A278221 -id:A278221 - cp's OEIS frontend

A278220 Filtering sequence (related to prime factorization): a(n) = A046523(A241909(n)).

1, 2, 4, 2, 8, 4, 16, 2, 6, 8, 32, 4, 64, 16, 12, 2, 128, 6, 256, 8, 24, 32, 512, 4, 12, 64, 6, 16, 1024, 12, 2048, 2, 48, 128, 36, 6, 4096, 256, 96, 8, 8192, 24, 16384, 32, 12, 512, 32768, 4, 24, 12, 192, 64, 65536, 6, 72, 16, 384, 1024, 131072, 12, 262144, 2048, 24, 2, 144, 48, 524288, 128, 768, 36, 1048576, 6, 2097152, 4096, 30, 256, 72, 96, 4194304, 8, 6, 8192

Offset: 1

Antti Karttunen, Nov 16 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..410

Cf. A046523, A075158, A241909.

Cf. also A278219, A278221.

(define (A278220 n) (A046523 (A241909 n)))

a(n) = A046523(A241909(n)).

a(n) = A278219(A075158(n-1)).

A336146 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000035(i) = A000035(j) and A000265(i) = A000265(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 12 2020

nonn

Restricted growth sequence transform of the ordered pair [A000035(n), A000265(n)] (parity and the odd part of n), or equally, of the ordered pair [A000265(n), A278221(n)].
For all i, j:
  A324400(i) = A324400(j) => a(i) = a(j),
  a(i) = a(j) => A336126(i) = A336126(j),
  a(i) = a(j) => A336147(i) = A336147(j),
  a(i) = a(j) => A336148(i) = A336148(j),
  a(i) = a(j) => A336149(i) = A336149(j).

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

Cf. A000035, A000265, A046523, A064989, A122111, A278221.

Cf. also A286621, A324400, A336126, A336147, A336148, A336149.

up_to = 65537;
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; };
A000035(n) = (n%2);
A000265(n) = (n>>valuation(n,2));
Aux336146(n) = [A000035(n), A000265(n)];
v336146 = rgs_transform(vector(up_to, n, Aux336146(n)));
A336146(n) = v336146[n];

A286454 Compound filter (prime signature & prime signature of conjugated prime factorization): a(n) = P(A101296(n), A286621(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 8, 9, 12, 32, 23, 20, 13, 49, 38, 51, 47, 82, 49, 35, 68, 51, 80, 72, 124, 140, 122, 74, 18, 175, 26, 111, 155, 334, 192, 65, 257, 280, 82, 116, 255, 329, 355, 99, 327, 570, 380, 177, 72, 469, 437, 132, 31, 72, 532, 216, 498, 74, 257, 144, 599, 634, 597, 448, 632, 745, 159, 119, 784, 1044, 782, 331, 907, 570, 863, 186, 905, 1039, 72, 384, 140, 1335, 1037

Offset: 1

Antti Karttunen, May 14 2017

nonn

Here, instead of A046523 and A278221 we use as the components of a(n) their rgs-versions A101296 and A286621 because of the latter sequence's moderate growth rates.

For all i, j: a(i) = a(j) => A286356(i) = A286356(j).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A046523, A101296, A122111, A278221, A285334, A286621, A286356, A286455, A286456.

(define (A286454 n) (* (/ 1 2) (+ (expt (+ (A101296 n) (A286621 n)) 2) (- (A101296 n)) (- (* 3 (A286621 n))) 2)))

a(n) = (1/2)*(2 + ((A101296(n)+A286621(n))^2) - A101296(n) - 3*A286621(n)).

A286455 Compound filter (smallest prime dividing n & prime signature of conjugated prime factorization): a(n) = P(A055396(n), A286621(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 2, 8, 2, 18, 11, 40, 2, 8, 22, 71, 11, 97, 46, 30, 2, 143, 11, 179, 22, 93, 92, 262, 11, 18, 121, 8, 46, 335, 154, 417, 2, 212, 211, 69, 11, 540, 254, 302, 22, 679, 326, 794, 92, 30, 379, 918, 11, 40, 22, 467, 121, 1051, 11, 234, 46, 530, 529, 1242, 154, 1344, 631, 93, 2, 744, 704, 1615, 211, 822, 326, 1790, 11, 1912, 904, 30, 254, 140, 947, 2167, 22, 8

Offset: 1

Antti Karttunen, May 14 2017

nonn

Note that as the other component of a(n) we use A286621 instead of A278221, because of latter sequence's unwieldy large terms.
For all i, j: a(i) = a(j) => A243055(i) = A243055(j).
For all i, j: a(i) = a(j) => A286470(i) = A286470(j).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A055396, A122111, A243055, A278221, A285334, A286621, A286454, A286456.

(define (A286455 n) (* (/ 1 2) (+ (expt (+ (A055396 n) (A286621 n)) 2) (- (A055396 n)) (- (* 3 (A286621 n))) 2)))

a(n) = (1/2)*(2 + ((A055396(n)+A286621(n))^2) - A055396(n) - 3*A286621(n)).

A279350 a(n) = A046523(A279352(n)).

Original entry on oeis.org

1, 2, 4, 2, 8, 6, 16, 2, 4, 12, 32, 6, 64, 24, 12, 2, 128, 6, 256, 12, 4, 48, 512, 6, 8, 96, 36, 24, 1024, 30, 2048, 2, 12, 192, 24, 6, 4096, 384, 72, 12, 8192, 60, 16384, 48, 4, 768, 32768, 6, 16, 12, 12, 96, 65536, 6, 8, 24, 36, 1536, 131072, 30, 262144, 3072, 144, 2, 72, 120, 524288, 192, 12, 60, 1048576, 6, 2097152, 6144, 288, 384, 48, 240, 4194304, 12, 72

Offset: 1

Antti Karttunen, Dec 12 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..736

Cf. A046523, A249818, A278221, A278523, A279352, A279354.

a(n) = A046523(A279352(n)).

a(n) = A278221(A249818(n)).

A285334 a(n) = A046523(A243505(n)).

Original entry on oeis.org

1, 2, 4, 8, 2, 16, 32, 6, 64, 128, 12, 256, 4, 2, 512, 1024, 24, 12, 2048, 48, 4096, 8192, 6, 16384, 8, 96, 32768, 36, 192, 65536, 131072, 12, 72, 262144, 384, 524288, 1048576, 6, 24, 2097152, 2, 4194304, 144, 768, 8388608, 72, 1536, 288, 16777216, 24, 33554432, 67108864, 30, 134217728, 268435456, 3072, 536870912, 576, 48, 216, 16, 6144, 4, 1073741824, 12288

Offset: 1

Antti Karttunen, Apr 24 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024

Cf. A046523, A064216, A122111, A243505, A278221.

(define (A285334 n) (A046523 (A243505 n)))

a(n) = A046523(A243505(n)).

a(n) = A278221(A064216(n)).

A278525 Filtering sequence (related to prime factorization): a(n) = A046523(A241916(n)).

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 8, 6, 4, 2, 8, 2, 4, 6, 16, 2, 12, 2, 8, 6, 4, 2, 16, 6, 4, 12, 8, 2, 12, 2, 32, 6, 4, 6, 24, 2, 4, 6, 16, 2, 12, 2, 8, 12, 4, 2, 32, 6, 12, 6, 8, 2, 36, 6, 16, 6, 4, 2, 24, 2, 4, 12, 64, 6, 12, 2, 8, 6, 12, 2, 48, 2, 4, 30, 8, 6, 12, 2, 32, 24, 4, 2, 24, 6, 4, 6, 16, 2, 36, 6, 8, 6, 4, 6, 64, 2, 12, 12, 24, 2, 12, 2, 16, 30, 4, 2, 72, 2

Offset: 1

Antti Karttunen, Nov 30 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000040, A046523, A241916, A278220, A278221.

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A209229(n) = (n && !bitand(n,n-1));
A241916(n) = if(1==A209229(n), n, my(f = factor(2*n), nbf = #f~, igp = primepi(f[nbf,1]), g = f); for(i=1,nbf,g[i,1] = prime(1+igp-primepi(f[i,1]))); factorback(g)/2);
A278525(n) = A046523(A241916(n)); \\ Antti Karttunen, Jul 02 2018

(define (A278525 n) (A046523 (A241916 n)))

a(n) = A046523(A241916(n)).
Other identities. For all n:
a(2^n) = 2^n.
a(A000040(n)) = 2.

A336925 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336147(1+sigma(i)) = A336147(1+sigma(j)), for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 5, 1, 6, 7, 4, 8, 9, 2, 2, 1, 7, 10, 11, 12, 13, 14, 2, 15, 1, 12, 16, 17, 18, 19, 13, 1, 3, 20, 3, 21, 22, 15, 17, 23, 12, 24, 9, 25, 26, 19, 3, 2, 27, 28, 19, 13, 20, 29, 19, 29, 5, 23, 15, 4, 11, 24, 30, 1, 25, 31, 32, 33, 24, 31, 19, 6, 9, 34, 2, 35, 24, 4, 5, 36, 37, 33, 25, 9, 38, 39, 29, 40, 23, 41, 42, 4, 43, 31, 29, 44, 13, 45, 46, 47, 48, 49, 30

Offset: 1

Antti Karttunen, Aug 10 2020

nonn

Restricted growth sequence transform of the function f(n) = A336147(A088580(n)).
For all i, j:
  A324400(i) = A324400(j) => a(i) = a(j),
  a(i) = a(j) => A336691(i) = A336691(j),
  a(i) = a(j) => A336924(i) = A336924(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. also A336926.

up_to = 65537;
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; };
A020639(n) = if(1==n, n, factor(n)[1, 1]);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A278221(n) = A046523(A122111(n));
Aux336147(n) = [A020639(n),A278221(n)];
v336925 = rgs_transform(vector(up_to, n, Aux336147(1+sigma(n))));
A336925(n) = v336925[n];

cp's OEIS Frontend

A278220 Filtering sequence (related to prime factorization): a(n) = A046523(A241909(n)).

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A336146 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000035(i) = A000035(j) and A000265(i) = A000265(j), for all i, j >= 1.

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A286454 Compound filter (prime signature & prime signature of conjugated prime factorization): a(n) = P(A101296(n), A286621(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A286455 Compound filter (smallest prime dividing n & prime signature of conjugated prime factorization): a(n) = P(A055396(n), A286621(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A279350 a(n) = A046523(A279352(n)).

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A285334 a(n) = A046523(A243505(n)).

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A278525 Filtering sequence (related to prime factorization): a(n) = A046523(A241916(n)).

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A336925 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336147(1+sigma(i)) = A336147(1+sigma(j)), for all i, j >= 1.

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