A177355 -id:A177355 - 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

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)).

A177458 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 2.

Original entry on oeis.org

9, 22, 23, 22, 42, 37, 40, 90, 63, 96, 147, 120, 111, 134, 237, 166, 219, 304, 214, 279, 254, 252, 369, 484, 399, 520, 429, 270, 519, 481, 709, 426, 793, 581, 611, 734, 661, 691, 1003, 615, 1087, 914, 1129, 647, 707, 1094, 1339, 1130, 1032, 1423, 915, 1140

Offset: 3

Vladimir Shevelev, May 09 2010, May 10 2010

nonn

This gives the number of rows in A115627 for which the n-th and (n+1)st column are both in {1,2,4,8,16,..}.
For n=2 the corresponding value is not known and >=25; moreover, we do not know if this value is finite.
A more general result concerning the cases for non-adjacent primes and a finite search interval for the values of m is in the 2007 publication.

			For n=3, the 9 values of m are 7, 8, 9, 10, 11, 12, 13, 14, and 20.
m=6, for example, is not counted because 6!=2^4*3^2*5 does not contain prime(4)=7.
m=15, for example, is not counted because 15!=2^11*3^6*5^3*7^2*11*13 contains a third power of prime(3)=5.

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

Cf. A000142, A177355, A177349, A177378, A177436.

tp[n_] := Flatten[Position[FoldList[Plus, 0, IntegerExponent[Range[100000], n]], ?(IntegerQ[Log[2, #]] &)]]; Table[s = Intersection[tp[Prime[n]], tp[Prime[n + 1]]] - 1; Length[s], {n, 3, 60}] (* _T. D. Noe, Apr 10 2012 *)

Edited, example and relation to A115627 added, terms after 120 added by R. J. Mathar, Oct 29 2010

Extended by T. D. Noe, Apr 10 2012

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.

A177498 a(n) is the maximal positive integer m for which exponents of prime(n) and prime(n+1) in the prime power factorization of m! are both powers of 2.

Original entry on oeis.org

20, 98, 54, 38, 152, 94, 68, 260, 154, 332, 696, 386, 234, 476, 1002, 548, 1138, 2342, 656, 1342, 746, 800, 1648, 3332, 1750, 3530, 1852, 1016, 2158, 2226, 8904, 1250, 9684, 2566, 2668, 5378, 2838, 2940, 11634, 3076, 12414, 6368, 12804, 3382, 3586, 7358, 14754

Offset: 3

Vladimir Shevelev, May 10 2010

nonn

For n=2 the corresponding value is not known; moreover, we do not know if this value is finite (in any case, it is not less than 524306). See also comment to A177458.

If a(2) exists, then it is at least 81129638414606681695789005144146. - Charles R Greathouse IV, Apr 10 2012

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

Cf. A000142, A177458, A177459, A177355, A177349, A177378, A177436, A050376, A168655, A169661.

tp[n_] := Flatten[Position[FoldList[Plus, 0, IntegerExponent[Range[100000], n]], ?(IntegerQ[Log[2, #]] &)]]; Table[s = Intersection[tp[Prime[n]], tp[Prime[n + 1]]] - 1; s[[-1]], {n, 3, 60}] (* _T. D. Noe, Apr 10 2012 *)

The maximal m with the considered property is in interval [q, 2*(-1+q^2*(log(2)/(2*log(q)-1)+1))), where q=prime(n+1).

Extended by T. D. Noe, Apr 10 2012

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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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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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A177458 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 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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A177498 a(n) is the maximal positive integer m for which exponents of prime(n) and prime(n+1) 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.