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.

A323244 a(1) = 0; and for n > 1, a(n) = A033879(A156552(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 6, 0, 5, 1, 10, 1, 16, 2, 6, 1, 12, 1, 18, -3, 18, 1, 22, -4, 46, 4, 22, 1, 10, 1, 30, 14, 82, -2, 14, 1, 256, -12, 22, 1, 36, 1, 66, 8, 226, 1, 46, -12, 19, 8, 130, 1, 28, -19, 70, -12, 748, 1, 42, 1, 1362, 16, 22, 10, 42, 1, 214, 254, 40, 1, 38, 1, 3838, 10, 406, -10, 106, 1, 78, -12, 5458, 1, 26, -72, 12250, -348, 30, 1, 12
Offset: 1

Views

Author

Antti Karttunen, Jan 10 2019

Keywords

Comments

After a(1) = 0, the other zeros occur for k >= 1, at A005940(1+A000396(k)), which, provided no odd perfect numbers exist, is equal to A324201(k) = A062457(A000043(k)): 9, 125, 161051, 410338673, ..., etc.
There are 2321 negative terms among the first 10000 terms.

Links

Crossrefs

Cf. A000043, A000396, A033879, A064989, A156552, A297112, A323240, A323243, A323245, A323248, A324115, A324051, A324103, A324396, A324398, A324543, A324713.
Cf. A324201 (positions of zeros, conjectured), A324551 (of negative terms), A324720 (of nonnegative terms), A324721 (of positive terms), A324731, A324732.
Cf. A329644 (Möbius transform).
Cf. A323174, A324055, A324185, A324546 for other permutations of deficiency, and also A324574, A324575, A324654.

Programs

  • Mathematica
    Array[2 # - If[# == 0, 0, DivisorSigma[1, #]] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]] &, 90] (* Michael De Vlieger, Apr 21 2019 *)
  • PARI
    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) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A323244(n) = if(1==n, 0, my(k=A156552(n)); (2*k)-sigma(k));
  • Python
    from sympy import divisor_sigma, primepi, factorint
def A323244(n): return (lambda n: (n<<1)-divisor_sigma(n))(sum((1< 1 else 0 # Chai Wah Wu, Mar 10 2023

Formula

a(n) = 2*A156552(n) - A323243(n).
a(1) = 0; and for n > 1, a(n) = A033879(A156552(n)).
a(n) = A323248(n) + A001222(n) = (A323247(n) - A323243(n)) + A001222(n).
From Antti Karttunen, Mar 12 2019 & Nov 23 2019: (Start)
a(n) = Sum_{d|n} (2*A297112(d) - A324543(d)) = Sum_{d|n} A329644(d).
A002487(a(n)) = A324115(n).
a(n) = A329638(n) - A329639(n).
a(n) = A329645(n) - A329646(n).
(End)

A323902 a(n) = A002487(A156552(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 5, 3, 4, 1, 5, 1, 7, 4, 6, 1, 7, 2, 7, 3, 9, 1, 8, 1, 5, 5, 8, 3, 8, 1, 9, 6, 10, 1, 11, 1, 11, 5, 10, 1, 9, 2, 7, 7, 13, 1, 7, 4, 13, 8, 11, 1, 13, 1, 12, 7, 6, 5, 14, 1, 15, 9, 11, 1, 11, 1, 13, 5, 17, 3, 17, 1, 13, 4, 14, 1, 18, 6, 15, 10, 16, 1, 12, 4, 19, 11, 16, 7, 11, 1, 9, 9, 12, 1, 20, 1, 19, 8
Offset: 1

Views

Author

Antti Karttunen, Feb 09 2019

Keywords

Comments

Even though certain subset of terms of A156552 soon grow quite big, this sequence still has a quite moderate growth rate, thanks to the compensating effect of A002487.

Links

Crossrefs

Cf. A002487, A156552, A322993.
Cf. also A323240, A323244, A323901, A323903.

Programs

  • PARI
    A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
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) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A323902(n) = A002487(A156552(n));

Formula

a(n) = A002487(A156552(n)) = A002487(A322993(n)).
a(p) = 1 for all primes p.

A323168 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = [A322867(n), A323174(n)] for n > 1, and f(1) = 0.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 10 2019

Keywords

Comments

Restricted growth sequence transform of function f, with f(1) = 0 and f(n) = [A322867(n), A323174(n)] for n > 1.
Equally, restricted growth sequence transform of function f, with f(1) = 0 and f(n) = A318310(A122111(n)) for n > 1.
For all i, j:
a(i) = a(j) => A322867(i) = A322867(j),
a(i) = a(j) => A323167(i) = A323167(j),
a(i) = a(j) => A323174(i) = A323174(j).

Links

Crossrefs

Cf. A000203, A005187, A122111, A294898, A318310, A322867, A323167, A323173, A323174.
Cf. also A323240.

Programs

  • PARI
    up_to = 4096;
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; };
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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n)-sigma(n));
A318310aux(n) = [hammingweight(n), A294898(n)];
A323168aux(n) = if(1==n,0,A318310aux(A122111(n)));
v323168 = rgs_transform(vector(up_to, n, A323168aux(n)));
A323168(n) = v323168[n];
Showing 1-3 of 3 results.

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