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.

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A286251 Compound filter: a(n) = P(A001511(1+n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

3, 2, 9, 7, 5, 16, 14, 29, 12, 16, 9, 67, 5, 16, 50, 121, 5, 67, 9, 67, 23, 16, 14, 277, 12, 16, 48, 67, 5, 436, 27, 497, 23, 16, 31, 631, 5, 16, 40, 277, 5, 436, 9, 67, 80, 16, 20, 1129, 12, 67, 31, 67, 5, 277, 40, 277, 23, 16, 9, 1771, 5, 16, 160, 2017, 23, 436, 9, 67, 23, 436, 14, 2557, 5, 16, 94, 67, 23, 436, 20, 1129, 138, 16, 9, 1771, 23, 16, 40, 277, 5
Offset: 1

Views

Author

Antti Karttunen, May 07 2017

Keywords

Links

Crossrefs

Cf. A000027, A001511, A046523, A286161, A286252, A286253, A286254.

Programs

  • PARI
    A001511(n) = (1+valuation(n,2));
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
A286251(n) = (2 + ((A001511(1+n)+A046523(n))^2) - A001511(1+n) - 3*A046523(n))/2;
for(n=1, 10000, write("b286251.txt", n, " ", A286251(n)));
  • Python
    from sympy import factorint
def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
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(a001511(n + 1), a046523(n)) # Indranil Ghosh, May 07 2017
  • Scheme
    (define (A286251 n) (* (/ 1 2) (+ (expt (+ (A001511 (+ 1 n)) (A046523 n)) 2) (- (A001511 (+ 1 n))) (- (* 3 (A046523 n))) 2)))

Formula

a(n) = (1/2)*(2 + ((A001511(1+n)+A046523(n))^2) - A001511(1+n) - 3*A046523(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)).

A286152 Compound filter: a(n) = T(A051953(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 2, 2, 12, 2, 40, 2, 59, 18, 61, 2, 179, 2, 86, 73, 261, 2, 265, 2, 265, 100, 148, 2, 757, 33, 185, 129, 367, 2, 1297, 2, 1097, 166, 271, 131, 1735, 2, 320, 205, 1105, 2, 1741, 2, 619, 517, 430, 2, 3113, 52, 850, 295, 769, 2, 1747, 205, 1517, 346, 625, 2, 5297, 2, 698, 730, 4497, 248, 2821, 2, 1117, 460, 2821, 2, 7069, 2, 941, 1070, 1315, 248, 3457, 2, 4513
Offset: 1

Views

Author

Antti Karttunen, May 04 2017

Keywords

Links

Crossrefs

Cf. A000027, A046523, A051953, A285729, A286142, A286143, A286144, A286149, A286154, A286160, A286161, A286162, A286163, A286164.

Programs

  • Mathematica
    Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ {n - EulerPhi@ n, Times @@ MapIndexed[Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]] - Boole[n == 1]}, {n, 80}] (* Michael De Vlieger, May 04 2017 *)
  • PARI
    A051953(n) = (n - eulerphi(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
A286152(n) = (2 + ((A051953(n)+A046523(n))^2) - A051953(n) - 3*A046523(n))/2;
for(n=1, 10000, write("b286152.txt", n, " ", A286152(n)));
  • Python
    from sympy import factorint, totient
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(n - totient(n), a046523(n)) # Indranil Ghosh, May 05 2017
  • Scheme
    (define (A286152 n) (* (/ 1 2) (+ (expt (+ (A051953 n) (A046523 n)) 2) (- (A051953 n)) (- (* 3 (A046523 n))) 2)))

Formula

a(n) = (1/2)*(2 + ((A051953(n)+A046523(n))^2) - A051953(n) - 3*A046523(n)).

A286573 Compound filter: a(n) = P(A007733(n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 5, 7, 14, 23, 9, 29, 42, 40, 65, 80, 90, 31, 40, 121, 44, 142, 189, 109, 61, 115, 77, 302, 273, 148, 318, 94, 434, 532, 20, 497, 115, 86, 148, 826, 702, 271, 148, 355, 230, 601, 119, 220, 265, 131, 299, 1178, 297, 485, 86, 265, 1430, 838, 320, 328, 271, 556, 1769, 1957, 1890, 50, 142, 2017, 148, 751, 2277, 179, 373, 832, 665, 2932, 54, 856, 485
Offset: 1

Views

Author

Antti Karttunen, May 26 2017

Keywords

Links

Crossrefs

Cf. A000027, A007733, A046523, A286160, A286161.

Programs

  • PARI
    A007733(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ This function from Michel Marcus, Apr 11 2015
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
A286573(n) = (1/2)*(2 + ((A007733(n)+A046523(n))^2) - A007733(n) - 3*A046523(n));
  • Python
    from sympy import divisors, factorint
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a002326(n):
    m=1
    while True:
        if (2**m - 1)%(2*n + 1)==0: return m
        else: m+=1
def a000265(n): return max(list(filter(lambda i: i%2 == 1, divisors(n))))
def a007733(n): return a002326((a000265(n) - 1)/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(a007733(n), a046523(n)) # Indranil Ghosh, May 26 2017

Formula

a(n) = (1/2)*(2 + ((A007733(n)+A046523(n))^2) - A007733(n) - 3*A046523(n)).

A286154 Compound filter: a(n) = T(A055396(n), A000010(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 1, 5, 2, 18, 2, 40, 7, 23, 7, 96, 7, 142, 16, 38, 29, 238, 16, 308, 29, 80, 46, 444, 29, 234, 67, 173, 67, 676, 29, 791, 121, 212, 121, 328, 67, 1093, 154, 302, 121, 1339, 67, 1499, 191, 302, 232, 1785, 121, 994, 191, 530, 277, 2227, 154, 864, 277, 668, 379, 2718, 121, 2944, 436, 668, 497, 1228, 191, 3505, 497, 992, 277, 3936, 277, 4207, 631, 822, 631
Offset: 1

Views

Author

Antti Karttunen, May 04 2017

Keywords

Links

Crossrefs

Cf. A000010, A000027, A055396, A286142, A286143, A286144, A286152, A286160, A286161, A286162, A286163, A286164.

Programs

  • Mathematica
    Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ {If[n == 1, 0, PrimePi[ FactorInteger[n][[1, 1]] ]], EulerPhi@ n}, {n, 76}] (* Michael De Vlieger, May 04 2017 *)
  • PARI
    A000010(n) = eulerphi(n);
A055396(n) = if(n==1, 0, primepi(factor(n)[1, 1])); \\ This function from Charles R Greathouse IV, Apr 23 2015
A286154(n) = (2 + ((A055396(n)+A000010(n))^2) - A055396(n) - 3*A000010(n))/2;
for(n=1, 10000, write("b286154.txt", n, " ", A286154(n)));
  • Python
    from sympy import primepi, isprime, primefactors, totient
def a049084(n): return primepi(n)*(1*isprime(n))
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a(n): return T(a055396(n), totient(n)) # Indranil Ghosh, May 05 2017
  • Scheme
    (define (A286154 n) (* (/ 1 2) (+ (expt (+ (A055396 n) (A000010 n)) 2) (- (A055396 n)) (- (* 3 (A000010 n))) 2)))

Formula

a(n) = (1/2)*(2 + ((A055396(n)+A000010(n))^2) - A055396(n) - 3*A000010(n)).

A286463 Compound filter (3-adic valuation & prime-signature): a(n) = P(A051064(n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 5, 7, 2, 23, 2, 29, 18, 16, 2, 80, 2, 16, 23, 121, 2, 94, 2, 67, 23, 16, 2, 302, 7, 16, 59, 67, 2, 467, 2, 497, 23, 16, 16, 706, 2, 16, 23, 277, 2, 467, 2, 67, 94, 16, 2, 1178, 7, 67, 23, 67, 2, 355, 16, 277, 23, 16, 2, 1832, 2, 16, 94, 2017, 16, 467, 2, 67, 23, 436, 2, 2704, 2, 16, 80, 67, 16, 467, 2, 1129, 195, 16, 2, 1832, 16, 16, 23, 277, 2, 1894, 16
Offset: 1

Views

Author

Antti Karttunen, May 10 2017

Keywords

Links

Crossrefs

Cf. A000027, A046523, A051064, A286161, A286461, A286462, A286464.

Programs

  • PARI
    A051064(n) = if(n<1, 0, 1+valuation(n, 3));
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
A286463(n) = (1/2)*(2 + ((A051064(n)+A046523(n))^2) - A051064(n) - 3*A046523(n));
for(n=1, 10000, write("b286463.txt", n, " ", A286463(n)));
  • Python
    from sympy import factorint, divisors, divisor_count, mobius
def a051064(n): return -sum([mobius(3*d)*divisor_count(n/d) for d in divisors(n)])
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 T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a(n): return T(a051064(n), a046523(n)) # Indranil Ghosh, May 11 2017
  • Scheme
    (define (A286463 n) (* (/ 1 2) (+ (expt (+ (A051064 n) (A046523 n)) 2) (- (A051064 n)) (- (* 3 (A046523 n))) 2)))

Formula

a(n) = (1/2)*(2 + ((A051064(n)+A046523(n))^2) - A051064(n) - 3*A046523(n)).

A286254 Compound filter: a(n) = P(A001511(n), A055396(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 1, 13, 1, 12, 1, 14, 1, 17, 1, 31, 1, 5, 1, 60, 1, 38, 1, 9, 1, 47, 1, 19, 1, 5, 1, 69, 1, 68, 1, 27, 1, 8, 1, 94, 1, 5, 1, 124, 1, 107, 1, 9, 1, 122, 1, 33, 1, 5, 1, 156, 1, 8, 1, 14, 1, 155, 1, 193, 1, 5, 1, 43, 1, 192, 1, 9, 1, 212, 1, 280, 1, 5, 1, 18, 1, 255, 1, 20, 1, 278, 1, 13, 1, 5, 1, 355, 1, 12, 1, 9, 1, 8, 1, 441, 1, 5, 1, 381, 1, 380, 1, 14
Offset: 1

Views

Author

Antti Karttunen, May 07 2017

Keywords

Links

Crossrefs

Cf. A000027, A001511, A055396, A253790, A286161, A286251, A286252, A286253.

Programs

  • PARI
    A001511(n) = (1+valuation(n,2));
A055396(n) = if(n==1, 0, primepi(factor(n)[1, 1])); \\ This function from Charles R Greathouse IV, Apr 23 2015
A286254(n) = (2 + ((A001511(n)+A055396(1+n))^2) - A001511(n) - 3*A055396(1+n))/2;
for(n=1, 10000, write("b286254.txt", n, " ", A286254(n)));
  • Python
    from sympy import primepi, isprime, primefactors
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a049084(n): return primepi(n)*(1*isprime(n))
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a(n): return T(a001511(n), a055396(n + 1)) # Indranil Ghosh, May 07 2017
  • Scheme
    (define (A286254 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A055396 (+ 1 n))) 2) (- (A001511 n)) (- (* 3 (A055396 (+ 1 n)))) 2)))

Formula

a(n) = (1/2)*(2 + ((A001511(n)+A055396(1+n))^2) - A001511(n) - 3*A055396(1+n)).

A286149 Compound filter: a(n) = T(A046523(n), A109395(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 8, 14, 17, 34, 30, 44, 19, 51, 68, 103, 93, 72, 196, 152, 155, 103, 192, 132, 72, 126, 278, 349, 32, 159, 53, 165, 437, 976, 498, 560, 709, 237, 786, 739, 705, 282, 159, 402, 863, 660, 948, 243, 337, 384, 1130, 1273, 49, 132, 1546, 288, 1433, 349, 126, 459, 282, 567, 1772, 2761, 1893, 636, 165, 2144, 2421, 1921, 2280, 390, 2707, 2046, 2558, 2773, 2703
Offset: 1

Views

Author

Antti Karttunen, May 04 2017

Keywords

Links

Crossrefs

Cf. A000027, A046523, A109395, A285729, A286142, A286143, A286144, A286152, A286154, A286160, A286161, A286162, A286163, A286164.

Programs

  • Mathematica
    Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ {Times @@ MapIndexed[ Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]] - Boole[n == 1], Denominator[EulerPhi[n]/n]}, {n, 73}] (* Michael De Vlieger, May 04 2017 *)
  • PARI
    A109395(n) = n/gcd(n, eulerphi(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
A286149(n) = (1/2)*(2 + ((A046523(n)+A109395(n))^2) - A046523(n) - 3*A109395(n));
for(n=1, 10000, write("b286149.txt", n, " ", A286149(n)));
  • Python
    from sympy import factorint, totient, gcd
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), n/gcd(n, totient(n))) # Indranil Ghosh, May 05 2017
  • Scheme
    (define (A286149 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A109395 n)) 2) (- (A046523 n)) (- (* 3 (A109395 n))) 2)))

Formula

a(n) = (1/2)*(2 + ((A046523(n)+A109395(n))^2) - A046523(n) - 3*A109395(n)).

A286362 Compound filter: a(n) = P(A089309(n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 5, 7, 2, 23, 9, 29, 7, 16, 5, 80, 2, 31, 40, 121, 2, 67, 5, 67, 16, 23, 9, 302, 7, 16, 38, 94, 2, 532, 20, 497, 16, 16, 23, 631, 2, 23, 31, 277, 2, 436, 5, 80, 67, 31, 14, 1178, 7, 67, 23, 67, 2, 302, 31, 328, 16, 16, 5, 1957, 2, 50, 142, 2017, 16, 436, 5, 67, 16, 467, 9, 2557, 2, 16, 80, 80, 16, 499, 14, 1129, 121, 16, 5, 1771, 16, 23, 31, 302, 2, 1771
Offset: 1

Views

Author

Antti Karttunen, May 08 2017

Keywords

Comments

For all odd i, odd j: a(i) = a(j) <=> A286251(i) = A286251(j).

Links

Crossrefs

Cf. A000027, A046523, A089309, A286161, A286251, A286365.

Programs

Formula

a(n) = (1/2)*(2 + ((A089309(n)+A046523(n))^2) - A089309(n) - 3*A046523(n)).

A286367 Compound filter: a(n) = P(A001511(n), A286364(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 2, 10, 22, 8, 2, 9, 4, 5, 11, 15, 4, 30, 2, 13, 121, 5, 2, 14, 46, 8, 407, 9, 4, 17, 2, 21, 121, 8, 11, 39, 4, 5, 11, 19, 4, 138, 2, 9, 67, 5, 2, 20, 22, 57, 11, 13, 4, 437, 11, 14, 121, 8, 2, 24, 4, 5, 2212, 28, 211, 138, 2, 13, 121, 17, 2, 49, 4, 8, 92, 9, 121, 17, 2, 26, 7261, 8, 2, 156, 211, 5, 11, 14, 4, 80, 11, 9, 121, 5, 11, 27, 4, 30
Offset: 1

Views

Author

Antti Karttunen, May 08 2017

Keywords

Comments

This sequence contains, in addition to the information contained in A286364 (which packs the values of A286361(n) and A286363(n) to a single value with the pairing function A000027), also the highest power of 2 dividing n. Note that this is more information than A286365, as it stores only the parity of the exponent of 2.
For all i, j: a(i) = a(j) => A286161(i) = A286161(j).

Links

Crossrefs

Cf. A000027, A001511, A286161, A286361, A286363, A286364, A286365.

Programs

  • Python
    from sympy import factorint
from operator import mul
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, k):
    f = factorint(n)
    return 1 if n == 1 else reduce(mul, [1 if i%4==k else i**f[i] for i in f])
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a286364(n): return T(a046523(n/A(n, 1)), a046523(n/A(n, 3)))
def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a(n): return T(a001511(n), a286364(n)) # Indranil Ghosh, May 09 2017
  • Scheme
    (define (A286367 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A286364 n)) 2) (- (A001511 n)) (- (* 3 (A286364 n))) 2)))

Formula

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