A278223 -id:A278223 - cp's OEIS frontend

A378223 Inverse Möbius transform of A345182.

1, 1, 2, 2, 2, 4, 2, 4, 4, 4, 2, 10, 2, 4, 6, 8, 2, 12, 2, 10, 6, 4, 2, 24, 4, 4, 8, 10, 2, 20, 2, 16, 6, 4, 6, 36, 2, 4, 6, 24, 2, 20, 2, 10, 16, 4, 2, 56, 4, 12, 6, 10, 2, 32, 6, 24, 6, 4, 2, 62, 2, 4, 16, 32, 6, 20, 2, 10, 6, 20, 2, 100, 2, 4, 16, 10, 6, 20, 2, 56, 16, 4, 2, 62, 6, 4, 6, 24, 2, 72, 6, 10, 6, 4, 6

Offset: 1

Antti Karttunen, Nov 25 2024

nonn

Apparently the Dirichlet convolution of A002131 and A323910. - Antti Karttunen, Nov 30 2024

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A002131, A323910, A345182, A378224 (Dirichlet inverse).
Cf. also A067824.
Odd bisection is not equal to A278223.

memoA345182 = Map();
A345182(n) = if(n<=2, n%2, my(v); if(mapisdefined(memoA345182,n,&v), v, v = sumdiv(n,d,if(dA345182(d),0)); mapput(memoA345182,n,v); (v)));
A378223(n) = sumdiv(n,d,A345182(d));

up_to = 20000;
A378223list(up_to_n) = { my(v=vector(up_to_n)); v[1] = 1; v[2] = 0; for(n=3,up_to_n,v[n] = 1+sumdiv(n,d,(dA378223list(up_to);
A378223(n) = v378223[n];

a(n) = Sum_{d|n} A345182(d).

For n > 2, a(n) = 2*A345182(n).

A292249 Compound filter (multiplicative order of 2 mod 2n+1 & prime signature of 2n+1): a(n) = P(A002326(n), A046523(2n+1)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 14, 9, 42, 65, 90, 40, 44, 189, 61, 77, 273, 318, 434, 20, 115, 148, 702, 148, 230, 119, 265, 299, 297, 86, 1430, 320, 271, 1769, 1890, 142, 148, 2277, 373, 665, 54, 485, 625, 819, 2400, 3485, 86, 556, 77, 148, 115, 856, 1224, 850, 5150, 1377, 832, 5777, 702, 856, 434, 1220, 265, 430, 6438, 320, 5771, 35, 185, 8645, 271

Offset: 0

Antti Karttunen, Oct 02 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A000027, A002326, A046523, A278223, A286573, A291769 (rgs-version of the same filter).

Cf. also A291755, A292268.

A002326(n) = if(n<0, 0, znorder(Mod(2, 2*n+1))); \\ This function from Michael Somos, Mar 31 2005
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A292249(n) = (1/2)*(2 + ((A002326(n)+A046523(n+n+1))^2) - A002326(n) - 3*A046523(n+n+1));

(define (A292249 n) (* 1/2 (+ (expt (+ (A002326 n) (A046523 (+ 1 n n))) 2) (- (A002326 n)) (- (* 3 (A046523 (+ 1 n n)))) 2)))

a(n) = (1/2)*(2 + ((A002326(n) + A046523(2n+1))^2) - A002326(n) - 3*A046523(2n+1)).

A286461 Compound filter (2-adic valuation of n & 4k+1,4k+3 prime-signature combination of 2n-1): a(n) = P(A001511(n), A286364((2*n)-1)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 4, 9, 22, 5, 4, 32, 4, 5, 121, 9, 46, 437, 4, 20, 121, 17, 4, 24, 4, 5, 67, 14, 22, 17, 4, 24, 121, 5, 4, 2562, 211, 5, 121, 9, 4, 107, 121, 14, 7261, 5, 211, 24, 4, 17, 121, 41, 4, 2280, 4, 9, 254, 5, 4, 32, 4, 17, 67, 24, 22, 17, 631, 35, 121, 5, 121, 783, 4, 5, 121, 32, 211, 2280, 4, 9, 67, 17, 4, 41, 121, 5, 254, 9, 46, 2280, 4, 140, 121, 5, 4, 24

Offset: 1

Antti Karttunen, May 10 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function
Indranil Ghosh, Python program to generate the sequence

Cf. A000027, A001511, A046523, A278223, A286161, A286364, A286371, A286372, A286463, A286465.

(define (A286461 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A286364 (+ n n -1))) 2) (- (A001511 n)) (- (* 3 (A286364 (+ n n -1)))) 2)))

a(n) = (1/2)*(2 + ((A001511(n)+A286364((2*n)-1))^2) - A001511(n) - 3*A286364((2*n)-1)).

A291767 Odd bisection of A291761.

Original entry on oeis.org

1, 3, 3, 3, 7, 3, 3, 9, 3, 3, 9, 3, 7, 12, 3, 3, 9, 9, 3, 9, 3, 3, 16, 3, 7, 9, 3, 9, 9, 3, 3, 16, 9, 3, 9, 3, 3, 16, 9, 3, 21, 3, 9, 9, 3, 9, 9, 9, 3, 16, 3, 3, 23, 3, 3, 9, 3, 9, 16, 9, 7, 9, 12, 3, 9, 3, 9, 26, 3, 3, 9, 9, 9, 16, 3, 3, 16, 9, 3, 9, 9, 3, 23, 3, 7, 16, 3, 16, 9, 3, 3, 9, 9, 9, 26, 3, 3, 23, 3, 3, 9, 9, 9, 16, 9

Offset: 1

Antti Karttunen, Sep 11 2017

nonn

Records occur at positions: 1, 2, 5, 8, 14, 23, 41, 53, 68, 113, 122, 158, 203, 338, 365, ... (= (A147516(n)+1)/2) that give also all distinct values in this sequence: 1, 3, 7, 9, 12, 16, 21, 23, 26, 32, 34, 37, 40, 46, 48, 53, 58, 59, 64, 69, 72, 77, 81, ... Note that the terms of A291768 are all from the complementary sequence: 2, 4, 5, 6, 8, 10, 11, 13, 14, 15, 17, ...

Antti Karttunen, Table of n, a(n) for n = 1..16385

Cf. A147516, A278223, A291761, A291768.

(define (A291767 n) (A291761 (+ n n -1)))

a(n) = A291761(2n - 1).

A286452 Compound filter (largest prime factor of n & prime signature of 2n-1): a(n) = P(A061395(n), A046523(2n-1)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 2, 5, 2, 18, 5, 14, 16, 5, 9, 50, 5, 42, 59, 9, 2, 73, 23, 44, 31, 14, 20, 199, 5, 18, 61, 5, 40, 115, 9, 77, 67, 50, 35, 40, 5, 90, 179, 61, 9, 391, 14, 185, 50, 9, 100, 205, 23, 14, 94, 35, 27, 1006, 5, 20, 40, 44, 115, 395, 31, 228, 131, 59, 2, 61, 20, 295, 442, 54, 14, 320, 23, 346, 265, 9, 44, 125, 61, 275, 31, 23, 104, 1349, 14, 52, 314, 65, 125, 430

Offset: 1

Antti Karttunen, May 14 2017

nonn

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

Cf. A000027, A046523, A061395, A278223, A285334, A286356, A286372, A286453.

from sympy import primepi, primefactors, factorint
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a061395(n): return 0 if n==1 else primepi(primefactors(n)[-1])
def a(n): return T(a061395(n), a046523(2*n - 1)) # Indranil Ghosh, May 14 2017

(define (A286452 n) (* (/ 1 2) (+ (expt (+ (A061395 n) (A046523 (+ n n -1))) 2) (- (A061395 n)) (- (* 3 (A046523 (+ n n -1)))) 2)))

a(n) = (1/2)*(2 + ((A061395(n)+A046523(2n-1))^2) - A061395(n) - 3*A046523(2n-1)).

A286453 Compound filter: a(n) = P(A061395(n), A286465(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 2, 5, 11, 94, 5, 14, 254, 17, 9, 195, 47, 259, 500, 9, 11, 413, 138, 44, 303, 32, 20, 2784, 47, 354, 216, 5, 329, 506, 9, 77, 3161, 356, 35, 175, 107, 202, 2709, 216, 24, 11188, 14, 420, 356, 24, 285, 450, 498, 70, 2349, 35, 51, 115937, 5, 20, 329, 74, 310, 3420, 864, 1243, 336, 500, 11, 384, 20, 580, 47285, 87, 14, 615, 498, 1296, 3015, 9, 74, 3491, 216

Offset: 1

Antti Karttunen, May 14 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Indranil Ghosh, Python program to generate the sequence

Cf. A000027, A046523, A061395, A112049, A278223, A286356, A286372, A286452.

(define (A286453 n) (* (/ 1 2) (+ (expt (+ (A061395 n) (A286465 n)) 2) (- (A061395 n)) (- (* 3 (A286465 n))) 2)))

a(n) = (1/2)*(2 + ((A061395(n)+A286465(n))^2) - A061395(n) - 3*A286465(n)).

A359600 The least odd number with the same prime signature as n.

Original entry on oeis.org

1, 3, 3, 9, 3, 15, 3, 27, 9, 15, 3, 45, 3, 15, 15, 81, 3, 45, 3, 45, 15, 15, 3, 135, 9, 15, 27, 45, 3, 105, 3, 243, 15, 15, 15, 225, 3, 15, 15, 135, 3, 105, 3, 45, 45, 15, 3, 405, 9, 45, 15, 45, 3, 135, 15, 135, 15, 15, 3, 315, 3, 15, 45, 729, 15, 105, 3, 45, 15, 105, 3, 675, 3, 15, 45, 45, 15, 105, 3, 405, 81

Offset: 1

Antti Karttunen, Jan 12 2023

nonn

Cf. A003961, A046523, A101296.

Cf. also A278223.

a:= n-> (l-> mul(ithprime(i+1)^l[i][2], i=1..nops(l)))
        (sort(ifactors(n)[2], (x, y)->x[2]>y[2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 12 2023

a(n) = { my(f=vecsort(factor(n)[, 2], , 4)); prod(i=1, #f, prime(i+1)^f[i]) } \\ Andrew Howroyd, Jan 12 2023

from math import prod
from sympy import prime, factorint
def A359600(n): return prod(prime(i)**e for i, e in enumerate(sorted(factorint(n).values(), reverse=True),2)) # Chai Wah Wu, Jan 12 2023

a(n) = A003961(A046523(n)).

A290083 Odd bisection of A289626.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 19 2017

nonn

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

Cf. A037225, A278223, A289626.

(define (A290083 n) (A289626 (+ n n -1)))

a(n) = A289626(2n-1).

cp's OEIS Frontend

A378223 Inverse Möbius transform of A345182.

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A292249 Compound filter (multiplicative order of 2 mod 2n+1 & prime signature of 2n+1): a(n) = P(A002326(n), A046523(2n+1)), where P(n,k) is sequence A000027 used as a pairing function.

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A286461 Compound filter (2-adic valuation of n & 4k+1,4k+3 prime-signature combination of 2n-1): a(n) = P(A001511(n), A286364((2*n)-1)), where P(n,k) is sequence A000027 used as a pairing function.

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A291767 Odd bisection of A291761.

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A286452 Compound filter (largest prime factor of n & prime signature of 2n-1): a(n) = P(A061395(n), A046523(2n-1)), where P(n,k) is sequence A000027 used as a pairing function.

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A286453 Compound filter: a(n) = P(A061395(n), A286465(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A359600 The least odd number with the same prime signature as n.

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Maple

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A290083 Odd bisection of A289626.

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