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 1-3 of 3 results.

A329603 a(n) = A005940(1+(1+(3*A156552(n)))) = (1/2) * A005940(1+(3*A156552(2*n))).

Original entry on oeis.org

2, 5, 8, 15, 18, 11, 50, 45, 20, 125, 98, 33, 242, 245, 32, 135, 338, 77, 578, 375, 72, 605, 722, 99, 42, 845, 60, 735, 1058, 17, 1682, 405, 200, 1445, 162, 231, 1922, 1805, 392, 1125, 2738, 1331, 3362, 1815, 44, 2645, 3698, 297, 110, 275, 968, 2535, 4418, 539, 450, 2205, 1352, 4205, 5618, 51, 6962, 4805, 500, 1215, 882, 1859, 7442, 4335, 2312
Offset: 1

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Author

Antti Karttunen, Nov 21 2019

Keywords

Comments

Function n -> 3n+1 (A016777) conjugated by A156552. - Antti Karttunen, Aug 21 2021

Links

Crossrefs

Permutation of A329604.
Cf. A000035, A005940, A016777, A064989, A156552, A329903, A332449, A332461, A332814, A332823, A341354, A341510, A341515, A341516, A347117 (Möbius transform).
A skewed diagonal of A341510.

Programs

  • PARI
    A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A156552(n) = { my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A329603(n) = ((1/2)*A005940(1+(3*A156552(2*n))));
  • Python
    from math import prod
from itertools import accumulate
from collections import Counter
from sympy import prime, primepi, factorint
def A329603(n): return prod(prime(len(a)+1)**b for a, b in Counter(accumulate(bin(1+3*sum((1<Chai Wah Wu, Mar 11 2023

Formula

a(n) = (1/2) * A005940(1+(3*A156552(2*n))).
From Antti Karttunen, Feb 14 2021: (Start)
A156552(2*a(n)) = 3*A156552(2*n) = 3*(1+2*A156552(n)) = 3 + 6*A156552(n).
a(n) = A341510(n,2n) = A005940(1+A156552(n)+A156552(2n)) = A005940(1+(1+(3*A156552(n)))).
a(n) = A005940(1+A016777(A156552(n))).
For all n >= 1, A329903(a(n)) = A332814(a(n)) = A332823(A332461(a(n))) = 1.
For all n >= 1, A341354(a(n)) > 0.
For all n >= 1, A000035(a(n)) = 1 - A000035(n). [Flips the parity of n]
(End)
a(n) = A332449(2*n)/2, a(n) = Sum_{d|n} A347117(d). - Antti Karttunen, Aug 21 2021

Extensions

New primary definition added by Antti Karttunen, Feb 14 2021

A347115 Möbius transform of A341515.

Original entry on oeis.org

1, 4, 1, 10, 2, 5, 4, 30, 2, 118, 6, 12, 10, 236, 2, 90, 12, 64, 16, 240, 4, 594, 18, 36, 6, 830, 4, 480, 22, -116, 28, 270, 6, 1428, 8, 132, 30, 1784, 10, 720, 36, 1076, 40, 1200, 4, 2622, 42, 108, 20, 144, 12, 1680, 46, 458, 12, 1440, 16, 4178, 52, -228, 58, 4772, 8, 810, 20, 1242, 60, 2880, 18, 2752, 66, 396
Offset: 1

Views

Author

Antti Karttunen, Aug 20 2021

Keywords

Links

Crossrefs

Cf. A008683, A064989, A329603, A341515.
Cf. A285702 (odd bisection), A347116 (even bisection).
Cf. also A332463, A347117, A347118, A348045.

Programs

  • PARI
    A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
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)};
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));
A347115(n) = sumdiv(n,d,moebius(n/d)*A341515(d));

Formula

a(n) = A008683(n/d) * A341515(d).

A347118 Möbius transform of A332449.

Original entry on oeis.org

1, 3, 8, 6, 24, 4, 48, 20, 12, 8, 120, 0, 168, 48, 48, 60, 288, 12, 360, 208, 168, 72, 528, 24, 30, 312, 84, 384, 840, -32, 960, 180, 312, 384, 552, 108, 1368, 792, 912, 480, 1680, -136, 1848, 1008, -54, 912, 2208, 72, 42, 18, 1224, 1200, 2808, -4, 1080, 960, 2232, 1272, 3480, -244, 3720, 2400, 792, 540, 2832, -120
Offset: 1

Views

Author

Antti Karttunen, Aug 21 2021

Keywords

Links

Crossrefs

Cf. A008683, A156552, A332449.
Cf. also A332463, A347115, A347117.

Programs

  • PARI
    A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A332449(n) = A005940(1+(3*A156552(n)));
A347118(n) = sumdiv(n,d,moebius(n/d)*A332449(d));

Formula

a(n) = Sum_{d|n} A008683(n/d) * A332449(d).
Showing 1-3 of 3 results.

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