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.

A294898 Deficiency minus binary weight: a(n) = A033879(n) - A000120(n) = A005187(n) - A000203(n).

Original entry on oeis.org

0, 0, 0, 0, 2, -2, 3, 0, 3, 0, 7, -6, 9, 1, 2, 0, 14, -5, 15, -4, 7, 5, 18, -14, 16, 7, 10, -3, 24, -16, 25, 0, 16, 12, 19, -21, 33, 13, 18, -12, 37, -15, 38, 1, 8, 16, 41, -30, 38, 4, 26, 3, 48, -16, 33, -11, 30, 22, 53, -52, 55, 23, 16, 0, 44, -14, 63, 8, 39, -7, 66, -53, 69, 31, 22, 9, 54, -16, 73, -28, 38, 35, 78, -59, 58
Offset: 1

Views

Author

Antti Karttunen, Nov 25 2017

Keywords

Comments

"Least deficient numbers" or "almost perfect numbers" are those k for which A033879(k) = 1, or equally, for which a(k) = -A048881(k-1). The only known solutions are powers of 2 (A000079), all present also in A295296. See also A235796 and A378988. - Antti Karttunen, Dec 16 2024

Links

Crossrefs

Cf. A000120, A000203, A001065, A005187, A011371, A013661, A033879, A048881, A235796, A294896, A294899, A297114 (Möbius transform), A317844 (difference from a(n)), A326133, A326138, A324348 (a(n) applied to Doudna sequence), A379008 (a(n) applied to prime shift array), A378988.
Cf. A295296 (positions of zeros), A295297 (parity of a(n)).

Programs

Formula

a(n) = A005187(n) - A000203(n).
a(n) = A011371(n) - A001065(n).
a(n) = A033879(n) - A000120(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - zeta(2)/2 = 0.177532... . - Amiram Eldar, Feb 22 2024

Extensions

Name edited by Antti Karttunen, Dec 16 2024

A297114 Möbius transform of A294898, where A294898 is deficiency minus binary weight.

Original entry on oeis.org

0, 0, 0, 0, 2, -2, 3, 0, 3, -2, 7, -4, 9, -2, 0, 0, 14, -6, 15, -4, 4, -2, 18, -8, 14, -2, 7, -4, 24, -14, 25, 0, 9, -2, 14, -12, 33, -2, 9, -8, 37, -18, 38, -4, 3, -2, 41, -16, 35, -10, 12, -4, 48, -18, 24, -8, 15, -2, 53, -28, 55, -2, 6, 0, 33, -26, 63, -4, 21, -22, 66, -24, 69, -2, 6, -4, 44, -30, 73, -16, 28, -2
Offset: 1

Views

Author

Antti Karttunen, Dec 26 2017

Keywords

Links

Crossrefs

Cf. A000203, A005187, A051953, A083254, A294898, A297111, A297115, A297117, A317844, A324397.
Cf. A378986 (even bisection, -2*A176095).

Programs

  • Mathematica
    Table[DivisorSum[n, MoebiusMu[n/#] (2 # - DigitCount[2 #, 2, 1] - DivisorSigma[1, #]) &], {n, 82}] (* Michael De Vlieger, Mar 11 2019 *)
  • PARI
    A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A297114(n) = sumdiv(n,d,moebius(n/d)*(A005187(d)-sigma(d)));

Formula

a(n) = Sum_{d|n} A008683(n/d)*A294898(d).
a(n) = A297111(n) - n.
a(n) = A297117(n) - A051953(n).
a(n) = A083254(n) - A297115(n).
a(2n) = A083254(2n) = A378986(n) = -2*A176095(n).
a(n) = A294898(n) - A317844(n).

A318448 a(n) = Sum_{d|n} A294898(d), where A294898(d) = A005187(d) - sigma(d).

Original entry on oeis.org

0, 0, 0, 0, 2, -2, 3, 0, 3, 2, 7, -8, 9, 4, 4, 0, 14, -4, 15, -2, 10, 12, 18, -22, 18, 16, 13, 1, 24, -14, 25, 0, 23, 26, 24, -31, 33, 28, 27, -14, 37, -6, 38, 13, 15, 34, 41, -52, 41, 22, 40, 19, 48, -10, 42, -10, 45, 46, 53, -76, 55, 48, 29, 0, 55, 12, 63, 34, 57, 18, 66, -98, 69, 64, 42, 37, 64, 16, 73, -42, 51, 72, 78, -74, 74, 74, 73, 6
Offset: 1

Views

Author

Antti Karttunen, Aug 27 2018

Keywords

Comments

Inverse Möbius transform of A294898.

Links

Crossrefs

Cf. A000203, A005187, A007429, A093653, A294898, A296075, A318446, A318447.
Cf. also A297114, A317844.

Programs

Formula

a(n) = Sum_{d|n} A294898(d).
a(n) = A318447(n) + A294898(n).
a(n) = A318446(n) - A007429(n).
a(n) = A296075(n) - A093653(n).
Showing 1-3 of 3 results.

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