A177498 -id:A177498 - 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

A177355 The number of positive integers m for which the exponents of prime(n) and prime(n+1) in the prime power factorization of m! are both powers of 3.

Original entry on oeis.org

3, 1, 3, 4, 14, 10, 26, 22, 22, 61, 38, 59, 97, 77, 70, 82, 156

Offset: 1

Vladimir Shevelev, May 07 2010

			If n=1, then 3<=m<2*(-1+9*(log(2)/(2*log(3)-1)+1))=26.4... In interval [3,26.3) we find only 3 numbers m=3,4,5 with required property. Therefore, a(1)=3.

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

Cf. A000142, A177349, A177378, A177436, A177458, A177459, A177498.

All such m belong to interval [q, 2*(-1+q^2*(log(2)/(2*log(q)-1)+1))), where q=p_(n+1).

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.

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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A177355 The number of positive integers m for which the exponents of prime(n) and prime(n+1) in the prime power factorization of m! are both powers of 3.

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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.

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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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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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A177355 The number of positive integers m for which the exponents of prime(n) and prime(n+1) in the prime power factorization of m! are both powers of 3.

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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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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.