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For k = 5 or 15 it is true for primes p > m/k with p == 1 (mod 4).
Given positive integer k, let m = A001554(k).
If p is a prime > m/k and A001554(p*k) == 29, m (mod k), then a(p*k) = 28m.
If p is a prime >= 71, a(2*p) = 140.
If p is a prime >= 263, a(3*p) = 784.
If p is a prime >= 1171, a(4*p) = 4676.
This is true for all primes p > m/k for k = 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 14, ...
If p For k = 5 it is a prime true for primes p >= 5807 and m/k with p == 1 (mod 4), a(5*p) = 29008.
If p is a prime >= 30809, a(6*p) = 184820.
If p is a prime >= 171473, a(7*p) = 1200304. (End)
For k = 11 it is true for primes p > m/k with p == 1 or 7 (mod 10).
For k = 13 it is true for primes p > m/k with p == 1 (mod 12).
(End)
If p is a prime >= 5807 and p == 1 (mod 4), a(5*p) = 29008. (End)
If p is a prime >= 30809, a(6*p) = 184820.
If p is a prime >= 171473, a(7*p) = 1200304. (End)
From Robert Israel, Feb 09 2023: (Start)
If n p is a prime >= 29, a(np) = 28. _Robert Israel_, Feb 09 2023
If p is a prime >= 71, a(2*p) = 140.
If p is a prime >= 263, a(3*p) = 784.
If p is a prime >= 1171, a(4*p) = 4676.
If p is a prime >= 5807 and p == 1 (mod 4), a(5*p) = 29008. (End)
Robert Israel, <a href="/A341413/b341413.txt">Table of n, a(n) for n = 1..10000</a>
If n is a prime >= 29, a(n) = 28. Robert Israel, Feb 09 2023
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a:= n-> add(i&^n, i=1..7) mod n:
seq(a(n), n=1..100); # Alois P. Heinz, Feb 11 2021
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