cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A286360 Compound filter (prime signature & sum of the divisors): a(n) = P(A046523(n), A000203(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 8, 12, 49, 23, 142, 38, 239, 124, 259, 80, 753, 107, 412, 412, 1051, 173, 1237, 212, 1390, 672, 826, 302, 3427, 565, 1087, 1089, 2223, 467, 5080, 530, 4403, 1384, 1717, 1384, 7911, 743, 2086, 1836, 6352, 905, 7780, 992, 4477, 3928, 2932, 1178, 14583, 1774, 5368, 2932, 5898, 1487, 10177, 2932, 10177, 3576, 4471, 1832, 25711, 1955, 5056, 6567, 18019, 3922
Offset: 1

Views

Author

Antti Karttunen, May 10 2017

Keywords

Links

Crossrefs

Cf. A000027, A286359, A286460.
Cf. A007503, A065608 (sequences matching to this filter), also A000203, A046523, A161942, A286034, A286357.

Programs

  • PARI
    A000203(n) = sigma(n);
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
A286360(n) = (1/2)*(2 + ((A046523(n)+A000203(n))^2) - A046523(n) - 3*A000203(n));
for(n=1, 10000, write("b286360.txt", n, " ", A286360(n)));
  • Python
    from sympy import factorint, divisor_sigma as D
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 a(n): return T(a046523(n), D(n)) # Indranil Ghosh, May 12 2017
  • Scheme
    (define (A286360 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A000203 n)) 2) (- (A046523 n)) (- (* 3 (A000203 n))) 2)))

Formula

a(n) = (1/2)*(2 + ((A046523(n)+A000203(n))^2) - A046523(n) - 3*A000203(n)).

A286260 Compound filter: a(n) = P(A001511(n), A161942(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 8, 1, 39, 4, 8, 1, 157, 79, 47, 4, 39, 22, 8, 4, 600, 37, 782, 11, 256, 1, 47, 4, 157, 466, 233, 11, 39, 106, 47, 1, 2284, 4, 380, 4, 4281, 172, 122, 22, 1132, 211, 8, 56, 256, 742, 47, 4, 600, 1597, 4373, 37, 1278, 352, 122, 37, 157, 11, 1037, 106, 256, 466, 8, 79, 8785, 211, 47, 137, 2083, 4, 47, 37, 19507, 667, 1655, 466, 669, 4, 233, 11, 4661, 7261
Offset: 1

Views

Author

Antti Karttunen, May 07 2017

Keywords

Links

Crossrefs

Cf. A000027, A001511, A161942, A286161, A286162, A286259, A286034.

Programs

  • PARI
    A001511(n) = (1+valuation(n,2));
A000265(n) = (n >> valuation(n, 2));
A161942(n) = A000265(sigma(n));
A286260(n) = (2 + ((A001511(n)+A161942(n))^2) - A001511(n) - 3*A161942(n))/2;
for(n=1, 16384, write("b286260.txt", n, " ", A286260(n)));
  • Python
    from sympy import factorint, divisors, divisor_sigma
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a000265(n): return max(list(filter(lambda i: i%2 == 1, divisors(n))))
def a161942(n): return a000265(divisor_sigma(n))
def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a(n): return T(a001511(n), a161942(n)) # Indranil Ghosh, May 07 2017
  • Scheme
    (define (A286260 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A161942 n)) 2) (- (A001511 n)) (- (* 3 (A161942 n))) 2)))

Formula

a(n) = (1/2)*(2 + ((A001511(n)+A161942(n))^2) - A001511(n) - 3*A161942(n)).

A296090 Filter combining the sum of divisors (A000203) and prime-signature (A101296) of n; restricted growth sequence transform of A286360.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Dec 07 2017

Keywords

Comments

For all i, j:
a(i) = a(j) => A286034(i) = A286034(j).
a(i) = a(j) => A295880(i) = A295880(j).

Links

Crossrefs

Cf. A000203, A101296, A046523, A286034, A286360, A295300, A295888, A296089.
Differs from related A295880 for the first time at n=135, where a(135) = 123, while A295880(135) = 104.

Programs

  • PARI
    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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A000203(n) = sigma(n);
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
A286360(n) = (1/2)*(2 + ((A046523(n)+A000203(n))^2) - A046523(n) - 3*A000203(n));
write_to_bfile(1,rgs_transform(vector(up_to,n,A286360(n))),"b296090.txt");

A286358 Compound filter: a(n) = P(A286357(n), A161942(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 4, 6, 22, 8, 13, 10, 106, 79, 47, 13, 39, 30, 19, 19, 466, 47, 742, 24, 233, 21, 58, 19, 139, 466, 233, 32, 49, 122, 70, 21, 1954, 26, 380, 26, 4096, 192, 139, 49, 1037, 233, 34, 81, 256, 782, 70, 26, 531, 1597, 4279, 70, 1227, 380, 157, 70, 157, 41, 1037, 139, 280, 498, 34, 124, 8002, 256, 83, 174, 2018, 34, 83, 70, 18916, 705, 1655, 531, 669, 34, 280, 41
Offset: 1

Views

Author

Antti Karttunen, May 10 2017

Keywords

Comments

Partitions natural numbers to the same equivalence classes as A000203. That is, for all i, j: a(i) = a(j) <=> A000203(i) = A000203(j). This follows because both A161942(n) and A286357(n) can be (are) defined as functions of A000203, and on the other hand, A000203(n) can be uniquely reconstructed from A161942(n) and A286357(n), thus from a(n).

Links

Crossrefs

Cf. A000027, A000203, A161942, A286034, A286260, A286357, A286359, A286360, A286460.

Programs

Formula

a(n) = (1/2)*(2 + ((A286357(n)+A161942(n))^2) - A286357(n) - 3*A161942(n)).

A286259 Compound filter: a(n) = P(A001511(n), A049820(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 1, 6, 4, 5, 11, 25, 16, 23, 37, 31, 56, 57, 56, 110, 106, 80, 137, 123, 137, 173, 211, 175, 232, 255, 254, 279, 352, 255, 407, 471, 407, 467, 466, 409, 596, 597, 596, 599, 742, 597, 821, 783, 742, 905, 991, 866, 1036, 992, 1082, 1131, 1276, 1083, 1276, 1279, 1379, 1487, 1597, 1228, 1712, 1713, 1597, 1960, 1831, 1713, 2081, 2019, 2081, 1955, 2347, 1957
Offset: 1

Views

Author

Antti Karttunen, May 07 2017

Keywords

Links

Crossrefs

Cf. A000027, A001511, A049820, A286161, A286162, A286260, A286034.

Programs

  • PARI
    A001511(n) = (1+valuation(n,2));
A049820(n) = (n-numdiv(n));
A286259(n) = (2 + ((A001511(n)+A049820(n))^2) - A001511(n) - 3*A049820(n))/2;
for(n=1, 10000, write("b286259.txt", n, " ", A286259(n)));
  • Python
    from sympy import divisor_count as d
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a(n): return T(a001511(n), n - d(n)) # Indranil Ghosh, May 07 2017
  • Scheme
    (define (A286259 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A049820 n)) 2) (- (A001511 n)) (- (* 3 (A049820 n))) 2)))

Formula

a(n) = (1/2)*(2 + ((A001511(n)+A049820(n))^2) - A001511(n) - 3*A049820(n)).
Showing 1-5 of 5 results.

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