A293434 - OEIS

A293434

a(n) is the sum of the proper divisors of n that are Jacobsthal numbers (A001045).

0, 1, 1, 1, 1, 4, 1, 1, 4, 6, 1, 4, 1, 1, 9, 1, 1, 4, 1, 6, 4, 12, 1, 4, 6, 1, 4, 1, 1, 9, 1, 1, 15, 1, 6, 4, 1, 1, 4, 6, 1, 25, 1, 12, 9, 1, 1, 4, 1, 6, 4, 1, 1, 4, 17, 1, 4, 1, 1, 9, 1, 1, 25, 1, 6, 15, 1, 1, 4, 6, 1, 4, 1, 1, 9, 1, 12, 4, 1, 6, 4, 1, 1, 25, 6, 44, 4, 12, 1, 9, 1, 1, 4, 1, 6, 4, 1, 1, 15, 6, 1, 4, 1, 1, 30

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OFFSET

1,6

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..21845

Index entries for sequences related to sums of divisors

FORMULA

a(n) = Sum_{d|n, d<n} A147612(d)*d.

a(n) = A293432(n) - (A147612(n)*n).

EXAMPLE

For n = 15, whose proper divisors are [1, 3, 5], all of them are in A001045, thus a(15) = 1 + 3 + 5 = 9.

For n = 21, whose proper divisors are [1, 3, 7], both 1 and 3 are in A001045, thus a(21) = 1 + 3 = 4.

For n = 21845, whose proper divisors are [1, 5, 17, 85, 257, 1285, 4369], only 1, 5, 85 are in A001045, thus a(21845) = 1 + 5 + 85 = 91.

MATHEMATICA

With[{s = LinearRecurrence[{1, 2}, {0, 1}, 24]}, Table[DivisorSum[n, # &, And[MemberQ[s, #], # != n] &], {n, 105}]] (* Michael De Vlieger, Oct 09 2017 *)

PROG

(PARI)

A147612aux(n, i) = if(!(n%2), n, A147612aux((n+i)/2, -i));

A147612(n) = 0^(A147612aux(n, 1)*A147612aux(n, -1));

A293434(n) = sumdiv(n, d, (d<n)*A147612(d)*d);

(Python)

from sympy import divisors

def A293434(n): return sum(d for d in divisors(n, generator=True) if d<n and (m:=3*d+1).bit_length()>(m-3).bit_length()) # Chai Wah Wu, Apr 18 2025

CROSSREFS

Cf. A001045, A147612, A293432, A293433.

Cf. also A293436.

Sequence in context: A222371 A222479 A285788 * A091570 A116669 A016523

Adjacent sequences: A293431 A293432 A293433 * A293435 A293436 A293437

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 09 2017

STATUS

approved