A392604 Numbers k such that A380459(A019565(k)) has no divisors of the form p^p, for any prime p.

0, 1, 2, 3, 4, 8, 9, 16, 32, 33, 64, 128, 129, 256, 512, 1024, 1025, 1027, 1033, 2048, 2049, 4096, 8192, 8193, 16384, 32768, 65536, 131072, 131073, 131081, 132097, 262144, 262145, 524288, 524307, 1048576, 1048577, 2097152, 2097153, 4194304, 8388608, 16777216, 16777217, 33554432, 67108864, 67108865, 134217728

Offset

1,3

Comments

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

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

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.

Links

Antti Karttunen, Table of n, a(n) for n = 1..52; all terms less than 570425344

Index entries for sequences related to binary expansion of n.

Index entries for sequences related to primorial base.

Formula

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

Example

1027 = A048675(186) = 2^10 + 2^1 + 2^0 = "10000000011" in binary, is a term.
524307 = A048675(A380476(1)) = A048675(4686) = 2^19 + 2^4 + 2^1 + 2^0 = "10000000000000010011" in binary, is a term. See example in A380476.

Prog

(PARI) is_A392604(n) = A380467(A019565(n));
(PARI)
search_up_to_2_e = 20;
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; };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); };
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); };
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); };
v392604 = A392604_list(search_up_to_2_e);
A392604(n) = v392604[n];

Crossrefs

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

Sequence in context: A006899 A256179 A078830 * A304521 A026489 A126312

Adjacent sequences: A392601 A392602 A392603 * A392605 A392606 A392607

Keyword

nonn,base

Author

Antti Karttunen, Jan 21 2026

Status

approved