A047235 -id:A047235 - cp's OEIS frontend

A319674 a(n) = 1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 - 10 - 11 - 12 + ... - (up to n).

1, 3, 6, 2, -3, -9, -2, 6, 15, 5, -6, -18, -5, 9, 24, 8, -9, -27, -8, 12, 33, 11, -12, -36, -11, 15, 42, 14, -15, -45, -14, 18, 51, 17, -18, -54, -17, 21, 60, 20, -21, -63, -20, 24, 69, 23, -24, -72, -23, 27, 78, 26, -27, -81, -26, 30, 87, 29, -30, -90, -29

Offset: 1

Wesley Ivan Hurt, Sep 25 2018

In general, for sequences that add the first k natural numbers and then subtract the next k natural numbers, and continue to alternate in this way up to n, we have a(n) = Sum_{i=1..n} i*(-1)^floor((i-1)/k). Here, k=3.

			a(1) = 1;
a(2) = 1 + 2 = 3;
a(3) = 1 + 2 + 3 = 6;
a(4) = 1 + 2 + 3 - 4 = 2;
a(5) = 1 + 2 + 3 - 4 - 5 = -3;
a(6) = 1 + 2 + 3 - 4 - 5 - 6 = -9;
a(7) = 1 + 2 + 3 - 4 - 5 - 6 + 7 = -2;
a(8) = 1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 = 6;
a(9) = 1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 = 15;
a(10) = 1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 - 10 = 5; etc.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,-2,2,0,-1,1).

Cf. A001057 (k=1), A077140 (k=2), this sequence (k=3).

Table[Sum[i (-1)^Floor[(i - 1)/3], {i, n}], {n, 60}]
Accumulate[Flatten[If[EvenQ[#[[1]]],-#,#]&/@Partition[Range[70],3]]] (* or *) LinearRecurrence[{1,0,-2,2,0,-1,1},{1,3,6,2,-3,-9,-2},70] (* Harvey P. Dale, Sep 15 2021 *)

Vec(x*(1 + 2*x + 3*x^2 - 2*x^3 - x^4) / ((1 - x)*(1 + x)^2*(1 - x + x^2)^2) + O(x^60)) \\ Colin Barker, Sep 26 2018

a(n) = Sum_{i=1..n} i*(-1)^floor((i-1)/3).
From Colin Barker, Sep 26 2018: (Start)
G.f.: x*(1 + 2*x + 3*x^2 - 2*x^3 - x^4) / ((1 - x)*(1 + x)^2*(1 - x + x^2)^2).
a(n) = a(n-1) - 2*a(n-3) + 2*a(n-4) - a(n-6) + a(n-7) for n>7.
(End)
Conjectures from Bill McEachen, Dec 19 2024: (Start)
For a(n)>0 and n=A047235(m), a(n) = n/2 + 2*Mod(m,2), otherwise a(n) = 3*(n+1)/2.
For a(n)<0 and n=A007310(m), a(n)= 1 + (1-n)/2 + 2*(Mod(m,2)-1), otherwise a(n) = -3*n/2. (End)

A355017 a(n) is the number of bases in 2..n in which the sum of the digits of n is prime.

Original entry on oeis.org

0, 1, 1, 3, 4, 4, 3, 5, 6, 7, 7, 8, 7, 8, 5, 11, 9, 10, 8, 13, 8, 12, 9, 13, 11, 12, 10, 15, 11, 16, 10, 17, 10, 20, 12, 20, 14, 18, 13, 21, 13, 22, 13, 20, 14, 25, 14, 22, 18, 22, 15, 26, 12, 29, 17, 25, 15, 27, 15, 30, 19, 26, 14, 32, 17, 33, 19, 27, 19, 31, 18, 34, 19, 29, 19, 37, 16, 33, 21, 30, 24, 39, 20, 38

Offset: 2

Samuel Harkness, Jun 15 2022

The graph of (n,a(n)) shows an interesting structure, somewhat resembling a comet with four tails. Starting at the bottom tail and going upwards:
Observations:
The bottom "tail" contains all n with both 2 and 3 as prime factors, i.e., numbers n in A008588 (1/6 of all n).
The second "tail" contains all n with 2 as a prime factor but not 3, i.e., numbers n in A047235 (1/3 of all n).
The third "tail" contains all n with 3 as a prime factor but 2, i.e., numbers n in A016945 (1/6 of all n).
The top "tail" contains all n with neither 2 nor 3 as a prime factor, i.e., numbers n in A007310 (1/3 of all n).
The bottom of each "tail" contains n with 5 as a prime factor. Moving up within each "tail," the prime factors of each n tend to increase.

			For n=7, express 7 in all bases from 2 to 7, then add the numbers, counting those which are prime:
  base 2: 1 1 1 --> 1+1+1=3 prime
  base 3: 2 1   --> 2+1=3   prime
  base 4: 1 3   --> 1+3=4   nonprime
  base 5: 1 2   --> 1+2=3   prime
  base 6: 1 1   --> 1+1=2   prime
  base 7: 1     --> 1=1     nonprime
The sum of the digits of the base-b expansion of 7 in 4 different bases b (2, 3, 5, and 6) from base 2 to 7 is prime, so a(7)=4.

Samuel Harkness, Table of n, a(n) for n = 2..10000
Samuel Harkness, Colored Scatterplot of the first 50000 terms, with n as multiples of 2 or 3
Samuel Harkness, Colored Scatterplot of the first 50000 terms, with the lowest factor of n among 5, 7, 11, or 13
Rémy Sigrist, Scatterplot of (n, b) such that the sum of digits of n in base b is prime (with n = 1..1000, b = 2..n).
Rémy Sigrist, Colored scatterplot of the first 50000 terms (where the color is function of gcd(n, 2*3*5)).

Cf. A008588, A047235, A016945, A007310, A028834, A052294.

a[n_] := Count[Range[2, n], ?(PrimeQ[Plus @@ IntegerDigits[n, #]] &)]; Array[a, 84, 2] (* _Amiram Eldar, Jun 17 2022 *)

a(n) = sum(b=2, n, isprime(sumdigits(n, b))); \\ Michel Marcus, Jun 16 2022

from sympy.ntheory import isprime, digits
def A355017(n): return sum(1 for b in range(2,n) if isprime(sum(digits(n,b)[1:]))) # Chai Wah Wu, Jun 17 2022

cp's OEIS Frontend

A319674 a(n) = 1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 - 10 - 11 - 12 + ... - (up to n).

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