A392604 - OEIS

%I #30 Feb 18 2026 08:18:44

%S 0,1,2,3,4,8,9,16,32,33,64,128,129,256,512,1024,1025,1027,1033,2048,

%T 2049,4096,8192,8193,16384,32768,65536,131072,131073,131081,132097,

%U 262144,262145,524288,524307,1048576,1048577,2097152,2097153,4194304,8388608,16777216,16777217,33554432,67108864,67108865,134217728

%N Numbers k such that A380459(A019565(k)) has no divisors of the form p^p, for any prime p.

%C Sequence A048675(A380468(.)) sorted into ascending order.

%C It is conjectured that A000120(a(n)) is always <= 4. See A380475.

%C Numbers k >= 1 such that 2^k + 1 is present in this sequence are given by {1} U A138968, and the corresponding primes, A000040(1+k) are given by A007645, i.e., 3 and primes of the form 6m + 1. Proof: A276086(2) = 3, and for any prime p in A047257, A276086(p) is a multiple of 9, thus 3*9 = 3^3 is already "over the quota" for prime factor 3. On the other hand, for any prime p in A045331, A276086(p) is not a multiple of 9, thus multiplying it with A276086(2) = 3 is allowed.

%H Antti Karttunen, <a href="/A392604/b392604.txt">Table of n, a(n) for n = 1..52; all terms less than 570425344</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>.

%F {k such that A380467(A019565(k)) = 1}.

%e 1027 = A048675(186) = 2^10 + 2^1 + 2^0 = "10000000011" in binary, is a term.

%e 524307 = A048675(A380476(1)) = A048675(4686) = 2^19 + 2^4 + 2^1 + 2^0 = "10000000000000010011" in binary, is a term. See example in A380476.

%o (PARI) is_A392604(n) = A380467(A019565(n));

%o (PARI)

%o search_up_to_2_e = 20;

%o choosebybits(v,m) = { my(s=vector(hammingweight(m)),i=j=1); while(m>0,if(m%2,s[j] = v[i];j++); i++; m >>= 1); s; };

%o A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

%o A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); };

%o is_it(ps) = { my(n=factorback(ps),d,m=1); for(i=1,#ps,d=ps[i]; m *= A276086(n/d); if(!A359550(m), return(0))); (1); };

%o A392604_list(up_to_2_e) = { my(lista=List([]),all_ps=primes(up_to_2_e)); for(w=0, (2^up_to_2_e)-1, if(is_it(choosebybits(all_ps,w)), listput(lista,w))); Vec(lista); };

%o v392604 = A392604_list(search_up_to_2_e);

%o A392604(n) = v392604[n];

%Y Cf. A000079 (subsequence), A000120, A019565, A048675, A276086, A276156, A359550, A380467, A380468, A380475, A380476, A392603 [= A276156(a(n))].

%K nonn,base

%O 1,3

%A _Antti Karttunen_, Jan 21 2026