A246277 -id:A246277 - cp's OEIS frontend

A246677 Permutation of natural numbers: a(1) = 1, a(2n) = A000079(A055396(2n+1)-1) * ((2A246277(2n+1))-1), a(2n+1) = 1 + 2a(n).

1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 17, 10, 15, 64, 13, 128, 19, 18, 33, 256, 23, 12, 65, 14, 35, 512, 21, 1024, 31, 26, 129, 20, 27, 2048, 257, 42, 39, 4096, 37, 8192, 67, 22, 513, 16384, 47, 24, 25, 50, 131, 32768, 29, 36, 71, 66, 1025, 65536, 43, 131072, 2049, 38, 63, 52, 53, 262144, 259, 74, 41

Offset: 1

Antti Karttunen, Sep 01 2014

nonn

See the comments in A246675. This is otherwise similar permutation, except for odd numbers, which are here recursively permuted by the emerging permutation itself. The even bisection halved gives A246679, the odd bisection from a(3) onward with one subtracted and then halved gives this sequence back.

Antti Karttunen, Table of n, a(n) for n = 1..2048
Index entries for sequences that are permutations of the natural numbers

Inverse: A246678. Variants: A246675, A246683.
Even bisection halved: A246679.
Cf. A000079, A055396, A246277.
a(n) differs from A156552(n+1) for the first time at n=32, where a(32) = 26, while A156552(33) = 34.

a(1) = 1, a(2n) = A000079(A055396(2n+1)-1) * ((2*A246277(2n+1))-1), a(2n+1) = 1 + 2*a(n).

A250247 Permutation of natural numbers: a(1) = 1, a(n) = A083221(a(A055396(n)),A246277(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 39, 34, 35, 36, 37, 38, 63, 40, 41, 42, 43, 44, 33, 46, 47, 48, 49, 50, 75, 52, 53, 54, 65, 56, 99, 58, 59, 60, 61, 62, 57, 64, 95, 66, 67, 68, 111, 70, 71, 72, 103, 74, 51, 76, 77, 78, 79, 80, 45, 82

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

The first 7-cycle occurs at: (33 39 63 57 99 81 45), which is mirrored at the cycle (137 167 307 269 523 419 197), consisting of primes (p_33, p_39, p_63, ...).

			As a(21) = 27, and A000040(21) = 73 and A000040(27) = 103, a(73) = 103.

Antti Karttunen, Table of n, a(n) for n = 1..4240
Index entries for sequences that are permutations of the natural numbers

Inverse: A250248.
Cf. A000040, A005843, A049076, A049084, A055396, A078442, A083221, A246277.
Differs from its inverse A250248 for the first time at n = 33, where a(33) = 39, while A250248(33) = 45.
Differs from the "vanilla version" A249817 for the first time at n=73, where a(73) = 103, while A249817(73) = 73.
Differs from "doubly recursed" version A250249 for the first time at n=42, where a(42) = 42, while A250249(42) = 54, thus the first prime where they get different values is p_42 = 181, where a(181) = 181, while A250249(181) = 251 = p_54.

a(1) = 1, a(n) = A083221(a(A055396(n)),A246277(n)).
Other identities. For all n >= 1:
a(A005843(n)) = A005843(n). [Fixes even numbers].
a(p_n) = p_{a(n)}, or equally, a(n) = A049084(a(A000040(n))). [Restriction to primes induces the same sequence].
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves the "order of primeness of n"].

A300247 Restricted growth sequence transform of A286457(n), filter combining A078898(n) and A246277(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 03 2018

For all i, j:
  a(i) = a(j) => A280492(i) = A280492(j).
  a(i) = a(j) => A300248(i) = A300248(j).
The latter follows because A046523(n) = A046523(2*A246277(n)).

			a(65) = a(119) (= 42) because A078898(65) = A078898(119) = 5 (both numbers occur in column 5 of A083221) and because A246277(65) = A246277(119) = 7 (both numbers occur in column 7 of A246278). Note that 65 = 5*13 = prime(3)*prime(6) and 119 = 7*17 = prime(4)*prime(7) = A003961(65). A246277(n) contains complete information about the (relative) differences between prime indices in the prime factorization of n.

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

Cf. A078898, A246277, A286457.

Cf. also A300248, A280492.

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,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])); }
A020639(n) = { if(1==n,n,vecmin(factor(n)[, 1])); };
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)};
A078898(n) = { if(n<=1,n, my(spf=A020639(n),k=1,m=n/spf); while(m>1,if(A020639(m)>=spf,k++); m--); (k)); }; \\ Antti Karttunen, Mar 03 2018
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };
A286457(n) = if(1==n,0,(1/2)*(2 + ((A078898(n)+A246277(n))^2) - A078898(n) - 3*A246277(n)));
write_to_bfile(1,rgs_transform(vector(65537,n,A286457(n))),"b300247.txt");

A249825 Permutation of natural numbers: a(n) = A246277(A084968(n)).

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 4, 41, 43, 47, 53, 59, 61, 6, 67, 71, 73, 10, 79, 83, 89, 97, 101, 103, 14, 9, 107, 109, 22, 113, 127, 15, 131, 137, 139, 26, 149, 151, 25, 157, 163, 167, 21, 173, 179, 181, 191, 34, 33, 193, 38, 35, 197, 199, 211, 223, 227, 229, 55, 233, 39, 239, 46, 241, 251, 257, 263, 269, 271, 58, 49

Offset: 1

Antti Karttunen, Dec 06 2014

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A249826.
Row 4 of A249821.
Cf. A084968, A246277, A251721, A064216, A249823, A249745, A250475.

(define (A249825 n) (A246277 (A084968 n)))

a(n) = A246277(A084968(n)).
As a composition of other permutations:
a(n) = A249823(A250475(n)).
a(n) = A064216(A249745(A250475(n))). [Composition of the first three rows of array A251721.]

A329620 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A046523(n), A246277(A324886(n))].

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 4, 2, 7, 2, 4, 8, 9, 2, 10, 2, 7, 8, 4, 2, 11, 12, 4, 13, 7, 2, 14, 2, 15, 8, 4, 16, 17, 2, 4, 8, 18, 2, 19, 2, 7, 20, 4, 2, 21, 22, 23, 8, 7, 2, 24, 25, 26, 8, 4, 2, 27, 2, 4, 28, 29, 30, 31, 2, 7, 8, 32, 2, 33, 2, 4, 34, 7, 35, 31, 2, 36, 37, 4, 2, 38, 39, 4, 8, 26, 2, 40, 41, 7, 8, 4, 42, 43, 2, 44, 45, 46, 2, 31, 2, 26, 47

Offset: 1

Antti Karttunen, Nov 18 2019

nonn

Restricted growth sequence transform of the ordered pair [A046523(n), A246277(A324886(n))].
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A101296(i) = A101296(j),
  a(i) = a(j) => A329345(i) = A329345(j),
  a(i) = a(j) => A329618(i) = A329618(j),
  a(i) = a(j) => A329619(i) = A329619(j).

Cf. A046523, A034386, A108951, A246277, A276086, A305800, A324886, A329345, A329618, A329619.

up_to = 8192;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(n));
A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2);
Aux329620(n) = [A046523(n), A246277(A324886(n))];
v329620 = rgs_transform(vector(up_to, n, Aux329620(n)));
A329620(n) = v329620[n];

A280492 a(1) = 0; for n > 1, a(n) = A246277(n) - A078898(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 7, 0, 0, 0, 0, 0, -1, 0, 2, 0, 0, 0, 7, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, -6, 0, 0, 0, 5, 0, 8, 0, 0, 0, 1, 0, 13, 0, 6, 0, 0, 0, -3, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 12, 0, 0, 0, 9, 0, 2, 0, 2

Offset: 1

Antti Karttunen, Jan 09 2017

sign

For n > 1, a(n) gives the difference of column positions of n's location in arrays A246278 and A083221. Note that any n occurs on the same row in both arrays.

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

Cf. A078898, A083221, A246277, A246278, A249820.

Cf. also A278523, A279350, A280497, A280498.

(define (A280492 n) (if (= 1 n) 0 (- (A246277 n) (A078898 n))))

a(1) = 0; for n > 1, a(n) = A246277(n) - A078898(n).

A304728 Restricted growth sequence transform of A246277(A303751(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 19 2018

nonn

For all i, j: a(i) = a(j) => A304729(i) = A304729(j).

Antti Karttunen, Table of n, a(n) for n = 1..76503
Index entries for sequences computed from indices in prime factorization

Cf. A246277, A303751, A304729, A304731.

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)};
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
v304728 = rgs_transform(vector(76503,n,A246277(A303751(n)))); \\ Needs also code from A303751
A304728(n) = v304728[n];

A305796 Dirichlet convolution of A246277 with itself.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 4, 1, 2, 0, 10, 0, 2, 2, 12, 0, 10, 0, 14, 2, 2, 0, 32, 1, 2, 4, 18, 0, 22, 0, 32, 2, 2, 2, 47, 0, 2, 2, 48, 0, 30, 0, 26, 10, 2, 0, 88, 1, 14, 2, 30, 0, 38, 2, 64, 2, 2, 0, 104, 0, 2, 14, 80, 2, 42, 0, 38, 2, 30, 0, 148, 0, 2, 10, 42, 2, 54, 0, 136, 12, 2, 0, 144, 2, 2, 2, 96, 0, 98, 2, 50, 2, 2, 2, 224, 0, 18, 18, 103, 0, 66, 0, 112, 22

Offset: 1

Antti Karttunen, Jun 13 2018

nonn

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

Cf. A246277, A305791, A305798.

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)};
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };
A305796(n) = sumdiv(n,d,A246277(d)*A246277(n/d));

a(n) = Sum_{d|n} A246277(d)*A246277(n/d).

A329038 a(n) = A246277(A276086(n)).

Original entry on oeis.org

0, 1, 1, 3, 2, 9, 1, 5, 3, 15, 6, 45, 2, 25, 9, 75, 18, 225, 4, 125, 27, 375, 54, 1125, 8, 625, 81, 1875, 162, 5625, 1, 7, 5, 21, 10, 63, 3, 35, 15, 105, 30, 315, 6, 175, 45, 525, 90, 1575, 12, 875, 135, 2625, 270, 7875, 24, 4375, 405, 13125, 810, 39375, 2, 49, 25, 147, 50, 441, 9, 245, 75, 735, 150, 2205, 18, 1225, 225, 3675, 450, 11025, 36

Offset: 0

Antti Karttunen, Nov 08 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..2310
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..32768
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to primorial base

Cf. A046523, A246277, A276086, A278226, A329048 (rgs-transform).

Cf. also A329345.

A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2);
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A329038(n) = A246277(A276086(n));

a(n) = A246277(A276086(n)).

For n >= 1, A046523(2*a(n)) = A278226(n).

A342457 Terms of A342456 prime-shifted so far towards lower primes that they become even: a(n) = 2*A246277(A342456(n)).

Original entry on oeis.org

2, 2, 2, 4, 2, 4, 6, 6, 2, 4, 64, 16, 324, 36, 10, 36, 2, 4, 64, 16, 2304, 96, 486, 24, 7290, 104976, 21600, 1296, 1708593750000, 100, 93750, 10, 2, 4, 64, 16, 144, 6, 216, 6, 172186884, 7776, 2160, 216, 216000000, 236196, 10497600, 54, 10935000000000, 53144100, 1476225000000, 7290, 122500000000, 10935000, 140, 360

Offset: 0

Antti Karttunen, Mar 15 2021

nonn

These terms have the same prime signature as the corresponding terms in A342456, thus applying omega and bigomega to these gives the same derived sequences A342461 and A342462.

Cf. A246277, A276086, A283980, A329038, A329886, A342456, A342461 [= A001221(a(n))], A342462 [= A001222(a(n))], A342464 [= A051903(a(n))].

A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2);
A342457(n) = 2*A246277(A342456(n)); \\ Uses also code from A342456.

a(n) = 2*A246277(A342456(n)) = 2*A329038(A329886(n)).

cp's OEIS Frontend

A246677 Permutation of natural numbers: a(1) = 1, a(2n) = A000079(A055396(2n+1)-1) * ((2*A246277(2n+1))-1), a(2n+1) = 1 + 2*a(n).

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A250247 Permutation of natural numbers: a(1) = 1, a(n) = A083221(a(A055396(n)),A246277(n)).

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A300247 Restricted growth sequence transform of A286457(n), filter combining A078898(n) and A246277(n).

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A249825 Permutation of natural numbers: a(n) = A246277(A084968(n)).

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A329620 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A046523(n), A246277(A324886(n))].

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A280492 a(1) = 0; for n > 1, a(n) = A246277(n) - A078898(n).

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A304728 Restricted growth sequence transform of A246277(A303751(n)).

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A305796 Dirichlet convolution of A246277 with itself.

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A329038 a(n) = A246277(A276086(n)).

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A246677 Permutation of natural numbers: a(1) = 1, a(2n) = A000079(A055396(2n+1)-1) * ((2A246277(2n+1))-1), a(2n+1) = 1 + 2a(n).