A240755 -id:A240755 - cp's OEIS frontend

A240764 Least k such that prime power factorization of A240751(k)! contains p^k when the smallest such p equals prime(n), or a(n)=0 if there is no such k.

1, 2, 12, 29, 186, 2865, 3265, 379852, 7172525

Offset: 1

Vladimir Shevelev, Apr 12 2014

The first position k in A240755 in which A240755(k) = prime(n), or a(n)=0 if prime(n) does not occur in A240755.

Conjecture: all a(n)>0.

			A240751(a(3))! = A240751(12)! = 50!. 50! is the least factorial having exponent 12 in its prime factorization. That exponent denotes the multiplicity of prime(3) = 5. - _David A. Corneth_, Mar 27 2017

David A. Corneth, A240751; a(n) is the smallest k such that in the prime power factorization of k! there exists at least one exponent n., Seqfan, (Mar 25 2017)

Cf. A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755.

a(5)-a(7) from Peter J. C. Moses, Apr 14 2014

a(8)-a(9) from David A. Corneth, Mar 27 2017

A240905 Smallest k such that the minimal factor in factorization of k! over distinct terms of A050376 is A050376(n), or a(n) = 0 if there is no such k.

Original entry on oeis.org

2, 12, 20, 6, 10, 130, 180, 240, 480, 597, 901, 40537, 15841, 23401, 36720, 112321, 20377, 177842, 101755, 855369, 2333282, 654549, 15078026, 9206403, 5829606, 75690506, 64946036, 650025768, 1100265039, 1966444000, 6660851430, 1505032794, 7305026242, 42063043872, 54868098259

Offset: 1

Vladimir Shevelev, Apr 14 2014

a(n) is the smallest k such that the minimal infinitary divisor of k! is A050376(n).
Conjecture: a(n) > 0 for all n.
a(28) > 2.5*10^8, if it exists. - Amiram Eldar, Jun 18 2025
a(34) > 2.8*10^10 if it exists. - David A. Corneth, Jun 19 2025

			Let n = 4. A050376(4)=5. For k = 2, 3, 4, 5, 6, we have the following factorizations over distinct terms of A050376: 2! = 2, 3! = 2*3, 4! = 2*3*4, 5! = 2*3*4*5, 6! = 5*9*16. Only the last factorization begins with 5. So a(4) = 6.
From _David A. Corneth_, Jun 19 2025: (Start)
a(6) = 130. Once we checked that a(6) is > 125 we try 126. The minimal factor of k! into distinct products must be A050376(6) = 8. For 126! we have the 5-adic valuation of 31 so the minimal factor is at most 5.
To get rid of the 5 we try the next candidate > 126 that is a multiple of 5. This is 130. We can just skip 127, 128 and 129 altogether. It turns out this smallest factor for 130! is 8 giving the value for a(6). (End)

David A. Corneth, PARI program
Amiram Eldar, PARI program

Cf. A050376, A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764.

PARI
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\\ See Corneth link
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PARI
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\\ See Eldar link
```

More terms from Peter J. C. Moses, Apr 19 2014
a(19)-a(27) from Amiram Eldar, Jun 18 2025
a(28)-a(35) from David A. Corneth, Jun 18, Jun 21 2025

A240906 Maximal A050376-factors in factorization of factorials over distinct terms of A050376.

Original entry on oeis.org

2, 3, 4, 5, 16, 81, 256, 65536, 4294967296, 18446744073709551616, 340282366920938463463374607431768211456, 115792089237316195423570985008687907853269984665640564039457584007913129639936

Offset: 2

Vladimir Shevelev, Apr 14 2014

nonn

In contrast to prime power factorization of n! where maximal prime does not exceed n, in the considered factorization of n!, the maximal A050376-factor could be much greater than n. For example, for n=18, it is 65536.

Essentially, a(n) is A177334 with duplicates removed. - Ivan Neretin, May 31 2016

			81 is in the sequence, since 9! = 2*4*5*7*16*81.

Cf. A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905.

a(9) from A177334 by Robert Price, Aug 31 2019

a(10)-a(12) from Amiram Eldar, Sep 17 2019

A241123 Smallest k such that the factorization of k! over distinct terms of A050376 contains exactly n primes.

Original entry on oeis.org

2, 3, 5, 13, 17, 21, 23, 37, 33, 42, 43, 56, 59, 57, 75, 84, 99, 101, 105, 109, 123, 119, 133, 139, 157, 162, 163, 182, 186, 183, 207, 208, 222, 219, 235, 220, 255, 257, 263, 268, 267, 303, 305, 307, 316, 315, 340, 344, 341, 343, 383, 385, 387, 397, 411, 425

Offset: 1

Vladimir Shevelev, Apr 16 2014

nonn

			Factorization of 5! over distinct terms of A050376 is 5! = 2*3*4*5. Thus 5 is the smallest k such that such a factorization contains 3 primes: 2,3,5. So a(3)=5.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 (Russian; MR 2000f: 11097, pp. 3912-3913).

Amiram Eldar, Table of n, a(n) for n = 1..500
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. A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906.

f[p_, e_] := Mod[e, 2]; b[1] = 0; b[n_] := Plus @@ (f @@@ FactorInteger[n]); m = 56; v = Table[0, {m}]; c = 0; p = 1; n = 2; While[c < m, p *= n; i = b[p]; If[i <= m && v[[i]] == 0, c++; v[[i]] = n]; n++]; v (* Amiram Eldar, Sep 17 2019 *)

nbp(n) = {f = factor(n); sum (i=1, #f~, f[i,2] % 2);}
a(n) = {k = 1; while(nbp(k!) != n, k++); k;} \\ Michel Marcus, Apr 27 2014

More terms from Peter J. C. Moses, Apr 17 2014

A241124 Smallest k such that the factorization of k! over distinct terms of A050376 contains at least n nonprime terms of A050376.

Original entry on oeis.org

4, 6, 8, 12, 14, 15, 16, 24, 25, 26, 30, 32, 46, 46, 48, 48, 62, 63, 63, 64, 64, 87, 91, 95, 96, 96, 96, 114, 114, 122, 124, 125, 128, 129, 160, 161, 176, 177, 178, 178, 188, 189, 190, 192, 192, 192, 194, 225, 226, 226, 240, 252, 254, 255, 256, 288, 288, 289, 290, 320

Offset: 1

Vladimir Shevelev, Apr 16 2014

nonn

			For k=2,3,4,5,6, we have the following factorizations of k! over distinct terms of A050376: 2!=2, 3!=2*3, 4!=2*3*4, 5!=2*3*4*5, 6!=5*9*16.
Therefore, a(1)=4, a(2)=6.

Cf. A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906, A241123.

f[n_] := DigitCount[n, 2, 1] - Mod[n, 2]; nb[n_] := Total@(f/@ FactorInteger[n][[;;,2]]); a[n_] := (k=1; While[nb[k!] < n, k++]; k); Array[a, 60] (* Amiram Eldar, Dec 16 2018 from the PARI code *)

nb(n) = {my(f = factor(n)); sum(k=1, #f~, hammingweight(f[k,2]) - (f[k,2] % 2));}
a(n) = {my(k=1); while (nb(k!) < n, k++); k;} \\ Michel Marcus, Dec 16 2018

More terms from Michel Marcus, Dec 16 2018

A241139 Number of nonprimes in factorization of n! over distinct terms of A050376.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6, 7, 7, 4, 4, 5, 5, 6, 6, 8, 9, 10, 10, 9, 9, 11, 11, 12, 12, 10, 9, 8, 8, 9, 10, 11, 11, 12, 12, 11, 12, 14, 14, 16, 15, 15, 15, 13, 13, 14, 14, 14, 14, 16, 16, 16, 16, 17, 19, 21, 21, 18, 18, 19, 16, 14, 14, 16, 16, 17

Offset: 2

Vladimir Shevelev, Apr 16 2014

nonn

			Factorization of 4! over distinct terms of A050376 is 4! = 2*3*4. This factorization contains only one A050376-nonprime. So a(4)=1.

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. A177329, A177333, A177334, A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906, A241123, A241124.

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}]; Length[Select[(b /@ v) // Flatten, # > 1 &]]]; Array[a, 73, 2]  (* Amiram Eldar, Sep 17 2019 *)

a(n)={my(f=factor(n!)[,2]); sum(i=1, #f~, hammingweight(f[i]>>1))} \\ Andrew Howroyd, Sep 17 2019

a(n) = A177329(n) - A055460(n).

More terms from Peter J. C. Moses, Apr 17 2014

A241148 Number of factorials k!, 0<=k<=n, relatively prime to n! in Fermi-Dirac arithmetic.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 5, 5, 2, 2, 7, 7, 4, 4, 4, 2, 2, 2, 5, 5, 7, 4, 3, 3, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 3, 3, 4, 4, 3, 2, 4, 4, 3, 3, 2, 4, 5, 5, 4, 4, 2, 2, 2, 2, 6, 5, 2, 2, 3, 3, 7, 7, 3, 2, 2, 2, 3, 3, 3, 4, 3, 3, 4, 4, 2, 2, 2, 2, 6, 6, 4, 4, 2, 2, 2, 3, 4

Offset: 0

Vladimir Shevelev, Apr 16 2014

nonn

Or, equivalently, the number of factorials k!, 0<=k<=n, for which k! and n! have no common A050376-factors in their factorizations over distinct terms of A050376.

Note that 1 (=0!=1!) corresponds to an empty subset of A050376.

			0!=1, 1!=1; further we have the following factorizations of k! over distinct terms of A050376 for k = 2,3,4,5,6:
2!=2, 3!=2*3, 4!=2*3*4, 5!=2*3*4*5, 6!=5*9*16. Thus, in the sense of the factorizations being considered, 6! is relatively prime to 0!,1!,2!,3!, and 4!. So a(6)=5.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 (Russian; MR 2000f: 11097, pp. 3912-3913).

Amiram Eldar, Table of n, a(n) for n = 0..500
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. A177329, A177333, A177334, A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906, A241123, A241124, A241139.

b[n_] := 2^(-1 + Position[Reverse@IntegerDigits[n, 2], ?(# == 1 &)]) // Flatten; infp[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}]; (Prime /@ Range[np])^(b /@ v) // Flatten]; infCoprimeQ[x_, y_] := Intersection[infp[x], infp[y]] == {}; a[n_] := Length @ Select[Range[0, n], infCoprimeQ[n, #] & ]; Array[a, 87, 0] (* Amiram Eldar, Sep 17 2019 *)

More terms from Peter J. C. Moses, Apr 18 2014

A284050 a(n) = floor(A240751(n) / n), where A240751(n) = the smallest k such that in the prime power factorization of k! there exists at least one exponent n.

Original entry on oeis.org

2, 3, 1, 1, 2, 2, 1, 1, 2, 1, 1, 4, 2, 2, 1, 1, 2, 1, 1, 4, 2, 1, 1, 4, 1, 1, 2, 2, 6, 2, 1, 1, 4, 1, 1, 2, 4, 1, 1, 2, 1, 1, 4, 2, 2, 1, 1, 2, 1, 1, 4, 4, 1, 1, 2, 1, 1, 2, 2, 6, 2, 2, 1, 1, 4, 1, 1, 2, 4, 1, 1, 2, 1, 1, 2, 2, 4, 1, 1, 2, 1, 1, 4, 2, 1, 1, 4

Offset: 1

David A. Corneth, Mar 19 2017

For n > 2, p = a(n) + 1 is the prime that has exponent n in A240751(n)! (see A240751 for an outline of a proof).

First occurrence of p-1: 1, 2, 12, 29, 186, 2865, 3265, 379852, 7172525, ..., (A240764). - Robert G. Wilson v, Apr 15 2017. Comment changed by David A. Corneth, Apr 15 2017

			For n = 5, p = a(n) + 1 = 3 is the prime such that A240751(5)! = 12! is the least factorial that has exponent 5.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A240751, A240755, A240764, A284051, A157928.

Table[k = 2; While[! MemberQ[FactorInteger[k!][[All, -1]], n], k++]; Floor[k/n], {n, 87}] (* Michael De Vlieger, Mar 24 2017 *)

a(n) = A240751(n)\n \\ (for computation of A240751(n), see A240751)

A240751(n) = n*a(n) + A284051(n). - Antti Karttunen, Mar 22 2017

a(n) = A240755(n) - 1 for n > 2 and a(n) = A240755(n) for n < 3. I.e., A240755(n) - A157928(n+1). - David A. Corneth, Mar 27 2017

A241289 Numbers n for which in the factorization of n! over distinct terms of A050376, the numbers of prime and nonprime terms are equal.

Original entry on oeis.org

7, 8, 9, 13, 18, 22, 37, 57, 71

Offset: 1

Vladimir Shevelev, Apr 18 2014

a(10), if it exists, should be more than 5000. Is a(9)=71 the last term of sequence? - Peter J. C. Moses, Apr 19 2014

One can prove that a(9)=71 indeed is the last term of this sequence. - Vladimir Shevelev, Apr 19 2014.

			7 is in the sequence, since 7! in the considered factorization is 5*7*9*16, and here we have 2 primes and 2 nonprimes.

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

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. A177329, A177333, A177334, A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906, A241123, A241124, A241139, A241148.

Terms a(7) - a(9) from Peter J. C. Moses, Apr 19 2014

A235623 Numbers n for which in the prime power factorization of n!, the numbers of exponents 1 and >1 are equal.

Original entry on oeis.org

0, 1, 4, 7, 8, 9, 13, 19, 20, 21

Offset: 1

Vladimir Shevelev, Apr 20 2014

Number n is in the sequence, if and only if pi(n) = 2*pi(n/2), where pi(x) is the number of primes<=x. Indeed, all primes from interval (n/2, n] appear in prime power factorization of n! with exponent 1, while all primes from interval (0, n/2] appear in n! with exponents >1. However, it follows from Ehrhart's link that, for n>=22, pi(n) < 2*pi(n/2). Therefore, a(9)=21 is the last term of the sequence.

m is in this sequence if and only if the number of prime divisors of [m/2]! equals the number of unitary prime divisors of m! - Peter Luschny, Apr 29 2014

			21! = 2^20*3^9*5^4*7^3*11*13*17*19. Here 4 primes with exponent 1 and 4 primes with exponents >1, so 21 is in the sequence.

Eugene Ehrhart, On prime numbers, Fibonacci Quarterly 26:3 (1988), pp. 271-274.

Cf. A056171, A177329, A177333, A177334, A240537, A240588, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906, A241123, A241124, A241139, A241148, A241289.

with(numtheory): a := proc(n) factorset(n!); factorset(iquo(n,2)!);
`if`(nops(%% minus %) = nops(%), n, NULL) end: seq(a(n), n=0..30); # Peter Luschny, Apr 28 2014

isok(n) = {f = factor(n!); sum(i=1, #f~, f[i,2] == 1) == sum(i=1, #f~, f[i,2] > 1);} \\ Michel Marcus, Apr 20 2014

cp's OEIS Frontend

A240764 Least k such that prime power factorization of A240751(k)! contains p^k when the smallest such p equals prime(n), or a(n)=0 if there is no such k.

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A240905 Smallest k such that the minimal factor in factorization of k! over distinct terms of A050376 is A050376(n), or a(n) = 0 if there is no such k.

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A240906 Maximal A050376-factors in factorization of factorials over distinct terms of A050376.

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A241123 Smallest k such that the factorization of k! over distinct terms of A050376 contains exactly n primes.

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A241124 Smallest k such that the factorization of k! over distinct terms of A050376 contains at least n nonprime terms of A050376.

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A241139 Number of nonprimes in factorization of n! over distinct terms of A050376.

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A241148 Number of factorials k!, 0<=k<=n, relatively prime to n! in Fermi-Dirac arithmetic.

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A284050 a(n) = floor(A240751(n) / n), where A240751(n) = the smallest k such that in the prime power factorization of k! there exists at least one exponent n.

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