A336825 a(n) is the smallest positive integer which is expressed by the greedy algorithm as the sum of exactly n prime-powers (including 1).

1, 6, 95, 360748

Offset: 1

Author

Jonathan Hoseana, Aug 04 2020

Keywords

• nonn
• more

Comments

Analogous to A066352 with prime-powers replacing primes.

Examples

			The greedy algorithm expresses every positive integer as a sum of prime-powers (including 1) by choosing the largest possible summand at each step. Consider the following initial data of such expressions:
1 = 1,
2 = 2,
3 = 3,
4 = 4,
5 = 5,
6 = 5 + 1,
7 = 7,
8 = 7 + 1,
9 = 9,
10 = 9 + 1.
The smallest positive integer which is expressed by the greedy algorithm as the sum of exactly 1 prime-power is a(1) = 1. The smallest positive integer which is expressed by the greedy algorithm as the sum of exactly 2 prime-powers is a(2) = 6. Similarly, a(3) = 95 (95 = 89 + 5 + 1) and a(4) = 360748 (360748 = 360653 + 89 + 5 + 1).
		

Links

• Steven and Jonathan Hoseana, The prime-power map, arXiv:2008.01368 [math.DS], 2020.

Crossrefs

Cf. A066352, A000961 (power of primes), A031218.

Programs

PARI

ispp(n) = isprimepower(n) || (n==1); \\ A000961
f(n) = while(!ispp(n), n--); n; \\ A031218
nbs(n) = my(nb=0); while(n, n -= f(n); nb++); nb;
a(n) = my(k=1); while (nbs(k) != n, k++); k; \\ Michel Marcus, Aug 05 2020

Formula

a(1) = 1 and, for every positive integer n, a(n+1) = a(n) + q1(n), where (q1(n), q2(n)) is the first pair of consecutive prime-powers with q2(n) - q1(n) >= a(n) + 1.