A307137 Terms of A194269 that are not squares of primes.

68, 17500, 5053176, 435034872, 3079148398, 107924794257, 122918945808, 63602175290616, 27232626132792608, 131685306017557752

Offset: 1

Author

Michel Marcus, Mar 26 2019

Keywords

• nonn
• more

Comments

A194269 lists the integers j that satisfy Sum_{i=1..k} d(i)^i = j+1 for some k where d() is the sorted list of divisors of j.

All squares of primes (A001248) are in A194269 (with k=2), but some terms of A194269 are not in A001248.

So the sequence consists of the integers that satisfy Sum_{i=1..k} d(i)^i = j+1 for k > 2.

a(1) = 68 = 1 + 2^2 + 4^3 is the only term with k = 3; the others should have k > 3. - Max Alekseyev, Nov 18 2024

Also in the sequence is 1125089196456707568267636780. - Daniel Suteu, Mar 28 2019

146891545244361186001164385301664828062299990188246080274986418 and 1074195891803336036687617104130500495489848428998310547357799403665 are also terms. - Giovanni Resta, Mar 29 2019

No other terms below 10^22. Also contains 1397818536728878029228901636216061803887128332853093401018283627886158780 and 187-digit number Sum_{i=1..52} A027750(734441344,i)^i. - Max Alekseyev, Nov 22 2024

Examples

			The divisors of 68 are 1, 2, 4, 17, 34, 68; 1^1 + 2^2 + 4^3 = 69, so 68 is a term.
		

Crossrefs

Cf. A001248, A194269.

Programs

PARI

isok1(n) = {my(d=divisors(n), s=0); for(k=1, #d, s += d[k]^k; if (s == n+1, return (k)); if (s > n+1, break););0;} \\ A194269
isok(n) = isok1(n) && isprimepower(n)!=2;

Extensions

a(6)-a(7) from Giovanni Resta, Mar 29 2019

a(8)-a(10) from Daniel Suteu confirmed by Max Alekseyev, Nov 18 2024