A266344 -id:A266344 - cp's OEIS frontend

A266345 Positions of records in A266344.

1, 72, 120, 1440, 1680, 2160, 2520, 37800, 45360, 50400, 55440, 65520, 75600, 83160, 1441440, 1663200, 1801800, 1884960, 1965600, 2106720, 2162160, 2751840, 2827440, 2882880, 3326400, 3603600

Offset: 1

Antti Karttunen, Dec 28 2015

nonn

The corresponding record values, A266344(a(n)) are: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 17, 19, 22, 24, 25, 26, 27, 28, 33, 34, 38, 40, 42, 48, ...

Cf. A266344.

A084558(n) = { my(m=1); if(0==n,n,while(m!<=n,m++);return(m-1)); }
A266344(n) = sumdiv(n, d, ((d <= (n/d)) && (A084558(d)==A084558(n/d))));
m=0; i=0; for(n=1, 1625621761, k = A266344(n); if(k > m, m = k; i++; write("b266345.txt", i, " ", n)));

A266342 a(n) = number of ways n can be expressed as a product of two natural numbers that have same number of significant digits in base-2 representation (up to the ordering of unequal factors).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2

Offset: 1

Antti Karttunen, Dec 27 2015

			For n=1 we have one possibility, 1*1 = 1, thus a(1) = 1.
For n=2 we have no choices, as the binary representation of 1 which is "1" is shorter than the binary representation of 2 which is "10", thus a(2) = 0 (and likewise for any prime).
For n=120 we have two choices, either 8*15 (in binary "1000" * "1111") or 10*12 ("1010" * "1100"), thus a(120) = 2. (15*8 and 8*15 are not counted separately.)

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000523.
Cf. A266346 (positions of nonzeros), A266347 (positions of zeros).
Cf. A266343 (positions of records).
Cf. also A266344.

Map[Length, Table[Flatten@ Map[Differences@ IntegerLength[#, 2] &, Transpose@ {#, n/#}] &@ TakeWhile[Divisors@ n, # <= Sqrt@ n &], {n, 120}] /. k_ /; k > 0 -> Nothing] (* Michael De Vlieger, Dec 30 2015, Version 7.0 *)

A000523(n) = if(n<1,0,#binary(n) - 1);
A266342(n) = sumdiv(n, d, ((d <= (n/d)) && (A000523(d)==A000523(n/d))));
for(n=1, 32768, write("b266342.txt", n, " ", A266342(n)));

a(n) = Sum_{d|n} [(d <= (n/d)) and (A000523(d) = A000523(n/d))].

(In the above formula [ ] stands for Iverson bracket, resulting in 1 only if d is less than or equal to n/d and the binary lengths of d and n/d are equal, and 0 otherwise.)

A078781 Numbers n such that n!-1 is a semiprime.

Original entry on oeis.org

5, 8, 10, 13, 16, 20, 23, 24, 26, 27, 34, 36, 40, 47, 50, 59, 68, 79, 85, 93, 137, 143, 151

Offset: 1

Jason Earls, Jan 09 2003

The next candidate for a continuation is 154!-1, which is composite with 272 decimal digits and unknown factorization. Further known terms are 157, 229, 381, 390, 392, 400, 814, 929; factorization unknown for 154, 196, 232, 271, 307, 322, 332, 333, 334, 350, 352, 386, 389, 443, 449, ...
Note that the two prime factors of 24!-1 = 620448401733239439359999 = 625793187653 * 991459181683 both have 12 decimal digits.
There is another term with prime factors with equal number of decimal digits: 34!-1 = 10398560889846739639*28391697867333973241 (20 digits each)
From Antti Karttunen, Dec 27 2015: (Start)
Furthermore, both factors of 24!-1 are in binary system 40 bits long (A070939), and in factorial base representation (A007623) they both have 14 digits: <7,2,6,5,4,8,2,3,0,0,2,0,2,1> and <11,5,2,10,1,5,6,3,4,1,1,3,0,1>. That is, A007623(625793187653) = 72654823002021, but the latter number cannot be represented reliably in such a more compact form, because it already contains digits > 9.
Factors of 34!-1 are 64 and 65 bits long, and their factorial base representations contain both 20 digits: <4,5,9,3,1,13,11,7,9,1,0,6,1,1,6,5,3,1,0,1> and  <11,13,7,10,0,12,3,4,6,11,1,8,1,4,2,2,1,2,2,1>.
Also the factors of 5!-1 = 119 = 7*17 are both of the same length in factorial base system: "101" and "221".
(End)
1338, 1447, 1788, 1824, 2805, 2881, 2960, 5824 are also terms of the sequence. - Chai Wah Wu, Feb 28 2020

FactorDB, Status of 154!-1.
Paul Leyland, Tables of factors of N!+1 and N!-1.
Andrew Walker, Factors of n!-1 for n>=400.

Cf. A001358, A080802.
Cf. A078778 (numbers such that n!+1 is a semiprime).
Cf. also A007623, A070939, A266344.

IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [3..50] | IsSemiprime(Factorial(n)-1)]; // Vincenzo Librandi, Dec 28 2015

Select[Range[50], PrimeOmega[#! - 1] == 2 &] (* Vincenzo Librandi, Dec 28 2015 *)

{ fm(a,b)=local(c,d,r); for(n=a,b,r=n!-1; c=vecmin(factor(r)[,1]~); d=vecmax(factor(r)[,1]~); if(bigomega(r)==2 && isprime(c) && isprime(d), print1(n" ");))} fp(2,100)

More terms from Hugo Pfoertner, Apr 05 2003

a(23) added by Daniel Suteu, Mar 30 2019

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A266345 Positions of records in A266344.

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A266342 a(n) = number of ways n can be expressed as a product of two natural numbers that have same number of significant digits in base-2 representation (up to the ordering of unequal factors).

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A078781 Numbers n such that n!-1 is a semiprime.

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