb 14 2022: (Start)
a(A253560(n)) = a(n)^2.
For n >= 2, a(A003961(n)) = A331590(a(n), 2^2^(A001222(n)-1)).
a(A350066(n, k)) = A331590(a(n), a(k)).
(End)

cp's OEIS Frontend

A331590 Square array A(n,k) = A225546(A225546(n) * A225546(k)), n >= 1, k >= 1, read by descending antidiagonals.

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A331591 a(n) is the number of distinct prime factors of A225546(n), or equally, number of distinct prime factors of A293442(n).

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A225547 Fixed points of A225546.

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A331733 a(n) = sigma(A225546(n)), where sigma is the sum of divisors.

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A331287 a(n) = gcd(n, A225546(n)).

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A331309 a(n) = A000005(A225546(n)), where A000005 gives the number of divisors of n.

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A340675 Exponential of Mangoldt function conjugated by Tek's flip: a(n) = A225546(A014963(A225546(n))).

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A334109 a(n) = A329697(A225546(n)).

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