A243071 -id:A243071 - cp's OEIS frontend

A364498 Odd numbers k such that k divides A243071(k).

1, 3, 43, 1177, 3503, 49477, 169413, 428015, 4394113, 33228911

Offset: 1

Antti Karttunen, Jul 27 2023

Primes p present are those that occur as factors of (2^A000720(p))-1: 3, 43, 49477, 4394113, 33228911, ...

			1177 = 11 * 107, with A243071(1177) = 536870895 = 3*5*11*47*107*647, therefore 1177 is present. Note that 536870895 = 11111111111111111111111101111 in binary, with four 1-bits at the least significant end, followed by 0, and then 24 more 1-bits at the most significant end, so A163511(536870895) = A000040(1+4) * A000040(4+24) = 11 * 107.

Index entries for sequences computed from indices in prime factorization

Odd terms in A364497.

Cf. A000720, A163511, A243071, A364256.

A054429(n) = ((3<<#binary(n\2))-n-1); \\ From A054429
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A243071(n) = if(n<=2, n-1, A054429(A156552(n)));
isA364498(n) = ((n%2)&&!(A243071(n)%n));

a(10) from Chai Wah Wu, Jul 27 2023

A297161 Restricted growth sequence transform of A297171, which is Möbius transform of A243071.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 27 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A243071, A297171.

Cf. also A297113, A297157, A297162.

up_to = 8192;
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
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)};
A243071(n) = if(n<=2, n-1, if(!(n%2), 2*A243071(n/2), 1+(2*A243071(A064989(n)))));
A297171(n) = sumdiv(n,d,moebius(n/d)*A243071(d));
write_to_bfile(1,rgs_transform(vector(up_to,n,A297171(n))),"b297161.txt");

A334859 a(n) = A243071(A225546(n)).

Original entry on oeis.org

0, 1, 2, 3, 8, 4, 128, 6, 5, 16, 32768, 12, 2147483648, 256, 32, 7, 9223372036854775808, 10, 170141183460469231731687303715884105728, 48, 512, 65536, 57896044618658097711785492504343953926634992332820282019728792003956564819968, 24, 17, 4294967296, 20, 768

Offset: 1

Antti Karttunen, Jun 08 2020

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse permutation of A334860. Composition of permutations A225546 and A243071, and also of A054429 and A334865.

Cf. A299090, A334871.

a(n) = A243071(A225546(n)).
a(n) = A054429(A334865(n)).
For n >= 1, A000120(a(n)) = A299090(n).
For n > 1, A070939(a(n)) = A334871(n).

A364256 a(n) = gcd(n, A243071(n)).

Original entry on oeis.org

1, 1, 3, 2, 1, 6, 1, 4, 1, 2, 1, 12, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 24, 1, 2, 9, 4, 1, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 43, 4, 5, 2, 1, 48, 1, 2, 1, 4, 1, 18, 1, 8, 1, 2, 1, 4, 1, 2, 3, 32, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 3, 4, 11, 2, 1, 16, 1, 2, 1, 4, 1, 86, 1, 8, 1, 10, 7, 4, 1, 2, 1, 96, 1, 2, 11, 4

Offset: 1

Antti Karttunen, Jul 17 2023

nonn

Primes p such that a(p) = p are those that occur as factors of (2^A000720(p))-1: 3, 43, 49477. Are there any more of them?

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

Cf. A243071.

Cf. also A364254, A364255.

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)};
A243071(n) = if(n<=2, n-1, if(!(n%2), 2*A243071(n/2), 1+(2*A243071(A064989(n)))));
A364256(n) = gcd(n, A243071(n));

A364290 Numbers k such that A243071(k) < k.

Original entry on oeis.org

1, 2, 4, 8, 9, 15, 16, 18, 25, 27, 30, 32, 35, 36, 45, 49, 50, 54, 60, 63, 64, 70, 72, 75, 77, 81, 90, 98, 100, 105, 108, 120, 121, 125, 126, 128, 135, 140, 143, 144, 147, 150, 154, 162, 165, 169, 175, 180, 189, 196, 200, 210, 216, 225, 231, 240, 242, 243, 245, 250, 252, 256, 270, 273, 275, 280, 286, 288, 289, 294

Offset: 1

Antti Karttunen, Jul 25 2023

nonn

If k is present, then 2*k is also present, and vice versa.

361 = 19^2 is the first square that is not present in this sequence.

Positions of strictly positive terms in A364288.
Subsequence of A364291.
Cf. A243071, A364289 (complement).
Cf. also A364287.

A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A243071(n) = if(n<=2, n-1, if(!(n%2), 2*A243071(n/2), 1+(2*A243071(A064989(n)))));
isA364290(n) = (A243071(n)

A253892 Permutation of natural numbers: a(n) = A243071(A245612(n)).

Original entry on oeis.org

0, 1, 7, 3, 30, 63, 4, 2, 8191, 57, 510, 11, 511, 10, 31, 6, 524286, 36893488147419103231, 131068, 65532, 1073741823, 16381, 8190, 262143, 508, 248, 65535, 125, 16, 60, 127, 15, 4194299, 633825300114114700748351602685, 2097134, 200867255532373784442745261542645325315275374222849104412671, 10141204801825835211973625643007, 442, 268435451, 32754, 190

Offset: 0

Antti Karttunen, Jan 17 2015

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse: A253891.

Cf. A243071, A245612, A054429, A253792, A253884.

(define (A253892 n) (A243071 (A245612 n)))

a(n) = A243071(A245612(n)).
As a composition of other related permutations:
a(n) = A054429(A253792(A054429(n))).

A332811 a(n) = A243071(A332808(n)).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 15, 4, 5, 14, 63, 12, 31, 30, 13, 8, 127, 10, 255, 28, 29, 126, 1023, 24, 11, 62, 9, 60, 511, 26, 4095, 16, 125, 254, 27, 20, 2047, 510, 61, 56, 8191, 58, 16383, 252, 25, 2046, 65535, 48, 23, 22, 253, 124, 32767, 18, 123, 120, 509, 1022, 262143, 52, 131071, 8190, 57, 32, 59, 250, 1048575, 508, 2045, 54, 4194303, 40, 524287, 4094, 21

Offset: 1

Antti Karttunen, Mar 05 2020

nonn

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

Cf. A332817 (inverse permutation).
Cf. A054429, A243071, A332808, A332816, A332895, A332896, A332899.
Cf. also A332215.

up_to = 26927;
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)};
A243071(n) = if(n<=2, n-1, if(!(n%2), 2*A243071(n/2), 1+(2*A243071(A064989(n)))));
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); };
A332811(n) = A243071(A332808(n));

a(n) = A243071(A332808(n)).
For n > 1, a(n) = A054429(A332816(n)).
a(n) = A332895(n) + A332896(n).
a(n) = A332895(n) OR A332896(n) = A332895(n) XOR A332896(n).
A000120(a(n)) = A332899(n).

A364291 Numbers k such that A243071(k) <= k.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 15, 16, 18, 24, 25, 27, 30, 32, 35, 36, 45, 48, 49, 50, 54, 60, 63, 64, 70, 72, 75, 77, 81, 90, 96, 98, 100, 105, 108, 120, 121, 125, 126, 128, 135, 140, 143, 144, 147, 150, 154, 162, 165, 169, 175, 180, 189, 192, 196, 200, 210, 216, 225, 231, 240, 242, 243, 245, 250, 252, 256, 270, 273

Offset: 1

Antti Karttunen, Jul 25 2023

nonn

If k is present, then 2*k is also present, and vice versa.

Positions of nonnegative terms in A364288.
Cf. A007283, A364290 (subsequences).
Cf. A243071.
Cf. also A364287.

A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A243071(n) = if(n<=2, n-1, if(!(n%2), 2*A243071(n/2), 1+(2*A243071(A064989(n)))));
isA364291(n) = (A243071(n)<=n);

A366276 Permutation of nonnegative integers: a(n) = A057889(A243071(n)).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 15, 4, 5, 14, 31, 12, 63, 30, 11, 8, 127, 10, 255, 28, 23, 62, 511, 24, 13, 126, 9, 60, 1023, 22, 2047, 16, 47, 254, 27, 20, 4095, 510, 95, 56, 8191, 46, 16383, 124, 19, 1022, 32767, 48, 29, 26, 191, 252, 65535, 18, 55, 120, 383, 2046, 131071, 44, 262143, 4094, 39, 32, 111, 94, 524287, 508, 767, 54

Offset: 1

Antti Karttunen, Oct 06 2023

nonn

Cf. A057889, A243071, A366275 (inverse map), A366277 (fixed points of map n -> a(n)).

A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2)));
A243071(n) = if(n<=2, n-1, my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p*p2*(2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); ((3<<#binary(res\2))-res-1)); \\ (Combining programs given in A156552 and A054429)
A366276(n) = A057889(A243071(n));

A275715 Permutation of natural numbers: a(n) = A243071(A249823(n)).

Original entry on oeis.org

0, 1, 3, 7, 15, 31, 63, 127, 2, 255, 511, 6, 1023, 2047, 4095, 8191, 5, 16383, 14, 32767, 65535, 30, 131071, 262143, 524287, 13, 1048575, 2097151, 62, 4194303, 29, 126, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455, 254, 61, 11, 4, 536870911, 1073741823, 125, 2147483647, 4294967295, 27, 510, 8589934591

Offset: 1

Antti Karttunen, Aug 06 2016

nonn

Note the indexing: the domain starts from 1, while the range includes also zero.

Antti Karttunen, Table of n, a(n) for n = 1..512
Index entries for sequences that are permutations of the natural numbers

Inverse: A275716.
Related or similar permutations: A243071, A249823, A245611.
Cf. also A273664, A273669.

(define (A275715 n) (A243071 (A249823 n)))

a(n) = A243071(A249823(n)).

cp's OEIS Frontend

A364498 Odd numbers k such that k divides A243071(k).

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A297161 Restricted growth sequence transform of A297171, which is Möbius transform of A243071.

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A334859 a(n) = A243071(A225546(n)).

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A364256 a(n) = gcd(n, A243071(n)).

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A364290 Numbers k such that A243071(k) < k.

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A253892 Permutation of natural numbers: a(n) = A243071(A245612(n)).

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A332811 a(n) = A243071(A332808(n)).

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A364291 Numbers k such that A243071(k) <= k.

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A366276 Permutation of nonnegative integers: a(n) = A057889(A243071(n)).

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A275715 Permutation of natural numbers: a(n) = A243071(A249823(n)).

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