A169655 -id:A169655 - cp's OEIS frontend

A177436 The number of positive integers m for which the exponents of 2 and prime(n) in the prime power factorization of m! are both powers of 2.

7, 7, 6, 3, 4, 4, 3, 4, 8, 10, 2, 2, 2, 4, 6, 8, 10, 3, 2, 2, 2, 2, 4, 4, 4, 5, 6, 6, 6, 14, 3, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 8, 8, 8, 8, 12, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6

Offset: 2

Vladimir Shevelev, May 08 2010

nonn

Or a(n) is the maximal m for which the Fermi-Dirac representation of m! (see comment in A050376) contains single power of 2 and single power of prime(n).

			For p_5 = 11, we have 11 = 2^3+3. Therefore a(5) = 3.
For p_27 = 103, we have 103 = (2^(4*2+1)+3)/5. Therefore a(27) = 5.
For p_31 = 127, a(31) = 2*(1+floor(log_2((127-5)/(128-127)))) = 14.

Robert Price, Table of n, a(n) for n = 2..127
Vladimir Shevelev, Compact integers and factorials, Acta Arithmetica 126 (2007), no. 3, 195-236.

Cf. A000142, A177378, A177355, A177349, A177458, A177459, A177498, A050376, A169655, A169661.

nlim = 127; mlim = (Prime[nlim] + 1)^2/2 + 3; f = Table[0, mlim]; c = Table[0, nlim];
For[m = 2, m <= mlim, m++,
  mf = FactorInteger[m];
  For[i = 1, i <= Length[mf], i++, f[[PrimePi@First@mf[[i]]]] += Last@mf[[i]]];
  If[! IntegerQ@Log[2, f[[1]]], Continue[]];
  For[p = 1, p <= nlim, p++, If[IntegerQ@Log[2, f[[p]]], c[[p]]++]];
]; c (* Robert Price, Jun 19 2019 *)

a(2) = a(3) = 7; a(4) = 6; if p_n has the form (2^(4*k+1)+3)/5, k>=2, then a(n) = 5; if p_n is a Fermat prime: p_n = 2^(2^(k-1))+1, k>=3, then a(n) = 4; if p_n has the form 2^k+3, k>=3, then a(n) = 3; otherwise, if 2^(k-1)+3 < p_n <= 2^k-1, then a(n) = 2*(1+floor(log_2((p_n-5)/(2^k-p_n)))), where p_n = prime(n).

a(32)-a(127) from Robert Price, Jun 19 2019

A169661 Compact factorials of positive integers.

Original entry on oeis.org

1, 2, 6, 720, 5040, 3628800, 39916800

Offset: 1

Vladimir Shevelev, Apr 05 2010, Jun 29 2010

A positive integer m is called a compact number if all factors of unique factorization of n over distinct terms of A050376 are relatively prime. It is convenient to suppose that 1 is compact number. Although the density of compact numbers is 0.872497..., it is easy to prove that the set of compact factorials is finite. Indeed, if n is sufficiently large, then the interval (n/4,n/3) contains a prime p and thus p^3||n! Therefore the factorization of n! over A050376 contains product p*p^2. Much more difficult to show that all compact factorials are: 1!,2!,3!,6!,7!,10!,11!. All these factorials are presented in the table.

T. M. Apostol, Review of "Compact integers and factorials" by V. Shevelev, zbMATH.
Vladimir Shevelev, Compact integers and factorials, Acta Arith. 126 (2007), no.3, 195-236.

Cf. A050376, A169655, A263881.

a(n) = A263881(n)!. - Jonathan Sondow, Nov 17 2015

A177378 a(n) is the smallest prime p>2 such that there are 2n or 2n+1 positive integers m for which the exponents of 2 and p in the prime power factorization of m! are both powers of 2.

Original entry on oeis.org

11, 13, 3, 29, 31, 251, 127, 509, 1021, 4091, 4093, 65519, 8191, 131063, 262133, 262139, 131071, 1048571, 524287, 8388593, 4194301, 67108837, 16777213, 67108861, 1073741789, 2147483587, 2147483629, 536870909

Offset: 1

Vladimir Shevelev, May 07 2010

nonn

			By the formula, for n=6, consider k >= 6. If k=6, then g(6,6) = 3, but 6 does not equal to 6 - floor(log_2(3)); if k=7, then g=15, but 6 does not equal to 7 - floor(log_2(15)); if k=8, then g=5 and we see that 6 = 8 - floor(log_2(5)). Therefore a(6) = 2^8 - 5 = 251.

V. Shevelev, Compact integers and factorials, Acta Arithmetica 126 (2007), no. 3, 195-236.

Cf. A000142, A050376, A169655, A169661, A177349, A177355, A177436, A177458, A177459, A177498.

For sufficiently large n, 2^n - 1 <= a(n) <= 2^ceiling(40*n/19). Let k >= n. Put g = g(n,k) = min{odd j >= 2^(k-n): 2^k - j is prime} and h(n) = min{k: k - n = floor(log_2(g))}. Then a(n) = 2^h(n) - g(n,h(n)).

A177459 The maximal positive integer m for which the exponents of 2 and prime(n) in the prime power factorization of m! are both powers of 2.

Original entry on oeis.org

19, 131, 34, 19, 35, 35, 35, 67, 259, 575, 67, 67, 67, 131, 259, 515, 1027, 131, 131, 131, 131, 131, 259, 259, 259, 514, 515, 515, 515, 8195

Offset: 2

Vladimir Shevelev, May 09 2010

Or a(n) is the maximal m for which the Fermi-Dirac representation of m! (see comment in A050376) contains single power of 2 and single power of prime(n).

			For n=31, prime(n)=127 is Mersenne primes. Thus a(31)=(1/2)*128^2+3=8195.

Vladimir Shevelev, Compact integers and factorials, Acta Arithmetica 126 (2007), no. 3, 195-236.

Cf. A000142, A177436, A177378, A177355, A177349, A177458, A177498, A050376, A169655, A169661.

a(2)=19, a(3)=131; if prime(n) has the form (2^(4k+1)+3)/5 for k>=1,then a(n)=5*prime(n)-1; if prime(n)>=17 is Fermat prime, then a(n)=2*prime(n)+1; if prime(n) has the form 2^k+3 for k>=3, then a(n)=2*prime(n)-3; otherwise, if prime(n) is in interval [2^(k-1)+5, 2^k) for k>=4, then a(n)=3+2^(k+floor(log_2((p_n-5)/(2^k-prime(n)))). In any case, a(n)<=(1/2)*(prime(n)+1)^2+3. Equality holds for Mersenne primes>=31.

A182005 Let s(n) = s_3(n) be digit sum of n in base 3. Consider iterations: a_1(n) = s(2^n), a_2(n) = s(2^n+a_1(n)),a_3(n)=s(2^n+a_2(n)),...The sequence lists those n for which these iterations are (eventually) periodic with period > 1.

Original entry on oeis.org

5, 7, 8, 17, 21, 23, 26, 31, 39, 40, 41, 45, 49, 51, 52, 53, 58, 62, 64, 67, 69, 78, 81, 82, 84, 87, 91, 93, 108, 113, 115, 116, 119, 121, 122, 128, 131, 135, 136, 139, 142, 151, 152, 155, 163, 170, 173, 174, 178, 181, 191, 193, 195, 198, 201

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Apr 06 2012

For every number n which is not in the sequences, there exists N=N(n) such that, for k>N, a_k(n)=constant(k).

Cf. A053735, A000079.

Enlarged on 1 numbers which are not in A169655.

cp's OEIS Frontend

A177436 The number of positive integers m for which the exponents of 2 and prime(n) in the prime power factorization of m! are both powers of 2.

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A169661 Compact factorials of positive integers.

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A177378 a(n) is the smallest prime p>2 such that there are 2*n or 2*n+1 positive integers m for which the exponents of 2 and p in the prime power factorization of m! are both powers of 2.

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A177459 The maximal positive integer m for which the exponents of 2 and prime(n) in the prime power factorization of m! are both powers of 2.

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A182005 Let s(n) = s_3(n) be digit sum of n in base 3. Consider iterations: a_1(n) = s(2^n), a_2(n) = s(2^n+a_1(n)),a_3(n)=s(2^n+a_2(n)),...The sequence lists those n for which these iterations are (eventually) periodic with period > 1.

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A177378 a(n) is the smallest prime p>2 such that there are 2n or 2n+1 positive integers m for which the exponents of 2 and p in the prime power factorization of m! are both powers of 2.