A064216 -id:A064216 - cp's OEIS frontend

A349398 Dirichlet convolution of A048673 with the Dirichlet inverse of its inverse permutation.

1, 0, 0, 0, 0, 1, -5, 8, 0, -6, -3, 2, 0, 19, -5, -4, -4, 20, -19, 22, 6, -15, 3, -8, 0, 0, 16, 16, -18, 24, -40, 70, 9, -24, 21, -7, -50, 55, 8, -24, 6, -41, -15, 58, 20, -17, -31, 108, 27, 70, -37, -24, 0, -20, -49, -98, 6, 26, -13, 21, -15, 62, 158, 84, -22, 9, -49, 130, -67, 12, -49, 62, -29, 112, 4, -60, 103, 16

Offset: 1

Antti Karttunen, Nov 19 2021

sign

Dirichlet convolution of A048673 with A349358, which is the Dirichlet inverse of A064216 (inverse permutation of A048673). Therefore, convolving A064216 with this sequence gives A048673.

Note how for n = 1 .. 35, a(n) = -A349397(n).

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

Cf. A003961, A048673, A064216, A064989, A323893, A349397 (Dirichlet inverse), A349399 (sum with it).

Cf. also A349376, A349377, A349385.

A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
A064216(n) = { my(f = factor(n+n-1)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
memoA349358 = Map();
A349358(n) = if(1==n,1,my(v); if(mapisdefined(memoA349358,n,&v), v, v = -sumdiv(n,d,if(dA064216(n/d)*A349358(d),0)); mapput(memoA349358,n,v); (v)));
A349398(n) = sumdiv(n,d,A048673(n/d)*A349358(d));

a(n) = Sum_{d|n} A048673(n/d) * A349358(d).

A245449 Fixed points of A245447 and A245448.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 13, 25, 26, 30, 33, 53, 93, 1023, 1034, 1203, 1330, 2657, 8584, 17159, 779212, 970225, 1558409, 8550146, 240902643, 244608573, 325422273, 414690595, 570131490, 1020233393, 1864797542, 2438037206

Offset: 1

Antti Karttunen, Jul 22 2014

First apply A003961(n), where the primes in the prime factorization of natural number n are shifted one step left [i.e. each p_i changes to p_{i+1}]. Then increment the resulting odd number by one to get an even number, which is divided by 2, and the same three operations are done second time to that quotient. This sequence consists of such numbers for which the final result is equal to the original n which we started from.
8550146 is the largest term <= 123456789.
Numbers which are in 1- and 2-cycles of A048673 and A064216.

			For n = 30 = 2*3*5 = p_1 * p_2 * p_3, the first shift operation results p_2 * p_3 * p_4 = 3*5*7 = 105, and (105+1)/2 = 53, which is the 16th prime, p_16. Shifting this once left results p_17 = 59, and (59+1)/2 = 30 again. Thus 30 is included in the sequence. For the same reason 53 is also included in the sequence.

A048674 is a subsequence.

Cf. A048673, A064216, A245447, A245448.

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
A048673(n) = (A003961(n)+1)/2;
isA245449(n) = ((A048673(A048673(n)) == n))
i=0; for(n=1, 123456789, if(isA245449(n), i++; write("b245449.txt", i, " ", n)))

;; With Antti Karttunen's IntSeq-library.
(define A245449 (FIXED-POINTS 1 1 A245447))

a(25)-a(32) added by Antti Karttunen, Sep 13 2014

A246376 Permutation of natural numbers: a(1) = 1, a(2n) = 2 * a(n), a(2n-1) = 1 + (2 * a(A064989(2n-1)-1)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 13, 12, 21, 18, 11, 16, 25, 14, 33, 20, 15, 26, 29, 24, 17, 42, 19, 36, 53, 22, 73, 32, 43, 50, 37, 28, 45, 66, 31, 40, 57, 30, 81, 52, 27, 58, 61, 48, 49, 34, 35, 84, 117, 38, 41, 72, 87, 106, 169, 44, 213, 146, 67, 64, 65, 86, 89, 100, 91, 74, 173, 56, 149, 90, 51, 132, 101, 62, 113, 80, 23

Offset: 1

Antti Karttunen, Aug 27 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Inverse: A246375.
Similar or related permutations: A005940, A005941, A064216, A243071, A245605, A246377, A246380.
Cf. A000035, A064989.

a(1) = 1, a(2n) = 2 * a(n), a(2n-1) = 1 + (2 * a(A064989(2n-1)-1)).
As a composition of related permutations:
a(n) = A246377(A246380(n)).
Other identities. For all n >= 1 the following holds:
A000035(a(n)) = A000035(n). [Like A005940 & A005941, this also preserves the parity].

A250472 Permutation of natural numbers: a(n) = A250470(2*n - 1).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 11, 6, 13, 17, 8, 19, 9, 10, 23, 29, 12, 15, 31, 14, 37, 41, 16, 43, 25, 18, 47, 21, 20, 53, 59, 22, 27, 61, 24, 67, 71, 26, 35, 73, 28, 79, 33, 30, 83, 55, 32, 39, 89, 34, 97, 101, 36, 103, 107, 38, 109, 45, 40, 65, 49, 42, 51, 113, 44, 127, 85, 46, 131, 137, 48, 77, 57, 50, 139, 149, 52, 63, 151, 54, 95, 157, 56, 163, 121, 58, 167, 69, 60

Offset: 1

Antti Karttunen, Dec 06 2014

nonn

For n > 1, a(n) tells which number is located immediately above n in the sieve of Eratosthenes (see A083140, A083221) in the same column of the sieve that contains 2n - 1.

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences that are permutations of the natural numbers

Inverse: A250471.
Odd bisection of A250470. The other bisection: A250479.
Cf. A055396, A064216, A078898, A083140, A083221, A114881, A114883.

a(1) = 1, a(n) = A083221(A055396(2*n - 1)-1, A078898(2*n - 1)).

a(n) = A250470(2*n - 1).

A292245 Base-2 expansion of a(n) encodes the steps where numbers of the form 3k+1 are encountered when map x -> A253889(x) is iterated down to 1, starting from x=n.

Original entry on oeis.org

1, 2, 2, 5, 4, 4, 11, 4, 8, 17, 10, 18, 9, 8, 22, 17, 8, 8, 17, 22, 36, 41, 8, 42, 17, 16, 44, 21, 34, 32, 35, 20, 32, 33, 36, 64, 69, 18, 34, 73, 16, 74, 37, 44, 82, 33, 34, 34, 89, 16, 64, 69, 16, 68, 65, 34, 64, 33, 44, 64, 33, 72, 16, 65, 82, 68, 85, 16, 128, 137, 84, 72, 69, 34, 138, 145, 32, 84, 145, 88, 88, 149, 42, 162, 65, 68, 164, 45, 64

Offset: 1

Antti Karttunen, Sep 15 2017

			For n=1 (the termination value of the iteration), 1 is of the form 3k+1, thus a(1) = 1*(2^0) = 1.
For n=2, 2 is not of the form 3k+1, while A253889(2) = 1 is, thus a(2) = 0*(2^0) + 1*2(^1) = 2.
For n=4, 4 is of the form 3k+1, while A253889(4) = 2 is not, but then A253889(2) = 1 again is, thus a(4) = 1*(2^0) + 0*(2^1) + 1*(2^2) = 5.

Cf. A048673, A064216, A289813, A292243, A292244, A292246, A292248.

f[n_] := Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1]; g[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; Map[FromDigits[#, 2] &[IntegerDigits[#, 3] /. 2 -> 0] &, Array[a, 98]] (* Michael De Vlieger, Sep 16 2017 *)

a(1) = 1; for n > 1, a(n) = 2*a(A253889(n)) + [n ≡ 1 (mod 3)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 3k+1, and 0 otherwise.
a(n) = A289813(A292243(n)).
Other identities. For all n >= 1:
a(A048673(n)) = A292248(n).
a(n) + A292244(n) = A064216(n).
a(n) AND A292244(n) = a(n) AND A292246(n) = 0, where AND is a bitwise-AND (A004198).

A246373 Primes p such that if 2p-1 = product_{k >= 1} A000040(k)^(c_k), then p <= product_{k >= 1} A000040(k-1)^(c_k).

Original entry on oeis.org

2, 3, 7, 19, 29, 31, 37, 47, 67, 71, 79, 89, 97, 101, 103, 107, 109, 127, 139, 151, 157, 181, 191, 197, 199, 211, 223, 227, 229, 241, 251, 269, 271, 277, 283, 307, 317, 331, 337, 349, 359, 367, 373, 379, 397, 409, 421, 433, 439, 457, 461, 467, 487, 499, 521, 541, 547, 569, 571, 577, 601

Offset: 1

Antti Karttunen, Aug 25 2014

nonn

Primes p such that A064216(p) >= p, or equally, A064989(2p-1) >= p.

All primes of A005382 are present here, because if 2p-1 is prime q, Bertrand's postulate guarantees (after cases 2 and 3 which are in A048674) that there exists at least one prime r larger than p and less than q = 2p-1, for which A064989(q) = r.

			2 is present, as 2*2 - 1 = 3 = p_2, and p_{2-1} = p_1 = 2 >= 2.
3 is present, as 2*3 - 1 = 5 = p_3, and p_{3-1} = p_2 = 3 >= 3.
5 is not present, as 2*5 - 1 = 9 = p_2 * p_2, and p_1 * p_1 = 4, with 4 < 5.
7 is present, as 2*7 - 1 = 13 = p_6, and p_5 = 11 >= 7.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Wikipedia, Bertrand's postulate

Intersection of A000040 and A246372.
Subsequence: A005382.
A246374 gives the primes not here.
Cf. A064216, A064989, A246281.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
n = 0; forprime(p=2,2^31, if((A064989((2*p)-1) >= p), n++; write("b246373.txt", n, " ", p); if(n > 9999, break)));
(Scheme, with Antti Karttunen's IntSeq-library)
(define A246373 (MATCHING-POS 1 1 (lambda (n) (and (prime? n) (>= (A064216 n) n)))))

A249734 Even bisection of A003961: Replace in 2n each prime factor p(k) with prime p(k+1).

Original entry on oeis.org

3, 9, 15, 27, 21, 45, 33, 81, 75, 63, 39, 135, 51, 99, 105, 243, 57, 225, 69, 189, 165, 117, 87, 405, 147, 153, 375, 297, 93, 315, 111, 729, 195, 171, 231, 675, 123, 207, 255, 567, 129, 495, 141, 351, 525, 261, 159, 1215, 363, 441, 285, 459, 177, 1125, 273, 891, 345, 279, 183, 945, 201, 333, 825, 2187, 357, 585, 213, 513, 435, 693, 219, 2025, 237, 369

Offset: 1

Antti Karttunen, Nov 23 2014

nonn

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

Row 2 of A246278.
Cf. A249735 (the other bisection of A003961).
Cf. A016945, A048673, A064216, A064989, A249827, A243501, A243502.
Cf. also A000079, A000244.

a(n) = A003961(2*n).
a(n) = 3 * A003961(n).
a(n) = A064989(A249827(n)).
a(n) = A003961(A243501(A064216(n))).
a(n) = A003961(A243502(A048673(n))).
a(n) = A016945(A048673(n)-1). [Permutation of A016945, 6n+3.]
Other identities. For all n >= 1:
a(A000079(n-1)) = A000244(n). [Maps each 2^n to 3^(n+1).]

A292243 a(1) = 1; for n > 1, a(n) = 3*a(A253889(n)) + (n mod 3).

Original entry on oeis.org

1, 5, 3, 16, 17, 9, 49, 11, 33, 160, 50, 156, 52, 53, 147, 88, 29, 27, 82, 149, 474, 457, 35, 453, 106, 101, 441, 151, 482, 303, 265, 152, 483, 250, 470, 1449, 1441, 158, 480, 1429, 161, 1407, 469, 443, 1371, 298, 266, 318, 1348, 89, 969, 961, 83, 954, 910, 248, 897, 268, 449, 1455, 322, 1424, 99, 808, 1373, 738, 1366, 107

Offset: 1

Antti Karttunen, Sep 15 2017

a(n) encodes in its base-3 representation the succession of modulo 3 residues obtained when map x -> A253889(x), starting from x=n, is iterated down to the eventual 1.

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

Cf. A010872, A048673, A064216, A253889, A292244, A292245, A292246.

Cf. also A292384.

f[n_] := Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1]; g[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; a[1] = 1; a[n_] := a[n] = 3 a[Floor@ g[Floor[f[n]/2]]] + Mod[n, 3]; Array[a, 68] (* Michael De Vlieger, Sep 16 2017 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A048673(n) = (A003961(n)+1)/2;
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A064216(n) = A064989((2*n)-1);
A253889(n) = if(1==n,n,A048673(A064216(n)\2));
A292243(n) = if(1==n,n,((n%3) + 3*A292243(A253889(n))));

;; With memoization-macro definec.
(definec (A292243 n) (if (= 1 n) n (+ (modulo n 3) (* 3 (A292243 (A253889 n))))))

a(1) = 1; for n > 1, a(n) = 3*a(A253889(n)) + A010872(n).

A245707 Permutation of natural numbers, the odd bisection of A245605 incremented by one and halved: a(n) = (1+A245605((2*n)-1))/2.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 9, 6, 19, 13, 8, 35, 11, 10, 17, 21, 18, 27, 139, 16, 23, 33, 14, 555, 51, 22, 37, 75, 36, 29, 105, 26, 67, 147, 278, 71, 165, 38, 53, 587, 12, 107, 83, 28, 25, 2219, 72, 43, 73, 20, 87, 41, 34, 291, 277, 210, 163, 31, 66, 149, 131, 330, 15, 229, 24, 39, 2347, 70, 49, 101

Offset: 1

Antti Karttunen, Jul 30 2014

nonn

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

Inverse: A245708.

Cf. A064216, A064989, A245605, A245607, A245706.

A064989(n) = my(f = factor(n)); for(i=1, #f~, if((2 == f[i,1]),f[i,1] = 1,f[i,1] = precprime(f[i,1]-1))); factorback(f);
A245605(n) = if(1==n, 1, if(0==(n%2), 2*A245605(A064989(n-1)), 1+(2*A245605(A064989(n)-1))));
A245707(n) = (1+A245605((2*n)-1))/2;

(define (A245707 n) (* (/ 1 2) (+ 1 (A245605 (-1+ (* 2 n))))))

(define (A245707 n) (if (= 1 n) n (+ 1 (A245605 (-1+ (A064216 n))))))

a(n) = (1+A245605((2*n)-1))/2.
As a composition of related permutations:
a(1) = 1, and for n > 1, a(n) = 1+A245605(A064216(n)-1).
a(n) = A245706(A245607(n)).

A246380 Permutation of natural numbers: a(1) = 1, a(2n) = nthcomposite(a(n)), a(2n-1) = nthprime(a(A064989(2n-1)-1)), where nthprime = A000040, nthcomposite = A002808, and A064989(n) shifts the prime factorization of n one step towards smaller primes.

Original entry on oeis.org

1, 4, 2, 9, 7, 6, 23, 16, 3, 14, 13, 12, 43, 35, 17, 26, 37, 8, 101, 24, 5, 22, 19, 21, 53, 62, 83, 51, 79, 27, 233, 39, 191, 54, 149, 15, 103, 134, 11, 36, 47, 10, 151, 34, 41, 30, 29, 33, 73, 75, 241, 86, 113, 114, 89, 72, 1153, 108, 443, 40, 593, 296, 547, 56, 167, 245, 173, 76, 563, 194, 1553, 25

Offset: 1

Antti Karttunen, Aug 29 2014

nonn

Has an infinite number of infinite cycles. See comments in A246379.

Antti Karttunen, Table of n, a(n) for n = 1..3098
Index entries for sequences that are permutations of the natural numbers

Inverse: A246379.
Similar or related permutations: A246376, A246378, A246363, A246364, A246366, A246368, A064216, A246682.
Cf. A000035, A000040, A002808, A010051, A064989.

default(primelimit,(2^31)+(2^30));
A002808(n) = { my(k=-1); while( -n + n += -k + k=primepi(n), ); n }; \\ This function from M. F. Hasler
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A246380(n) = if(1==n, 1, if(!(n%2), A002808(A246380(n/2)), prime(A246380(A064989(n)-1))));
for(n=1, 3098, write("b246380.txt", n, " ", A246380(n)));
(Scheme, with memoization-macro definec)
(definec (A246380 n) (cond ((< n 2) n) ((even? n) (A002808 (A246380 (/ n 2)))) (else (A000040 (A246380 (- (A064989 n) 1))))))

a(1) = 1, a(2n) = nthcomposite(a(n)), a(2n-1) = nthprime(a(A064989(2n-1)-1)), where nthprime = A000040, nthcomposite = A002808, and A064989(n) shifts the prime factorization of n one step towards smaller primes.
As a composition of related permutations:
a(n) = A246378(A246376(n)).
Other identities. For all n > 1 the following holds:
A010051(a(n)) = A000035(n). [Maps odd numbers larger than one to primes, and even numbers to composites, in some order. Permutations A246378 & A246682 have the same property].

cp's OEIS Frontend

A349398 Dirichlet convolution of A048673 with the Dirichlet inverse of its inverse permutation.

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A245449 Fixed points of A245447 and A245448.

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A246376 Permutation of natural numbers: a(1) = 1, a(2n) = 2 * a(n), a(2n-1) = 1 + (2 * a(A064989(2n-1)-1)).

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A250472 Permutation of natural numbers: a(n) = A250470(2*n - 1).

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A292245 Base-2 expansion of a(n) encodes the steps where numbers of the form 3k+1 are encountered when map x -> A253889(x) is iterated down to 1, starting from x=n.

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A246373 Primes p such that if 2p-1 = product_{k >= 1} A000040(k)^(c_k), then p <= product_{k >= 1} A000040(k-1)^(c_k).

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A249734 Even bisection of A003961: Replace in 2n each prime factor p(k) with prime p(k+1).

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A292243 a(1) = 1; for n > 1, a(n) = 3*a(A253889(n)) + (n mod 3).

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A245707 Permutation of natural numbers, the odd bisection of A245605 incremented by one and halved: a(n) = (1+A245605((2*n)-1))/2.

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