Daniel Blaine McBride - cp's OEIS frontend

User: Daniel Blaine McBride

Daniel Blaine McBride has authored 2 sequences.

A318682 a(n) is the number of odd values minus the number of even values of the integer log of all positive integers up to and including n.

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

Offset: 1

Daniel Blaine McBride, Aug 30 2018

sign

a(n) = Sum_{k=1..n} (-1)^(sopfr(k)+1), with sopfr(n) the sum of the prime factors of n with repetition, also known as the integer log of n.

			a(4) = -1 - 1 + 1 - 1 = -2, since sopfr(1) = 0, sopfr(2) = 2, sopfr(3) = 3, and sopfr(4) = 4.

Daniel Blaine McBride, Table of n, a(n) for n = 1..100000
K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294.
Eric Weisstein's World of Mathematics, Sum of Prime Factors

Cf. A001414 (sum of prime divisors of n with repetition, sopfr(n)).

Cf. A036349 (numbers such that sopfr(n) is even).

Nest[Append[#, #[[-1]] + (-1)^(1 + Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[Length@ # + 1] ])] &, {-1}, 79] (* Michael De Vlieger, Sep 10 2018 *)

sopfr(n) = my(f=factor(n)); sum(k=1, #f~, f[k, 1]*f[k, 2]);
a(n) = sum(k=1, n, (-1)^(sopfr(k)+1)); \\ Michel Marcus, Sep 09 2018

from sympy import factorint
def A318682(n):
    a_n = 0
    for i in range(1, n+1):
        a_n += (-1)**(sum(p*e for p, e in factorint(i).items())+1)
    return a_n

a(n) = a(n-1) + (-1)^(sopfr(n)+1) with a(1) = (-1)^(sopfr(1)+1) = -1.

A298156 Composite numbers n of which the sum of prime divisors of n (with repetition) equals the concatenation of two integers k and k + 1.

Original entry on oeis.org

35, 42, 50, 60, 64, 72, 76, 81, 86, 93, 136, 145, 153, 159, 164, 174, 253, 273, 289, 325, 365, 385, 390, 416, 438, 462, 468, 488, 494, 497, 549, 550, 555, 559, 592, 644, 658, 660, 664, 666, 686, 703, 704, 710, 737, 747, 792, 836, 852, 884, 885, 891, 920, 940, 944, 946, 980

Offset: 1

Daniel Blaine McBride, Jan 13 2018

Composite numbers n, of which A001414(n) (sum of prime divisors of n with repetition, sopfr(n)) is in sequence A001704 (numbers m which are the concatenation of k and k+1).

			35 = 5*7, sopfr(35) = 5+7 = 12, 12 =k||k+1 when k = 1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294.
Eric Weisstein's World of Mathematics, Sum of Prime Factors

Cf. A001414 (sum of prime divisors of n with repetition, sopfr(n)).

Cf. A001704 (numbers which are the concatenation of k and k+1).

Select[Range[10^3], And[CompositeQ@ #, Subtract @@ Map[FromDigits, TakeDrop[#, Floor[Length[#]/2]]] == -1 &@ IntegerDigits@ Total[Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger[#]]]] &] (* Michael De Vlieger, Jan 14 2018 *)

is(n)=my(f = factor(n), sopfr = sum(i = 1, #f~, f[i, 1] * f[i, 2]); d = digits(sopfr), v); if((#d) % 2 == 0, v = vector(#d / 2); v[#v] = -1; return(vector(#d / 2, j, d[j]) - vector(#d / 2, #d / 2 + j, d[j]) == v), return(d == concat(digits(10^(#d \ 2) - 1), digits(10^(#d \ 2))))) \\ David A. Corneth, Jan 13 2018

sopfr(n,f=factor(n))=sum(i=1,#f~, f[i,1]*f[i,2])
has(n)=my(d=digits(n),k=#d); digits(fromdigits(d[1..k\2])+1) == d[k\2+1..k]
list(lim)=my(v=List()); forfactored(n=35,lim\1, if(n[2][,2]!=[1]~ && has(sopfr(0,n[2])), listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Jan 15 2018

sopfr(n) = k||k+1 when n is not prime and k is a positive integer.

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User: Daniel Blaine McBride

A318682 a(n) is the number of odd values minus the number of even values of the integer log of all positive integers up to and including n.

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A298156 Composite numbers n of which the sum of prime divisors of n (with repetition) equals the concatenation of two integers k and k + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula