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 A305024 Minimal number of squares, not all equal to 1, having as sum prime(n), such that their squares also sum to a prime; 0 if no such decomposition exists. 1
 0, 0, 2, 4, 3, 2, 2, 3, 4, 2, 4, 2, 2, 3, 7, 2, 5, 4, 4, 4, 2, 4, 3, 2, 3, 3, 5, 3, 4, 5, 4, 3, 2, 3, 2, 4, 2, 4, 4, 3, 3, 2, 4, 4, 4, 4, 3, 4, 3, 4, 3, 4, 3, 3, 2, 4, 2, 4, 3, 2, 3, 2, 4, 4, 2, 4, 4, 4, 3, 2, 3, 4, 4, 2, 3, 4, 3, 2, 2, 2, 3, 2, 4, 3, 4, 3, 3, 4, 2, 4, 3, 4, 4, 3, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS It has been conjectured (cf. A126769) that any prime p >= 5 can be written in a nontrivial way as p = Sum (b_i)^2 such that Sum (b_i)^4 is also prime. This sequence lists the number of required terms b_i for each prime. The two initial zeros say that this decomposition is not possible for prime(1) = 2 and prime(2) = 3, and are thus conjectured to be the only zeros of the sequence. Since we are interested in the minimal number of terms, we can consider only nonzero b_i >= 1 and min{b_i} >= 2 to avoid the trivial solution b_i = 1 for all i <= k = prime(n). LINKS Robert Israel, Table of n, a(n) for n = 1..5000 FORMULA If EXAMPLE The first two primes, 2 and 3, cannot be written as a sum of squares not all equal to 1, because the smallest such sum is 1^2 + 2^2 = 5. (The empty sum and a one-term sum of a square cannot be prime, either.) Therefore a(1) = a(2) = 0. The third prime, 5, can be written in exactly one way as a nontrivial sum of two squares, 5 = 1^2 + 2^2, and the sum of the fourth powers is 1^4 + 2^4 = 17, which is again prime. Therefore, a(3) = 2. The fourth prime, 7, cannot be written as sum of 2 or 3 squares, but only 4 squares, as 7 = 1^2 + 1^2 + 1^2 + 2^2, and it turns out that sum of the fourth powers also yields a prime,  1^4 + 1^4 + 1^4 + 2^4 = 19. Therefore, a(4) = 4. prime(15) = 47 = 1^2 + 2^2 + 2^2 + 2^2 + 3^2 + 3^2 + 4^2, and the sum of the fourth powers gives the prime 467. Since no smaller number of terms has this property, a(15) = 7. prime(18) = 61 = 2^2 + 4^2 + 4^2 +5^2, and 2^4 + 4^4 + 4^4 +5^4 = 1153, a prime, and no smaller number of terms has this property, so a(18) = 4. prime(27) = 103 = 1^2 + 4^2 + 5^2 + 5^2 + 6^2 and 1^4 + 4^4 + 5^4 + 5^4 + 6^4 = 2803, a prime, and no smaller number of terms has this property, so a(27) = 5. The values 2, 3, 4, 5 appear for the first time at index n = 3, 5, 4, 17, and a(15) = 7. We don't know when the first 6 occurs, nor whether this happens at all. Conjecture: The sequence is bounded. Is it possible to show that no term of the sequence is larger than 7? MAPLE repss:= proc(n, k, i) option remember; # lists of k squares >= i^2 summing to n   if k = 1 then    if issqr(n) and n >= i^2 then {[sqrt(n)]}    else {}    fi   elif n < k then {}   else    `union`(seq(map(t -> [j, op(t)], procname(n-j^2, k-1, j)), j=i..floor(sqrt(n))))   fi end proc: f:= proc(n) local p, k, i, S; global Rep;   p:= ithprime(n);   for k from 2 do     S:= select(t -> isprime(convert(map(`^`, t, 4), `+`)), repss(p, k, 1));     if nops(S) > 0 then Rep[n]:= S; return k fi   od end proc: 0, 0, seq(f(n), n=3..100); # Robert Israel, Dec 12 2019 MATHEMATICA a[n_] := Block[{p = Prime@n, c, k=2}, c = Range[Sqrt[p]]^2; While[ k0 && isprime(s[i]^4+s[i]^4+t) && return(k))/*end for i*/     , 1/*forvec:increasing*/))}, [1..95]) \\ Bug fixed: M. F. Hasler, Dec 12 2019 CROSSREFS Cf. A126769, A128292. Sequence in context: A029717 A135567 A105972 * A064134 A238847 A011030 Adjacent sequences:  A305021 A305022 A305023 * A305025 A305026 A305027 KEYWORD nonn AUTHOR M. F. Hasler, May 23 2018 EXTENSIONS Corrected by Robert Israel, Dec 12 2019 STATUS approved

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Last modified September 23 04:52 EDT 2021. Contains 347609 sequences. (Running on oeis4.)