A177436 -id:A177436 - cp's OEIS frontend

A380388 Primes q for which A177436(pi(q)) is not equal to 2*(1 + floor(log_2(q/(2^ceiling(log_2(q))-q)))).

3, 5, 11, 13, 17, 19, 43, 67, 103, 131, 241, 257, 683, 2731, 4099, 32771, 43691, 61681, 65537, 65539, 174763, 262147, 2796203, 6710887, 15790321

Offset: 1

Michel Marcus, Jan 23 2025

Vladimir Shevelev, Compact integers and factorials, Acta Arithmetica 126 (2007), no. 3, 195-236. See p. 231.

Cf. A177436.

A169655 Numbers k such that 2^k is in A054861.

Original entry on oeis.org

0, 1, 2, 3, 5, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 21, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 41, 42, 43, 45, 46, 47, 49, 53, 54, 55, 56, 58, 59, 60, 62, 64, 65, 67, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 82, 84, 85, 87, 88, 89, 91, 93

Offset: 1

Vladimir Shevelev, Apr 05 2010

nonn

For a prime p, we call a number p-compact if the exponent of p in the factorization of the number is a power of two. However, if m=k!, then not all exponents of p of the form 2^t are possible. The sequence lists numbers t in possible exponents of the form 2^t of 3 in 3-compact factorials k!The question of description of the p-compact factorials is interesting since there exists only finite set of factorials compact over both 2 and an arbitrary fixed odd prime (cf. A177436). On the other hand, there exist infinitely many 2-compact factorials. However, up to now it is unknown, whether exist infinitely many p-compact factorials for a fixed odd prime p. It is expected that the answer to be in affirmative.

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

Cf. A050376, A054861.

A054861 := (Plus @@ Floor[#/3^Range[Length[IntegerDigits[#, 3]] - 1]] &);DeleteCases[Table[n - n Sign[2^n - A054861[2^(n + 1) + NestWhile[# + 1 &, 1, 2^n - A054861[2^(n + 1) + #] >= 0 &]  - 1]],{n, 1, 125}], 0] (* Peter J. C. Moses, Apr 10 2012 *)

More terms given by Peter J. C. Moses, Apr 07 2012

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)).

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

cp's OEIS Frontend

A380388 Primes q for which A177436(pi(q)) is not equal to 2*(1 + floor(log_2(q/(2^ceiling(log_2(q))-q)))).

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A169655 Numbers k such that 2^k is in A054861.

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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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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.