A176472 -id:A176472 - cp's OEIS frontend

A177329 Number of factors in the representation of n! as a product of distinct terms of A050376.

1, 2, 3, 4, 3, 4, 6, 6, 4, 5, 7, 8, 9, 10, 11, 12, 8, 9, 9, 11, 12, 13, 13, 14, 15, 16, 14, 15, 16, 17, 19, 21, 17, 16, 15, 16, 17, 18, 19, 20, 22, 23, 21, 21, 21, 22, 23, 22, 23, 25, 22, 23, 22, 24, 26, 28, 28, 29, 27, 28, 29, 30, 32, 34, 30, 31, 31, 28, 27, 28, 29, 30, 31, 33, 31, 31, 30

Offset: 2

Vladimir Shevelev, May 06 2010

nonn

Vladimir S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].

Chai Wah Wu, Table of n, a(n) for n = 2..10000 (terms 2..1000 from Amiram Eldar)
Simon Litsyn and Vladimir Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Cf. A000120, A001358, A050292, A050376, A064380, A064547, A176472, A176509, A176525.

read("transforms") ; A064547 := proc(n) f := ifactors(n)[2] ; a := 0 ; for p in f do a := a+wt(op(2,p)) ; end do: a ; end proc:
A177329 := proc(n) A064547(n!) ; end proc: seq(A177329(n),n=2..80) ; # R. J. Mathar, May 28 2010

f[p_, e_] := DigitCount[e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n!]; Array[a, 100, 2] (* Amiram Eldar, Aug 24 2024 *)

a(n) = vecsum(apply(x -> hammingweight(x), factor(n!)[,2])); \\ Amiram Eldar, Aug 24 2024

from collections import Counter
from sympy import factorint
def A177329(n): return sum(map(int.bit_count,sum((Counter(factorint(i)) for i in range(2,n+1)),start=Counter()).values())) # Chai Wah Wu, Jul 18 2024

a(n) = Sum_{i} A000120(e_i), where n! = Product_{i} p_i^e_i is the prime factorization of n!.

a(n) = A064547(n!). - R. J. Mathar, May 28 2010

a(20)=10 inserted by Vladimir Shevelev, May 08 2010

Terms from a(14) onwards replaced according to the formula - R. J. Mathar, May 28 2010

A176509 Composite numbers m for which A064380(m) = A000010(m).

Original entry on oeis.org

8, 27, 125, 128, 343, 1331, 2187, 2197, 4913, 6859, 12167, 24389, 29791, 32768, 50653, 68921, 78125, 79507, 103823, 148877, 205379, 226981, 300763, 357911, 389017, 493039, 571787, 704969, 823543, 912673, 1030301, 1092727, 1225043, 1295029, 1442897, 2048383, 2248091

Offset: 1

Vladimir Shevelev, Apr 19 2010

nonn

Theorem. A064380(m) = A000010(m) iff m has the form m=p^(2^k-1), k>=1, p a prime. Eliminating the primes (k=1), the terms of the sequence have this form for k>1. All terms of A030078 (k=2) and A092759 (k=3) and prime powers of A010803 (k=4) are in the sequence, for example.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A010803, A030078, A050376, A056824, A064380, A092759, A176472.

seq[max_] := Module[{ps = Select[Range[Floor[Surd[max, 3]]], PrimeQ], e, k, s = {}}, Do[e = Floor[Log[ps[[i]], max]]; k = Floor[Log2[e + 1]]; s = Join[s, ps[[i]]^(2^Range[2, k] - 1)], {i, 1, Length[ps]}]; Sort[s]]; seq[3*10^6] (* Amiram Eldar, Mar 26 2023 *)

is(n)=my(e=isprimepower(n));e>2 && 2^valuation(e+1,2)==e+1 \\ Charles R Greathouse IV, Feb 19 2013

a(n) ~ n^3 log^3 n. - Charles R Greathouse IV, Feb 19 2013

Sum_{n>=1} 1/a(n) = Sum_{k>=2} 1/P(2^k-1) = 0.183077059924063305405..., where P(s) is the prime zeta function. - Amiram Eldar, Jul 11 2024

128 inserted, 1024 deleted, 2187 inserted, 32768 inserted, etc. - R. J. Mathar, Nov 21 2010

More terms from Amiram Eldar, Mar 26 2023

A177333 Smallest factor in the factorization of n! over distinct terms of A050376.

Original entry on oeis.org

2, 2, 2, 2, 5, 5, 2, 2, 7, 7, 3, 3, 2, 2, 2, 2, 5, 5, 4, 3, 2, 2, 4, 4, 2, 2, 2, 2, 4, 4, 2, 2, 3, 3, 3, 3, 2, 2, 4, 4, 2, 2, 2, 2, 3, 3, 4, 4, 2, 2, 2, 2, 4, 4, 2, 2, 3, 3, 7, 7, 2, 2, 2, 2, 3, 3, 3, 4, 2, 2, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 3, 4, 2, 2, 4, 4, 5, 3, 2, 2, 4, 4, 2, 2, 2, 2, 3, 3, 2, 2, 4

Offset: 2

Vladimir Shevelev, May 06 2010

nonn

			The factorization of 10! = 3628800 is 2^8*3^4*5^2*7^1, where 2^8 > 3^4 > 5^2 > 7, so a(10)=7 is the smallest of these 4 factors.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].

Amiram Eldar, Table of n, a(n) for n = 2..1000
S. Litsyn and V. S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Cf. A050376, A177329, A055460, A176525, A001358, A176472, A176509, A064380, A050292.

A177333 := proc(n) local a,p,pow2 ; a := n! ; for p in ifactors(n!)[2] do pow2 := convert( op(2,p),base,2) ; for j from 1 to nops(pow2) do if op(j,pow2) <> 0 then a := min(a,op(1,p)^(2^(j-1))) ; end if; end do: end do: return a ; end proc:
seq(A177333(n),n=2..120) ; # R. J. Mathar, Jun 16 2010

b[n_] :=2^(-1+Position[ Reverse@IntegerDigits[n, 2],?(#==1&)])//Flatten; a[n] := Module[{np = PrimePi[n]}, v=Table[0,{np}]; Do[p = Prime[k]; Do[v[[k]] += IntegerExponent[j, p], {j,2,n}],  {k,1,np}]; Min[(Prime/@Range[np])^(b/@v) // Flatten]]; Array[a, 105, 2] (* Amiram Eldar, Sep 17 2019 *)

Corrected from a(10) on and extended beyond a(30) by R. J. Mathar, Jun 16 2010

A176525 Fermi-Dirac semiprimes: products of two distinct terms of A050376.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 18, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, 36, 38, 39, 44, 45, 46, 48, 50, 51, 52, 55, 57, 58, 62, 63, 64, 65, 68, 69, 74, 75, 76, 77, 80, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 100, 106, 111, 112, 115, 116, 117, 118, 119, 122

Offset: 1

Vladimir Shevelev, Apr 19 2010, Apr 20 2010

nonn

The sequence essentially differs from A000379 beginning with a(108)=212 (not 210). All squarefree terms of A001358 are in the sequence.

Vladimir S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences, Vol. 4 (1996), pp. 28-43 [Russian].

Peter J. C. Moses, Table of n, a(n) for n = 1..10000
Simon Litsyn and Vladimir S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, Vol. 7 (2007), Article #A33, pp. 1-36.

Cf. A001358, A050376, A000379, A176472, A176509, A064380, A050292.

Select[Range[120], Plus @@ DigitCount[Last /@ FactorInteger[#], 2, 1] == 2 &] (* Amiram Eldar, Nov 27 2020 *)

If a(n)=u*v, uA050376 "Fermi-Dirac primes", then A064380(a(n))=a(n)-u-v+1+Sum{i>=1}(-1)^(i-1)*floor(v/u^i).

Effectively duplicate content (due to duplicate referenced sequence) removed by Peter Munn, Dec 19 2019

A186777 Solutions x of the equation A064380(x)-A000010(x)=1 in integers x>=2.

Original entry on oeis.org

4, 6, 10, 15, 35, 77, 91, 143, 187, 209, 221, 247, 299, 323, 391, 437, 493, 527, 551, 589, 667, 703, 713, 851, 899, 943, 989, 1073, 1147, 1189, 1247, 1271, 1333, 1363, 1457, 1517, 1537, 1591, 1643, 1739, 1763, 1829, 1891, 1927, 1961, 2021, 2173, 2183, 2257, 2279

Offset: 1

Vladimir Shevelev, Feb 26 2011

nonn

The defining equation has infinitely many solutions.

Equals {4, 6, 10} UNION A082663. - Eric M. Schmidt, Oct 04 2013

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric M. Schmidt, The Near Equivalence of A082663 and A186777

Cf. A000010, A064380, A082663, A176472, A176509.

More terms from Amiram Eldar, Sep 14 2019

A186778 Solutions n of the equation A064380(n)-A000010(n)=2 in integers n>=2.

Original entry on oeis.org

9, 14, 21, 24, 33, 40, 55, 65, 119, 133, 253, 319, 341, 377, 403, 481, 629, 697, 731, 779, 799, 817, 893, 1007, 1081, 1219, 1357, 1403, 1541, 1711, 1769, 1943, 2059, 2077, 2117, 2201, 2263, 2291, 2407, 2449, 2573, 2759, 2923, 3071, 3293, 3403, 3589, 3649, 3737

Offset: 1

Vladimir Shevelev, Feb 26 2011

nonn

The defining equation has infinitely many solutions.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43.

Amiram Eldar, Table of n, a(n) for n = 1..420

Cf. A000010, A064380, A176472, A176509, A186777.

More terms from Amiram Eldar, Sep 14 2019

A186779 Solutions n of the equation A064380(n)-A000010(n)=3 in integers n>=2.

Original entry on oeis.org

12, 39, 56, 85, 88, 95, 104, 161, 407, 451, 473, 533, 559, 611, 901, 1003, 1037, 1121, 1139, 1159, 1273, 1349, 1387, 1633, 1679, 1817, 1909, 2047, 2581, 2813, 2929, 2987, 3007, 3103, 3131, 3161, 3193, 3277, 3317, 3379, 3503, 4181, 4699, 4847, 5069, 5143, 5207

Offset: 1

Vladimir Shevelev, Feb 26 2011

nonn

The defining equation has infinitely many solutions.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43.

Amiram Eldar, Table of n, a(n) for n = 1..300

Cf. A000010, A064380, A176472, A176509.

More terms from Amiram Eldar, Sep 14 2019

A186780 Solutions n of equation A064380(n)-A000010(n)=4 in integers n>=2.

Original entry on oeis.org

22, 25, 26, 32, 51, 57, 115, 135, 136, 145, 189, 203, 217, 297, 517, 583, 689, 767, 793, 1207, 1241, 1343, 1411, 1501, 1577, 1691, 2231, 2323, 2369, 2461, 2507, 2599, 3683, 3799, 3937, 3973, 4031, 4061, 4247, 4309, 4619, 4681, 5513, 5587, 5809, 6031, 6179, 6401

Offset: 1

Vladimir Shevelev, Feb 26 2011

nonn

The sequence is infinite.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43.

Amiram Eldar, Table of n, a(n) for n = 1..250

Cf. A000010, A064380, A176472, A176509, A186777, A186778, A186779.

More terms from Amiram Eldar, Sep 14 2019

A186781 Integer solutions x to the equation A064380(x)-A000010(x)=5.

Original entry on oeis.org

18, 20, 30, 34, 42, 69, 152, 155, 259, 287, 351, 459, 513, 649, 671, 871, 923, 949, 1513, 1649, 1717, 1843, 1919, 1957, 2033, 2071, 2147, 2921, 3013, 3151, 4321, 4379, 4553, 4727, 4843, 4867, 5017, 5053, 5177, 5363, 5549, 5611, 7067, 7141, 7289, 7363, 7807, 8651

Offset: 1

Vladimir Shevelev, Feb 26 2011

nonn

The defining equation has infinitely many solutions, see A176472.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43.

Amiram Eldar, Table of n, a(n) for n = 1..200

Cf. A000010, A064380, A176472, A176509, A186777, A186778, A186779, A186780.

More terms from Amiram Eldar, Sep 14 2019

cp's OEIS Frontend

A177329 Number of factors in the representation of n! as a product of distinct terms of A050376.

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A176509 Composite numbers m for which A064380(m) = A000010(m).

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A177333 Smallest factor in the factorization of n! over distinct terms of A050376.

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A176525 Fermi-Dirac semiprimes: products of two distinct terms of A050376.

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A186777 Solutions x of the equation A064380(x)-A000010(x)=1 in integers x>=2.

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A186778 Solutions n of the equation A064380(n)-A000010(n)=2 in integers n>=2.

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A186779 Solutions n of the equation A064380(n)-A000010(n)=3 in integers n>=2.

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A186780 Solutions n of equation A064380(n)-A000010(n)=4 in integers n>=2.

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