cp's OEIS Frontend

A331547 Numbers k such that k and k! - 1 have the same number of divisors.

%I A331547 #43 Mar 07 2020 02:02:55
%S A331547 3,7,8,10,26,27,34,85,93,104,143,152
%N A331547 Numbers k such that k and k! - 1 have the same number of divisors.
%C A331547 The sequence also includes: 143, 152, 186, 230, 379, 381, 543, 573, 602. - _Daniel Suteu_, Jan 21 2020
%C A331547 The sequence also includes 2881. Even though the complete factorization of 136!-1 is not known, 136 is not a term, since 136!-1 is known to be the product of 2 distinct primes and a composite number, so it has at least 4 prime factors and 3 distinct prime factors, thus the number of divisors >= 12, whereas 136 has 8 divisors. - _Chai Wah Wu_, Feb 26 2020
%C A331547 Similar reasoning (considering only small prime factors of k! - 1) shows that the next terms (> a(12) = 152) can only be within the set {154, 160, 162, 164, 176, 180, 182, 186, 187, 188, 192, 195, 196, 198, 204, ...}. - _M. F. Hasler_, Feb 26 2020
%H A331547 factordb.com, <a href="http://factordb.com/index.php?query=154%21-1">Status of 154!-1</a>.
%F A331547 A331547 = { n > 1 | A000005(n) = A064145(n) }. - _M. F. Hasler_, Feb 26 2020
%t A331547 Select[Range[50], DivisorSigma[0, #] - DivisorSigma[0, Factorial[#] - 1] == 0 &]
%o A331547 (PARI) isok(k) = k>1 && numdiv(k)==numdiv(k!-1); \\ _Jinyuan Wang_, Jan 20 2020
%o A331547 (PARI) {is(n)=my(f); n>2&& numdiv(n)>=numdiv(f=factor(n!-1,0))&& if( ispseudoprime(vecmax(f[,1])), numdiv(n)==numdiv(f), numdiv(n)<2*numdiv(f), 0, numdiv(n)==numdiv(n!-1))} \\ Avoids complete factorization if possible. - _M. F. Hasler_, Feb 26 2020
%Y A331547 Supersequence of A103317.
%Y A331547 Cf. A000005, A002982, A064145.
%K A331547 nonn,hard,more
%O A331547 1,1
%A A331547 _Matthew Niemiro_, Jan 20 2020
%E A331547 a(8)-a(9) from _Jinyuan Wang_, Jan 20 2020
%E A331547 a(10) from _Amiram Eldar_, Jan 20 2020
%E A331547 a(11)-a(12) from _Chai Wah Wu_, Feb 26 2020