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 1-4 of 4 results.

A285729 Compound filter: a(n) = T(A032742(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 2, 12, 2, 31, 2, 59, 18, 50, 2, 142, 2, 73, 50, 261, 2, 199, 2, 220, 73, 131, 2, 607, 33, 166, 129, 314, 2, 961, 2, 1097, 131, 248, 73, 1396, 2, 295, 166, 923, 2, 1246, 2, 550, 340, 401, 2, 2509, 52, 655, 248, 692, 2, 1252, 131, 1303, 295, 590, 2, 3946, 2, 661, 517, 4497, 166, 1924, 2, 1024, 401, 2051, 2, 5707, 2, 898, 655, 1214, 131, 2317, 2, 3781, 888
Offset: 1

Views

Author

Antti Karttunen, May 04 2017

Keywords

Links

Crossrefs

Cf. A000027, A032742, A046523, A286142, A286143, A286144, A286149, A286152, A286154, A286160, A286161, A286162, A286163, A286164.

Programs

  • Mathematica
    Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ {Sort[Flatten@ Apply[ TensorProduct, # /. {p_, e_} /; p > 1 :> p^Range[0, e]]][[-2]], Times @@ MapIndexed[Prime[First@ #2]^#1 &, Sort[#[[All, -1]], Greater]] - Boole[n == 1]} &@ FactorInteger@ n, {n, 81}] (* Michael De Vlieger, May 04 2017 *)
  • PARI
    A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
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
A285729(n) = (1/2)*(2 + ((A032742(n)+A046523(n))^2) - A032742(n) - 3*A046523(n));
for(n=1, 10000, write("b285729.txt", n, " ", A285729(n)));
  • Python
    from sympy import divisors, factorint
def a032742(n): return 1 if n==1 else max(divisors(n)[:-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(a032742(n), a046523(n)) # Indranil Ghosh, May 05 2017
  • Scheme
    (define (A285729 n) (* (/ 1 2) (+ (expt (+ (A032742 n) (A046523 n)) 2) (- (A032742 n)) (- (* 3 (A046523 n))) 2)))

Formula

a(n) = (1/2)*(2 + ((A032742(n)+A046523(n))^2) - A032742(n) - 3*A046523(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)).

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)).

A286568 Compound filter (phi(n) & 2-adic valuation of sigma(n)): a(n) = P(A000010(n), A286357(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 1, 8, 3, 14, 8, 42, 10, 21, 14, 76, 19, 90, 42, 63, 36, 152, 21, 208, 44, 148, 76, 322, 53, 210, 90, 228, 117, 434, 63, 625, 136, 296, 152, 402, 78, 702, 208, 375, 152, 860, 148, 988, 251, 324, 322, 1271, 169, 903, 210, 627, 324, 1430, 228, 943, 375, 816, 434, 1828, 187, 1890, 625, 777, 528, 1273, 296, 2344, 560, 1220, 402, 2698, 300, 2700, 702, 901
Offset: 1

Views

Author

Antti Karttunen, May 26 2017

Keywords

Links

Crossrefs

Cf. A000010, A000027, A286154, A286160, A286357, A286451, A286572.

Programs

  • PARI
    A000010(n) = eulerphi(n);
A001511(n) = (1+valuation(n,2));
A286357(n) = A001511(sigma(n));
A286568(n) = (1/2)*(2 + ((A000010(n)+A286357(n))^2) - A000010(n) - 3*A286357(n));
  • Python
    from sympy import divisor_sigma as D, totient
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a001511(n): return bin(n)[2:][::-1].index("1") + 1
def a286357(n): return a001511(D(n))
def a(n): return T(totient(n), a286357(n)) # Indranil Ghosh, May 26 2017
  • Scheme
    (define (A286568 n) (* (/ 1 2) (+ (expt (+ (A000010 n) (A286357 n)) 2) (- (A000010 n)) (- (* 3 (A286357 n))) 2)))

Formula

a(n) = (1/2)*(2 + ((A000010(n)+A286357(n))^2) - A000010(n) - 3*A286357(n)).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.