A286454 -id:A286454 - cp's OEIS frontend

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.

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)).

A318890 Filter sequence combining the prime signature of n (A046523) with the prime signature of its conjugated prime factorization (A278221).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 10, 15, 16, 12, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 14, 33, 34, 35, 36, 37, 38, 39, 40, 41, 18, 42, 43, 44, 45, 18, 46, 47, 48, 22, 31, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 39, 63, 64, 65, 66, 18, 67, 20, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 53, 36, 80, 81, 82, 83, 84, 85, 26, 86, 87, 88, 89, 90, 91, 39

Offset: 1

Antti Karttunen, Sep 16 2018

nonn

Restricted growth sequence transform of A286454.

For all i, j: a(i) = a(j) => A318891(i) = A318891(j).

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

Cf. A046523, A101296, A278221, A286454, A286621, A318891.

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; };
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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278221(n) = A046523(A122111(n));
A318890aux(n) = [A046523(n), A278221(n)];
v318890 = rgs_transform(vector(up_to,n,A318890aux(n)));
A318890(n) = v318890[n];

A286456 Compound filter: a(n) = P(A056239(n), A243503(n)), with a(1) = 0, where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 1, 5, 5, 13, 18, 25, 13, 19, 40, 41, 40, 61, 71, 41, 25, 85, 41, 113, 83, 72, 111, 145, 71, 42, 160, 42, 142, 181, 84, 221, 41, 112, 218, 73, 72, 265, 285, 161, 142, 313, 143, 365, 217, 85, 361, 421, 111, 74, 85, 219, 308, 481, 73, 113, 238, 286, 446, 545, 143, 613, 540, 144, 61, 162, 218, 685, 415, 362, 144, 761, 112, 841, 643, 86, 538, 114, 309, 925, 217

Offset: 1

Antti Karttunen, May 14 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192 (computed by using the b-files of A056239 and A243503)
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A056239, A243503, A278520, A286454, A286455.

(define (A286456 n) (if (= 1 n) 0 (* (/ 1 2) (+ (expt (+ (A056239 n) (A243503 n)) 2) (- (A056239 n)) (- (* 3 (A243503 n))) 2))))

a(1) = 0 and for n > 1, a(n) = (1/2)*(2 + ((A056239(n)+A243503(n))^2) - A056239(n) - 3*A243503(n)).

A289628 Compound filter (for the structure of the multiplicative group of integers modulo n & prime signature of n): a(n) = P(A289626(n), A101296(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 5, 8, 9, 12, 14, 41, 19, 18, 27, 50, 35, 25, 63, 99, 54, 40, 65, 86, 102, 42, 90, 203, 134, 52, 101, 131, 135, 128, 152, 342, 228, 75, 250, 221, 230, 88, 250, 399, 275, 182, 299, 271, 295, 117, 324, 517, 323, 185, 403, 295, 377, 146, 462, 623, 525, 168, 495, 549, 527, 187, 698, 728, 663, 343, 629, 460, 738, 370, 702, 889, 740, 273, 523, 590, 858, 370

Offset: 1

Antti Karttunen, Jul 19 2017

nonn

Here, instead of A046523 and A289625 we use as the components of a(n) their rgs-versions A101296 and A289626 because of the latter sequence's more moderate growth rate.
For all i, j: a(i) = a(j) => A286160(i) = A286160(j).
For all i, j: a(i) = a(j) => A289622(i) = A289622(j).

Antti Karttunen, Table of n, a(n) for n = 1..65536

Cf. A000027, A046523, A101296, A286160, A289622, A289626, A286454.

(define (A289628 n) (* (/ 1 2) (+ (expt (+ (A289626 n) (A101296 n)) 2) (- (A289626 n)) (- (* 3 (A101296 n))) 2)))

a(n) = (1/2)*(2 + ((A289626(n)+A101296(n))^2) - A289626(n) - 3*A101296(n)).

cp's OEIS Frontend

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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A318890 Filter sequence combining the prime signature of n (A046523) with the prime signature of its conjugated prime factorization (A278221).

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A286456 Compound filter: a(n) = P(A056239(n), A243503(n)), with a(1) = 0, where P(n,k) is sequence A000027 used as a pairing function.

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A289628 Compound filter (for the structure of the multiplicative group of integers modulo n & prime signature of n): a(n) = P(A289626(n), A101296(n)), where P(n,k) is sequence A000027 used as a pairing function.

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Formula