A108548 -id:A108548 - cp's OEIS frontend

A332817 a(n) = A108548(A163511(n)).

1, 2, 4, 3, 8, 9, 6, 5, 16, 27, 18, 25, 12, 15, 10, 7, 32, 81, 54, 125, 36, 75, 50, 49, 24, 45, 30, 35, 20, 21, 14, 13, 64, 243, 162, 625, 108, 375, 250, 343, 72, 225, 150, 245, 100, 147, 98, 169, 48, 135, 90, 175, 60, 105, 70, 91, 40, 63, 42, 65, 28, 39, 26, 11, 128, 729, 486, 3125, 324, 1875, 1250, 2401, 216, 1125, 750, 1715, 500

Offset: 0

Antti Karttunen, Mar 05 2020

This irregular table can be represented as a binary tree. Each child to the left is obtained by doubling the parent, and each child to the right is obtained by applying A332818 to the parent:
                                       1
                                       |
                    ...................2...................
                   4                                       3
         8......../ \........9                   6......../ \........5
        / \                 / \                 / \                 / \
       /   \               /   \               /   \               /   \
      /     \             /     \             /     \             /     \
    16       27         18       25         12       15         10       7
  32 81    54  125    36  75   50  49     24  45   30  35     20  21   14 13
etc.
This is the mirror image of the tree in A332815.

Cf. A332811 (inverse permutation).
Cf. A054429, A108548, A163511, A332815 (mirror image).
Cf. A108546 (the right edge of the tree from 2 downward).
Cf. also A332214.

up_to = 26927;
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A054429(n) = ((3<<#binary(n\2))-n-1); \\ From A054429
A163511(n) = if(!n,1,A005940(1+A054429(n)));
A108546list(up_to) = { my(v=vector(up_to), p,q); v[1] = 2; v[2] = 3; v[3] = 5; for(n=4,up_to, p = v[n-2]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[n] = q); (v); };
v108546 = A108546list(up_to);
A108546(n) = v108546[n]; \\ Antti Karttunen, Mar 05 2020
A108548(n) = { my(f=factor(n)); f[,1] = apply(A108546,apply(primepi,f[,1])); factorback(f); };
A332817(n) = A108548(A163511(n));

a(n) = A108548(A163511(n)).

For n >= 1, a(n) = A332815(A054429(n)).

A332815 a(n) = A108548(A005940(1+n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 25, 18, 27, 16, 13, 14, 21, 20, 35, 30, 45, 24, 49, 50, 75, 36, 125, 54, 81, 32, 11, 26, 39, 28, 65, 42, 63, 40, 91, 70, 105, 60, 175, 90, 135, 48, 169, 98, 147, 100, 245, 150, 225, 72, 343, 250, 375, 108, 625, 162, 243, 64, 17, 22, 33, 52, 55, 78, 117, 56, 77, 130, 195, 84

Offset: 0

Antti Karttunen, Feb 28 2020

This is variant of Doudna-sequence, A005940 and thus can be represented as a binary tree. Each child to the left is obtained by applying A332818 to the parent, and each child to the right is obtained by doubling the parent:
                                      1
                                      |
                   ...................2...................
                  3                                       4
        5......../ \........6                   9......../ \........8
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
    7       10         15       12         25       18         27       16
  13 14   21  20     35  30   45  24     49  50   75  36    125  54   81  32
etc.
Note the indexing: the sequence starts with a(0)=1, as is natural for sequences based on maps from base-2 expansion to prime factorization. This is
in contrast to A005940, which for historical reasons starts from offset 1.
For any n > 1, A332893(n) gives the value of the parent node. For any n >= 1, A332894(n) gives the distance to 1, and A332899(n) gives the number of odd numbers that occur (inclusively) on the path from 1 to n.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences that are permutations of the natural numbers

Cf. A332816 (inverse permutation).
Cf. A005940, A108548, A332818, A332893, A332894, A332897, A332898, A332899.
Cf. A108546 (the left edge of the tree from 2 downward).

up_to = 26927;
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A108546list(up_to) = { my(v=vector(up_to), p,q); v[1] = 2; v[2] = 3; v[3] = 5; for(n=4,up_to, p = v[n-2]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[n] = q); (v); };
v108546 = A108546list(up_to);
A108546(n) = v108546[n];
A108548(n) = { my(f=factor(n)); f[,1] = apply(A108546,apply(primepi,f[,1])); factorback(f); };
A332815(n) = A108548(A005940(1+n));

a(n) = A108548(A005940(1+n)).

A332818 a(n) = A108548(A003961(A332808(n))).

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 13, 27, 25, 21, 17, 45, 11, 39, 35, 81, 19, 75, 29, 63, 65, 51, 37, 135, 49, 33, 125, 117, 23, 105, 41, 243, 85, 57, 91, 225, 31, 87, 55, 189, 43, 195, 53, 153, 175, 111, 61, 405, 169, 147, 95, 99, 47, 375, 119, 351, 145, 69, 73, 315, 59, 123, 325, 729, 77, 255, 89, 171, 185, 273, 97, 675, 67, 93, 245, 261, 221, 165, 101

Offset: 1

Antti Karttunen, Feb 27 2020

Permutation of odd numbers. Preserves prime signature.

Antti Karttunen, Table of n, a(n) for n = 1..26921

Cf. A002144, A002145, A003961, A046523, A108548, A332808.

Cf. A332819 (a left inverse).

A332818(n) = A108548(A003961(A332808(n)));

Fully multiplicative with a(2) = 3, a(A002145(n)) = A002144(n) and a(A002144(n)) = A002145(1+n), for all n >= 1.
a(n) = A108548(A003961(A332808(n))).
A332819(a(n)) = n.
A046523(a(n)) = A046523(n).

A332819 a(n) = A108548(A064989(A332808(n))).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 13, 2, 7, 5, 6, 1, 11, 4, 17, 3, 10, 13, 29, 2, 9, 7, 8, 5, 19, 6, 37, 1, 26, 11, 15, 4, 23, 17, 14, 3, 31, 10, 41, 13, 12, 29, 53, 2, 25, 9, 22, 7, 43, 8, 39, 5, 34, 19, 61, 6, 47, 37, 20, 1, 21, 26, 73, 11, 58, 15, 89, 4, 59, 23, 18, 17, 65, 14, 97, 3, 16, 31, 101, 10, 33, 41, 38, 13, 67, 12, 35, 29

Offset: 1

Antti Karttunen, Feb 27 2020

Antti Karttunen, Table of n, a(n) for n = 1..26927

A left inverse of A332818.

Cf. A002144, A002145, A064989, A108548, A332808.

A332819(n) = A108548(A064989(A332808(n)));

Fully multiplicative with a(2) = 1, a(3) = 2, a(A002144(n)) = A002145(n), and a(A002145(1+n)) = A002144(n) for all n >= 1.
a(n) = A108548(A064989(A332808(n))).
a(A332818(n)) = n.

A108549 Fixed points for A108548.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 27, 28, 30, 32, 34, 35, 36, 38, 40, 41, 42, 43, 45, 48, 49, 50, 51, 54, 56, 57, 60, 63, 64, 68, 70, 72, 75, 76, 80, 81, 82, 84, 85, 86, 90, 95, 96, 98, 100, 102, 105, 108, 112, 114, 119, 120, 123, 125

Offset: 1

Reinhard Zumkeller, Jun 10 2005

nonn

A108548(a(n)) = a(n); multiplicative closure of A108547.

Cf. A108546.

A331137 a(n) = Sum_{primes p <= n} b(p-1), where b = A108548.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 1

N. J. A. Sloane, Jan 11 2020

nonn

Yoichi Motohashi, On the distribution of prime numbers which are of the form x^2+y^2+1, Acta Arith. 16 (1969/70), 351-363. MR0288086 (44 #5284).

Cf. A108548.

A267099 Fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes: a(1) = 1; a(prime(k)) = A267101(k), a(xy) = a(x)a(y) for x, y > 1.

Original entry on oeis.org

1, 2, 5, 4, 3, 10, 13, 8, 25, 6, 17, 20, 7, 26, 15, 16, 11, 50, 29, 12, 65, 34, 37, 40, 9, 14, 125, 52, 19, 30, 41, 32, 85, 22, 39, 100, 23, 58, 35, 24, 31, 130, 53, 68, 75, 74, 61, 80, 169, 18, 55, 28, 43, 250, 51, 104, 145, 38, 73, 60, 47, 82, 325, 64, 21, 170, 89, 44, 185, 78, 97, 200, 59, 46, 45, 116, 221, 70, 101

Offset: 1

Antti Karttunen, Feb 01 2016

Lexicographically earliest self-inverse permutation of natural numbers where each prime of the form 4k+1 is replaced by a prime of the form 4k+3 and vice versa, with the composite numbers determined by multiplicativity.
Fully multiplicative with a(p_n) = p_{A267100(n)} = A267101(n).
Maps each term of A004613 to some term of A004614, each (nonzero) term of A001481 to some term of A268377 and each term of A004431 to some term of A268378 and vice versa.
Sequences A072202 and A078613 are closed with respect to this permutation.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A000040, A000720, A010051, A020639, A032742, A267100, A267101, A354102 (Möbius transform), A354103 (inverse Möbius transform), A354192 (fixed points).
Cf. A002144, A002145, A005094, A065338, A072202, A078613, A079635.
Cf. A001481, A268377, A004431, A268378, A004613, A004614.
Cf. also A108548.

up_to = 2^16;
A267097list(up_to) = { my(v=vector(up_to),i=0,c=0); forprime(p=2,prime(up_to), if(1==(p%4), c++); i++; v[i] = c); (v); };
v267097 = A267097list(up_to);
A267097(n) = v267097[n];
A267098(n) = ((n-1)-A267097(n));
list_primes_of_the_form(up_to,m,k) = { my(v=vector(up_to),i=0); forprime(p=2,, if(k==(p%m), i++; v[i] = p; if(i==up_to,return(v)))); };
v002144 = list_primes_of_the_form(2*up_to,4,1);
A002144(n) = v002144[n];
v002145 = list_primes_of_the_form(2*up_to,4,3);
A002145(n) = v002145[n];
A267101(n) = if(1==n,2,if(1==(prime(n)%4),A002145(A267097(n)),A002144(A267098(n))));
A267099(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A267101(primepi(f[k,1]))); factorback(f); }; \\ Antti Karttunen, May 18 2022
(Scheme, with memoization-macro definec)
(definec (A267099 n) (cond ((<= n 1) n) ((= 1 (A010051 n)) (A267101 (A000720 n))) (else (* (A267099 (A020639 n)) (A267099 (A032742 n))))))

a(1) = 1; after which, if n is k-th prime [= A000040(k)], then a(n) = A267101(k), otherwise a(A020639(n)) * a(A032742(n)).
Other identities. For all n >= 1:
a(A000040(n)) = A267101(n).
a(2*n) = 2*a(n).
a(3*n) = 5*a(n).
a(5*n) = 3*a(n).
a(7*n) = 13*a(n).
a(11*n) = 17*a(n).
etc. See examples in A267101.
A000035(n) = A000035(a(n)). [Preserves the parity of n.]
A005094(a(n)) = -A005094(n).
A079635(a(n)) = -A079635(n).

Verbal description prefixed to the name by Antti Karttunen, May 19 2022

A108546 Lexicographically earliest permutation of primes such that for n>1 forms 4k+1 and 4k+3 alternate.

Original entry on oeis.org

2, 3, 5, 7, 13, 11, 17, 19, 29, 23, 37, 31, 41, 43, 53, 47, 61, 59, 73, 67, 89, 71, 97, 79, 101, 83, 109, 103, 113, 107, 137, 127, 149, 131, 157, 139, 173, 151, 181, 163, 193, 167, 197, 179, 229, 191, 233, 199, 241, 211, 257, 223, 269, 227, 277, 239, 281, 251, 293

Offset: 1

Reinhard Zumkeller, Jun 10 2005

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A002144, A002145, A102261, A108547 (fixed points), A108548, A111745, A332806 (inverse), A332807.

Cf. also A267101, A332211.

import Data.List (transpose)
a108546 n = a108546_list !! (n-1)
a108546_list =  2 : concat
   (transpose [a002145_list, a002144_list])
-- Reinhard Zumkeller, Nov 13 2014, Feb 22 2011

terms = 60; A111745 = Module[{prs = Prime[Range[2terms]], m3, m1, min}, m3 = Select[prs, Mod[#, 4] == 3&]; m1 = Select[prs, Mod[#, 4] == 1&]; min = Min[Length[m1], Length[m3]]; Riffle[Take[m3, min], Take[m1, min]]]; a[1] = 2; a[n_] := A111745[[n-1]]; Table[a[n], {n, 1, terms}] (* Jean-François Alcover, May 18 2017, using Harvey P. Dale's code for A111745 *)

up_to = 10000;
A108546list(up_to) = { my(v=vector(up_to), p,q); v[1] = 2; v[2] = 3; v[3] = 5; for(n=4,up_to, p = v[n-2]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[n] = q); (v); };
v108546 = A108546list(up_to);
A108546(n) = v108546[n]; \\ Antti Karttunen, Feb 27 2020

a(n) mod 4 = 3 - 2 * (n mod 2) for n>1.
For n > 1: a(n) = A111745(n-1).
a(2*n+1) - a(2*n) = A102261(n).
From Antti Karttunen, Feb 27 2020: (Start)
a(1) = 2, a(2n) = A002145(n), a(2n+1) = A002144(n).
a(n) = A000040(A332807(n)).
(End)

A332808 Fully multiplicative with a(p) = A332806(A000720(p)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 12, 11, 14, 15, 16, 17, 18, 19, 20, 21, 26, 29, 24, 25, 22, 27, 28, 23, 30, 37, 32, 39, 34, 35, 36, 31, 38, 33, 40, 41, 42, 43, 52, 45, 58, 53, 48, 49, 50, 51, 44, 47, 54, 65, 56, 57, 46, 61, 60, 59, 74, 63, 64, 55, 78, 71, 68, 87, 70, 79, 72, 67, 62, 75, 76, 91, 66, 89, 80, 81, 82, 101

Offset: 1

Antti Karttunen, Feb 27 2020

Antti Karttunen, Table of n, a(n) for n = 1..26927

Cf. A000720, A332806, A108549 (fixed points), A332818, A332819.

Inverse permutation is A108548, from which this differs for the first time at n=67, where a(67) = 71, while A108548(67) = 73.

up_to = 10000;
A332806list(up_to) = { my(v=vector(2), xs=Map(), lista=List([]), p,q,u); v[2] = 3; v[1] = 5; mapput(xs,1,1); mapput(xs,2,2); mapput(xs,3,3);  for(n=4,up_to, p = v[2-(n%2)]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[2-(n%2)] = q; mapput(xs,primepi(q),n)); for(i=1, oo, if(!mapisdefined(xs, i, &u), return(Vec(lista)), listput(lista, prime(u)))); };
v332806 = A332806list(up_to);
A332806(n) = v332806[n];
A332808(n) = { my(f=factor(n)); f[,1] = apply(A332806,apply(primepi,f[,1])); factorback(f); };

A354202 Fully multiplicative with a(p^e) = A354200(A000720(p))^e.

Original entry on oeis.org

1, 5, 7, 25, 13, 35, 11, 125, 49, 65, 19, 175, 17, 55, 91, 625, 29, 245, 23, 325, 77, 95, 31, 875, 169, 85, 343, 275, 37, 455, 43, 3125, 133, 145, 143, 1225, 41, 115, 119, 1625, 53, 385, 47, 475, 637, 155, 59, 4375, 121, 845, 203, 425, 61, 1715, 247, 1375, 161, 185, 67, 2275, 73, 215, 539, 15625, 221, 665, 71, 725

Offset: 1

Antti Karttunen, May 23 2022

Permutation of A007310. Preserves the prime signature.

Index entries for sequences computed from indices in prime factorization

Cf. A007310 (terms sorted into ascending order), A354200, A354203 (left inverse), A354204 (Möbius transform), A354205 (inverse Möbius transform).

Cf. also A003961, A108548, A267099, A332818, A348746, A354091 for similar constructions.

A354200(n) = if(1==n,5,my(p=prime(n), m=p%4); forprime(q=1+p,,if(m==(q%4),return(q))));
A354202(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354200(primepi(f[k,1]))); factorback(f); };

cp's OEIS Frontend

A332817 a(n) = A108548(A163511(n)).

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A332815 a(n) = A108548(A005940(1+n)).

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A332818 a(n) = A108548(A003961(A332808(n))).

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A332819 a(n) = A108548(A064989(A332808(n))).

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A108549 Fixed points for A108548.

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A331137 a(n) = Sum_{primes p <= n} b(p-1), where b = A108548.

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A267099 Fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes: a(1) = 1; a(prime(k)) = A267101(k), a(x*y) = a(x)*a(y) for x, y > 1.

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A108546 Lexicographically earliest permutation of primes such that for n>1 forms 4*k+1 and 4*k+3 alternate.

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A332808 Fully multiplicative with a(p) = A332806(A000720(p)).

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A267099 Fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes: a(1) = 1; a(prime(k)) = A267101(k), a(xy) = a(x)a(y) for x, y > 1.

A108546 Lexicographically earliest permutation of primes such that for n>1 forms 4k+1 and 4k+3 alternate.