A267098 -id:A267098 - cp's OEIS frontend

A038698 Excess of 4k-1 primes over 4k+1 primes, beginning with prime 2.

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

Offset: 1

Hans Havermann

a(n) < 0 for infinitely many values of n. - Benoit Cloitre, Jun 24 2002

First negative value is a(2946) = -1, which is for prime 26861. - David W. Wilson, Sep 27 2002

Stan Wagon, The Power of Visualization, Front Range Press, 1994, p. 2.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (first 10000 terms from T. D. Noe)
Vladimir Pletser, Chart for n=1..100000

Cf. A007350, A007351, A038691, A051024, A066520.
Cf. A112632 (race of 3k-1 and 3k+1 primes), A216057, A269364.
Cf. A156749 (sequence showing Chebyshev bias in prime races (mod 4)), A199547, A267097, A267098, A267107, A292378.
Cf. A163805, A212159.
List of primes p such that a(p) = 0 is A007351. List of primes p such that a(p) < 0 is A199547. List of primes p such that a(p) = -1 is A051025. List of integers k such that a(prime(k)) = -1 is A051024. - Ya-Ping Lu, Jan 18 2025

ans:=[0]; ct:=0; for n from 2 to 2000 do
p:=ithprime(n); if (p mod 4) = 3 then ct:=ct+1; else ct:=ct-1; fi;
ans:=[op(ans),ct]; od: ans; # N. J. A. Sloane, Jun 24 2016

FoldList[Plus, 0, Mod[Prime[Range[2,110]], 4] - 2]
Join[{0},Accumulate[If[Mod[#,4]==3,1,-1]&/@Prime[Range[2,110]]]] (* Harvey P. Dale, Apr 27 2013 *)

for(n=2,100,print1(sum(i=2,n,(-1)^((prime(i)+1)/2)),","))

from sympy import nextprime; a, p = 0, 2; R = [a]
for _ in range(2,88): p=nextprime(p); a += p%4-2; R.append(a)
print(*R, sep = ', ')  # Ya-Ping Lu, Jan 18 2025

a(n) = Sum_{k=2..n} (-1)^((prime(k)+1)/2). - Benoit Cloitre, Jun 24 2002
a(n) = (Sum_{k=1..n} prime(k) mod 4) - 2*n (assuming that x mod 4 > 0). - Thomas Ordowski, Sep 21 2012
From Antti Karttunen, Oct 01 2017: (Start)
a(n) = A267098(n) - A267097(n).
a(n) = A292378(A000040(n)).
(End)
From Ridouane Oudra, Nov 04 2024: (Start)
a(n) = Sum_{k=2..n} i^(prime(k)+1), where i is the imaginary unit.
a(n) = Sum_{k=2..n} sin(3*prime(k)*Pi/2).
a(n) = Sum_{k=2..n} A163805(prime(k)).
a(n) = Sum_{k=2..n} A212159(k). (End)
a(n) = a(n-1) + prime(n) (mod 4) - 2, n >= 2. - Ya-Ping Lu, Jan 18 2025

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

A292377 a(1) = 0, and for n > 1, a(n) = a(A252463(n)) + [n == 3 (mod 4)].

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 0, 0, 1, 3, 1, 3, 2, 2, 0, 3, 0, 4, 1, 1, 3, 5, 1, 0, 3, 1, 2, 5, 2, 6, 0, 2, 3, 3, 0, 6, 4, 4, 1, 6, 1, 7, 3, 1, 5, 8, 1, 0, 0, 4, 3, 8, 1, 2, 2, 3, 5, 9, 2, 9, 6, 2, 0, 2, 2, 10, 3, 4, 3, 11, 0, 11, 6, 1, 4, 3, 4, 12, 1, 0, 6, 13, 1, 4, 7, 6, 3, 13, 1, 3, 5, 5, 8, 5, 1, 13, 0, 3, 0, 13, 4, 14, 3, 2

Offset: 1

Antti Karttunen, Sep 17 2017

nonn

For numbers > 1, iterate the map x -> A252463(x) which divides even numbers by 2 and shifts every prime in the prime factorization of odd n one index step towards smaller primes. a(n) counts the numbers of the form 4k+3 encountered until 1 has been reached. The count includes also n itself if it is of the form 4k+3 (A004767).

In other words, locate the node which contains n in binary tree A005940 and traverse from that node towards the root, counting all numbers of the form 4k+3 that occur on the path.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000120, A005940, A061395, A064989, A252463, A292375, A292376, A292378, A292383.

a[1] = 0; a[n_] := a[n] = a[Which[n == 1, 1, EvenQ@ n, n/2, True, Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n]] + Boole[Mod[n, 4] == 3]; Array[a, 105]

a(1) = 0, and for n > 1, a(n) = a(A252463(n)) + floor((n mod 4)/3).
Equivalently, a(2n) = a(n), and for odd numbers n > 1, a(n) = a(A064989(n)) + [n == 3 (mod 4)].
a(n) = A000120(A292383(n)).
Other identities. For n >= 1:
a(n) >= A292376(n).
a(A000040(n)) = A267098(n).
1 + a(n) - A292375(n) = A292378(n).
For n >= 2, a(n) + A292375(n) = A061395(n).

A267097 a(n) = number of 4k+1 primes among first n primes; least monotonic left inverse of A080147.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 3, 3, 4, 4, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 11, 12, 12, 12, 13, 14, 14, 14, 15, 15, 16, 16, 17, 17, 17, 18, 18, 19, 19, 20, 21, 21, 21, 21, 21, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 28, 28, 29, 29, 29, 30, 31, 31, 32, 32, 33, 34, 34, 34, 35, 35, 35, 36, 37, 38, 39, 39, 40, 40, 41

Offset: 1

Antti Karttunen, Feb 01 2016

nonn

a(n) = number of 4k+1 primes (A002144) among primes in range 2 .. A000040(n).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000040, A002144, A080147, A267098.

Cf. also A267101.

a[n_]:=Length[Select[Prime[Range[n]],IntegerQ[(#-1)/4] &]]; Array[a,84] (* Stefano Spezia, May 01 2025 *)

Other identities. For all n >= 1:
a(A080147(n)) = n.
a(n) + A267098(n) = n-1.

A267101 2 followed by permutation of odd primes, where each n-th prime of the form 4k+1 (A002144) has been replaced with the n-th prime of the form 4k+3 (A002145) and vice versa.

Original entry on oeis.org

2, 5, 3, 13, 17, 7, 11, 29, 37, 19, 41, 23, 31, 53, 61, 43, 73, 47, 89, 97, 59, 101, 109, 67, 71, 79, 113, 137, 83, 103, 149, 157, 107, 173, 127, 181, 131, 193, 197, 139, 229, 151, 233, 163, 167, 241, 257, 269, 277, 179, 191, 281, 199, 293, 211, 313, 223, 317, 227, 239, 337, 251, 349, 353, 263, 271, 373, 283, 389

Offset: 1

Antti Karttunen, Feb 01 2016

nonn

After 2, for each n >= 1, swap the places of primes A002144(n) and A002145(n) in A000040.

			For n=2, for which A000040(2) = 3, the first prime of the form 4k+3, we select the first prime of the form 4k+1, which is 5, thus a(2) = 5.
For n=3, for which A000040(3) = 5, the first prime of the form 4k+1, we select the first prime of the form 4k+3, which is 3, thus a(3) = 3.
For n=4, for which A000040(4) = 7, the second prime of the form 4k+3, we select the second prime of the form 4k+1, which is 13, thus a(4) = 13.
For n=5, for which A000040(5) = 11, the third prime of the form 4k+3, we select the third prime of the form 4k+1, which is 17, thus a(5) = 17.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A000040, A002144, A002145, A267097, A267098, A267099, A267100.

Cf. also A108546.

(define (A267101 n) (cond ((= 1 n) 2) ((= 1 (modulo (A000040 n) 4)) (A002145 (A267097 n))) (else (A002144 (A267098 n)))))

a(1) = 2; after which, if prime(n) modulo 4 = 1, a(n) = A002145(A267097(n)), otherwise a(n) = A002144(A267098(n)).
a(n) = A000040(A267100(n)).
a(n) = A267099(A000040(n)).

A267107 "Chebyshev's bat permutation": a(1) = 1, a(A080147(n)) = A080148(a(n)), a(A080148(n)) = A080147(a(n)).

Original entry on oeis.org

1, 3, 2, 7, 6, 5, 4, 16, 13, 14, 12, 11, 9, 10, 35, 8, 29, 31, 30, 26, 23, 25, 21, 27, 22, 20, 24, 74, 17, 19, 18, 62, 67, 66, 15, 65, 54, 57, 51, 58, 55, 56, 45, 48, 43, 59, 50, 44, 53, 47, 39, 152, 49, 37, 41, 42, 38, 40, 46, 144, 130, 32, 139, 137, 36, 34, 33, 118, 136, 129, 128, 113, 121, 28, 108, 122, 125

Offset: 1

Antti Karttunen, Feb 01 2016

This is a self-inverse permutation of natural numbers.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Antti Karttunen, Entanglement Permutations, 2016-2017.
Index entries for sequences that are permutations of the natural numbers

Cf. A000040, A002144, A002145, A038698, A080147, A080148, A267097, A267098.
Cf. A268393 (record positions), A268394 (record values).
Cf. A267100, A267105, A267106 and also A270193, A270194, A270199, A270201, A270202 for other similarly constructed permutations based on prime distribution biases.

allocatemem(2^30);
default(primelimit,4294965247);
uplim = 2^20;
uplim2 = 366824; \\ Very ad hoc.
v080147 = vector(uplim);
v080148 = vector(uplim);
v267097 = vector(uplim);
v267107 = vector(uplim);
v267097[1] = 0; c = 0; v47i = 0; v48i = 0; for(n=2, uplim, if((1 == (prime(n)%4)), c++; v47i++; v080147[v47i] = n, v48i++; v080148[v48i] = n); v267097[n] = c; if(!(n%32768),print1(" n=",n)));
A080147(n) = v080147[n];
A080148(n) = v080148[n];
A267097(n) = v267097[n];
A267098(n) = (n - A267097(n))-1;
A267107(n) = v267107[n];
v267107[1] = 1; for(n=2, uplim2, if((1 == (prime(n) % 4)), v267107[n] = A080148(A267107(A267097(n))), v267107[n] = A080147(A267107(A267098(n))));  if(!(n%32768),print1(" n=",n)));
for(n=1, uplim2, write("b267107.txt", n, " ", A267107(n)));

;; With memoization-macro definec
(definec (A267107 n) (cond ((<= n 1) n) ((= 1 (modulo (A000040 n) 4)) (A080148 (A267107 (A267097 n)))) (else (A080147 (A267107 (A267098 n))))))

a(1) = 1; and for n > 1, if prime(n) modulo 4 = 1, a(n) = A080148(a(A267097(n))), otherwise a(n) = A080147(a(A267098(n))).

Name changed, the old name was "Manta moth permutation" - Antti Karttunen, Dec 10 2019

A267100 Self-inverse permutation of natural numbers: a(1) = 1, a(A080147(n)) = A080148(n), a(A080148(n)) = A080147(n).

Original entry on oeis.org

1, 3, 2, 6, 7, 4, 5, 10, 12, 8, 13, 9, 11, 16, 18, 14, 21, 15, 24, 25, 17, 26, 29, 19, 20, 22, 30, 33, 23, 27, 35, 37, 28, 40, 31, 42, 32, 44, 45, 34, 50, 36, 51, 38, 39, 53, 55, 57, 59, 41, 43, 60, 46, 62, 47, 65, 48, 66, 49, 52, 68, 54, 70, 71, 56, 58, 74, 61, 77, 63, 64, 78, 79, 67, 80, 82, 69, 72, 73, 75, 84, 76, 87, 81

Offset: 1

Antti Karttunen, Feb 01 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Cf. A000040, A000720, A080147, A080148, A267101, A267097, A267098, A267099.

Cf. also A267107 (a more recursed variant).

a(1) = 1; for n > 1, if prime(n) mod 4 = 1, then a(n) = A080148(A267097(n)), otherwise a(n) = A080147(A267098(n)).
Other identities. For all n >= 1:
a(n) = A000720(A267101(n)).

A267105 Permutation of natural numbers: a(1) = 1, a(A080147(n)) = 1+(2a(n)), a(A080148(n)) = 2a(n).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 7, 8, 12, 9, 10, 13, 11, 14, 16, 15, 24, 17, 18, 20, 25, 26, 22, 19, 21, 27, 28, 32, 23, 29, 30, 48, 33, 34, 31, 36, 49, 40, 50, 35, 52, 37, 44, 41, 51, 38, 42, 54, 56, 53, 45, 64, 39, 46, 43, 58, 55, 60, 57, 65, 96, 47, 66, 68, 59, 61, 62, 97, 72, 67, 69, 98, 80, 63, 100, 70, 73, 99, 81, 101, 104, 71, 74

Offset: 1

Antti Karttunen, Feb 01 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A267106.
Cf. A000040, A002144, A002145, A080147, A080148, A267097, A267098.
Cf. also A267107.

a(1) = 1; and for n > 1, if prime(n) mod 4 = 1, then a(n) = 1 + 2*a(A267097(n)), otherwise a(n) = 2*a(A267098(n)).

cp's OEIS Frontend

A038698 Excess of 4k-1 primes over 4k+1 primes, beginning with prime 2.

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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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A292377 a(1) = 0, and for n > 1, a(n) = a(A252463(n)) + [n == 3 (mod 4)].

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A267097 a(n) = number of 4k+1 primes among first n primes; least monotonic left inverse of A080147.

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A267101 2 followed by permutation of odd primes, where each n-th prime of the form 4k+1 (A002144) has been replaced with the n-th prime of the form 4k+3 (A002145) and vice versa.

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A267107 "Chebyshev's bat permutation": a(1) = 1, a(A080147(n)) = A080148(a(n)), a(A080148(n)) = A080147(a(n)).

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A267100 Self-inverse permutation of natural numbers: a(1) = 1, a(A080147(n)) = A080148(n), a(A080148(n)) = A080147(n).

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A267105 Permutation of natural numbers: a(1) = 1, a(A080147(n)) = 1+(2*a(n)), a(A080148(n)) = 2*a(n).

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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.

A267105 Permutation of natural numbers: a(1) = 1, a(A080147(n)) = 1+(2a(n)), a(A080148(n)) = 2a(n).