A391252 Odd composites k with more than 2 prime factors for which it holds that k = A048720(k/d, A065621(d)), for all divisor pairs (d, k/d), d <= k/d.

4953, 11049, 319449, 712617, 1136993, 1171313, 1498953, 2200913, 2271841, 2383581, 2776749, 4578769, 4627915, 4742589, 4797273, 5111769, 5496161, 5922093, 6479081, 8739797, 8838089, 9100201, 9755481, 9821009, 9870155, 10050357, 10144329, 11033277, 11057961, 11316873, 11328153, 11403177, 12507657, 13818217, 17389493

Offset

1,1

Comments

Sequence A391248 without any semiprimes.

The first terms with four prime factors are 1231936329, 4517099873, 4927773513, 5157774921.

Questions: Are all terms squarefree? Can A001222(a(n)) be larger than 4, or even arbitrarily large?

Links

Antti Karttunen, Table of n, a(n) for n = 1..505

Formula

{k such that A000035(k) = 1, A001222(k) > 2 and A391246(k) = A038548(k)}.

Example

4953 = 3*13*127, and its divisors are [1, 3, 13, 39, 127, 381, 1651, 4953]. Let A(n, k) = A277199(n, k) = A048720(n, A065621(k)), then A(4953,1) = A(1651,3) = A(381,13) = A(127,39) = 4953, so this holds for all A038548(4953) = 4 divisor pairs (n/d, d), d <= n/d of 4953, therefore 4953 is included in this sequence.
1136993 = 11*13*7951, and its divisors are [1, 11, 13, 143, 7951, 87461, 103363, 1136993], and for 4 divisor pairs we have A(1136993,1) = A(103363,11) = A(87461,13) = A(7951,143) = 1136993, therefore 1136993 is included.
1231936329 = 3*13*241*131071 has divisors [1, 3, 13, 39, 241, 723, 3133, 9399, 131071, 393213, 1703923, 5111769, 31588111, 94764333, 410645443, 1231936329], and for 8 divisor pairs we have A(1231936329,1) = A(410645443,3) = A(94764333,13) = A(31588111,39) = A(5111769,241) = A(1703923,723) = A(393213,3133) = A(131071,9399) = 1231936329, therefore 1231936329 is included in this sequence.

Prog

(PARI)
A048720(b, c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1, n+n-1);
A391246(n) = sumdiv(n, d, ((d <= (n/d)) && n==A391245_tr(n/d, d)));
is_A391252(n) = if(!(n%2) || bigomega(n)<=2, 0, fordiv(n, d, if(d>(n/d), return(1), if(n!=A048720(n/d, A065621(d)), return(0)))));

Crossrefs

Intersection of A391248 and A033942.

Cf. A001222, A048720, A065621, A277199.

Sequence in context: A260939 A188547 A037044 * A251847 A225718 A228302

Adjacent sequences: A391249 A391250 A391251 * A391253 A391254 A391255

Keyword

nonn

Author

Antti Karttunen, Dec 06 2025

Status

approved