A319627 -id:A319627 - cp's OEIS frontend

A307035 a(n) is the unique integer k such that A108951(k) = n!.

1, 1, 2, 3, 12, 20, 60, 84, 672, 1512, 5040, 7920, 47520, 56160, 157248, 393120, 6289920, 8225280, 37013760, 41368320, 275788800, 579156480, 1820206080, 2203407360, 26440888320, 73446912000, 173601792000, 585906048000, 3281073868800, 4137006182400, 20685030912000

Offset: 0

Allan C. Wechsler, Mar 20 2019

For all n, n! = A108951(k) for some unique k. This sequence gives that k for each n. In some sense this sequence tells how to factor factorials into primorials.

Represent n! as a product of primorials p#. Then replace each primorial with its base prime to calculate a(n).

			Represent 7! as a product of primorials:
7! = 2^4 * 3^2 * 5 * 7 = (2#)^2 * 3# * 7#
Replace primorials by primes:
2^2 * 3 * 7 = 84.
So a(7) = 84.

Alois P. Heinz, Table of n, a(n) for n = 0..1911 (terms n = 1..1000 from Vaclav Kotesovec)

Cf. A000142, A108951, A025487, A122111, A319626, A319627, A325508, A329900.

f:= proc(n) option remember; `if`(n<2, 0, f(n-1)+add(
      i[2]*x^numtheory[pi](i[1]), i=ifactors(n)[2]))
    end:
a:= proc(n) local d, p, r; p, r:= f(n), 1;
      do d:= degree(p); if d<1 then break fi;
         p, r:= p-add(x^i, i=1..d), ithprime(d)*r
      od: r
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Mar 21 2019

q[n_] := Apply[Times, Table[Prime[i], {i, 1, PrimePi[n]}]]; Flatten[{1, 1, Table[val = 1; fak = n!; Do[If[PrimeQ[k], Do[If[Divisible[fak, q[k]], val = val*k; fak = fak/q[k]], {j, 1, n}]], {k, n, 2, -1}]; val, {n, 2, 30}]}] (* Vaclav Kotesovec, Mar 21 2019 *)

g(n) = my(f=factor(n)); prod(k=1, #f~, my(p=f[k, 1]); (p/if(p>2, precprime(p-1), 1))^f[k, 2]); \\ A319626/A319627
a(n) = prod(k=1, n, g(k)); \\ Daniel Suteu, Mar 21 2019

A307035(n) = { my(m=1, pp=1); n=n!; while(1, forprime(p=2, ,if(n%p, if(2==p, return(m), break), n /= p; pp = p)); m *= pp); }; \\ Antti Karttunen, Dec 29 2019

a(0) = 1, a(n) = a(n-1) * (A319626(n) / A319627(n)), for n > 0. - Daniel Suteu, Mar 21 2019
a(n) = n! / Product_{k=1..n} A064989(k). - Vaclav Kotesovec, Mar 21 2019
a(n) = A122111(A325508(n)) = A319626(A000142(n)) = A329900(A000142(n)). - Antti Karttunen, Nov 19 & Dec 29 2019

a(12)-a(13) from Michel Marcus, Mar 21 2019
a(14)-a(15) from Vaclav Kotesovec, Mar 21 2019
a(0), a(16)-a(30) from Alois P. Heinz, Mar 21 2019

A346109 a(n) = A276085(A108951(A346097(n))), where A346097(n) gives the denominator of the primorial deflation of A276086(A108951(n)).

Original entry on oeis.org

0, 1, 3, 2, 9, 6, 39, 1, 3, 18, 249, 12, 2559, 78, 54, 2, 32589, 6, 543099, 36, 234, 498, 10242789, 9, 96, 5118, 42, 156, 233335659, 45, 6703028889, 10, 1494, 65178, 312, 12, 207263519019, 1086198, 15354, 9, 7628001653829, 39, 311878265181039, 996, 165, 20485578, 13394639596851069, 21, 1284, 192, 195534, 10236, 628284422185342479

Offset: 1

Antti Karttunen, Jul 08 2021

nonn

Cf. A002110, A064989, A108951, A276085, A324886, A319627, A346097, A346105, A346107, A346108.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A319627(n) = (A064989(n) / gcd(n, A064989(n)));
A346097(n) = A319627(A324886(n));
A346107(n) = A108951(A346097(n));
A002110(n) = prod(i=1,n,prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A346109(n) = A276085(A346107(n)); \\ Rest of program given in A324886.

a(n) = A346105(A346097(n)) = A276085(A346107(n)) = A276085(A108951(A346097(n))).

a(n) = A346108(n) - A108951(n).

A348993 a(n) = A064989(sigma(n) / gcd(sigma(n), A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, while A064989 shifts it back towards smaller primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 1, 5, 2, 1, 1, 3, 11, 2, 2, 5, 5, 1, 2, 29, 4, 11, 3, 1, 1, 2, 2, 1, 29, 5, 1, 5, 6, 2, 1, 5, 2, 4, 2, 55, 17, 3, 5, 3, 10, 1, 7, 5, 22, 2, 2, 29, 34, 29, 4, 25, 8, 1, 4, 3, 1, 6, 6, 1, 29, 1, 11, 113, 2, 2, 13, 5, 2, 2, 4, 11, 31, 17, 29, 15, 2, 5, 3, 29, 49, 10, 10, 5, 8, 7, 2, 3, 12, 22, 5, 5, 1, 2, 6, 5

Offset: 1

Antti Karttunen, Nov 10 2021

Cf. A000203, A000265, A003961, A064989, A161942, A342671, A348992, A349162, A349169 (gives odd k for which a(k) = A319627(k)).

Array[Times @@ Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#1/GCD[##]]] & @@ {DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &, 96] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A349162(n) = { my(s=sigma(n)); (s/gcd(s,A003961(n))); };
A348993(n) = A064989(A349162(n));

a(n) = A064989(A349162(n)) = A064989(A348992(n)).

A346107 a(n) = A108951(A346097(n)), where A346097(n) gives the denominator of the primorial deflation of A276086(A108951(n)).

Original entry on oeis.org

1, 2, 6, 4, 30, 36, 210, 2, 6, 900, 2310, 1296, 30030, 44100, 729000000, 4, 510510, 36, 9699690, 810000, 85766121000000, 5336100, 223092870, 216, 39690000, 901800900, 1260, 1944810000, 6469693230, 24300000, 200560490130, 60, 151939915084881000000, 260620460100, 3782285936100000000, 1296, 7420738134810, 94083986096100

Offset: 1

Antti Karttunen, Jul 08 2021

nonn

Cf. A064989, A108951, A276086, A319627, A324886, A346097, A346106, A346109.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A319627(n) = (A064989(n) / gcd(n, A064989(n)));
A346097(n) = A319627(A324886(n));
A346107(n) = A108951(A346097(n)); \\ Rest of program given in A324886.

a(n) = A108951(A346097(n)) = A108951(A319627(A324886(n))).

a(n) = A346106(n) / A324886(n).

A348994 a(n) = A003961(n) / gcd(n, A003961(n)), where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1).

Original entry on oeis.org

1, 3, 5, 9, 7, 5, 11, 27, 25, 21, 13, 15, 17, 33, 7, 81, 19, 25, 23, 63, 55, 39, 29, 45, 49, 51, 125, 99, 31, 7, 37, 243, 65, 57, 11, 25, 41, 69, 85, 189, 43, 55, 47, 117, 35, 87, 53, 135, 121, 147, 95, 153, 59, 125, 91, 297, 115, 93, 61, 21, 67, 111, 275, 729, 119, 65, 71, 171, 145, 33, 73, 75, 79, 123, 49, 207

Offset: 1

Antti Karttunen, Nov 10 2021

Numerator of ratio A003961(n) / n. This ratio is fully multiplicative, and a(n) / A348990(n) = A319626(A003961(n)) / A319627(A003961(n)) gives it in its lowest terms.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A319626, A319627, A348990 (denominators).

Array[#2/GCD[##] & @@ {#, If[# == 1, 1, Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]]} &, 76] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A348994(n) = (A003961(n) / gcd(n, A003961(n)));

a(n) = A003961(n) / gcd(n, A003961(n)).

a(n) = A319626(A003961(n)).

A348990 a(n) = n / gcd(n, A003961(n)), where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 10, 11, 4, 13, 14, 3, 16, 17, 6, 19, 20, 21, 22, 23, 8, 25, 26, 27, 28, 29, 2, 31, 32, 33, 34, 5, 4, 37, 38, 39, 40, 41, 14, 43, 44, 9, 46, 47, 16, 49, 50, 51, 52, 53, 18, 55, 56, 57, 58, 59, 4, 61, 62, 63, 64, 65, 22, 67, 68, 69, 10, 71, 8, 73, 74, 15, 76, 7, 26, 79, 80, 81, 82, 83, 28

Offset: 1

Antti Karttunen, Nov 10 2021

Denominator of ratio A003961(n) / n. This ratio is fully multiplicative, and A348994(n) / a(n) = A319626(A003961(n)) / A319627(A003961(n)) gives it in its lowest terms.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A000035, A000961, A002110, A003961, A319626, A319627, A319630 (fixed points), A322361, A349169 (where equal to A348992).

Cf. A348994 (numerators).

Array[#1/GCD[##] & @@ {#, Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &, 84] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A348990(n) = (n/gcd(n, A003961(n)));

a(n) = n / A322361(n) = n / gcd(n, A003961(n)).
a(n) = A319627(A003961(n)).
For all odd numbers n, a(n) = A003961(A319627(n)).
For all n >= 1, A000035(A348990(n)) = A000035(n). [Preserves the parity]

A355930 Sum of the prime indices of n minus the sum of the prime indices of the smallest number with same prime signature as n, when the sum is taken with multiplicity, as in A056239.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 2, 1, 4, 0, 5, 2, 2, 0, 6, 1, 7, 1, 3, 3, 8, 0, 4, 4, 3, 2, 9, 0, 10, 0, 4, 5, 4, 0, 11, 6, 5, 1, 12, 1, 13, 3, 3, 7, 14, 0, 6, 3, 6, 4, 15, 2, 5, 2, 7, 8, 16, 0, 17, 9, 4, 0, 6, 2, 18, 5, 8, 2, 19, 0, 20, 10, 4, 6, 6, 3, 21, 1, 4, 11, 22, 1, 7, 12, 9, 3, 23, 1, 7, 7, 10, 13, 8, 0, 24, 5, 5, 2, 25, 4, 26, 4, 3

Offset: 1

Antti Karttunen as suggested by Don Reble, Oct 25 2022

nonn

a(n) gives the signature excitation of n (a concept proposed by Allan C. Wechsler, indicating the distance of n from the terms of A025487), when the primes in the "excited state", i.e., those present in A328478(n), are de-excited one by one, and the prime signature of n is preserved. See the example.

			For n = 98 = 2*7*7, the other 7 is de-excited as 7 -> 5 -> 3 -> 2, and the other 7 is de-excited as 7 -> 5 -> 3, to get 2*2*3 = 12 = A046523(98). There are 3+2 de-excitations in total, therefore a(98) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A025487 (positions of zeros), A046523, A056239.
Cf. also A319627, A328478, A358218.
Differs from A325799 for the first time at n=18, where a(18) = 1, while A325799(18) = 0.

{0}~Join~Array[Total@ Flatten[ConstantArray[PrimePi[#1], #2] & @@@ #] - Total@ Flatten@ MapIndexed[ConstantArray[First[#2], #1] &, ReverseSort[#[[All, -1]]]] &@ FactorInteger[#] &, 104, 2] (* Michael De Vlieger, Nov 02 2022 *)

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A355930(n) = (A056239(n) - A056239(A046523(n)));

a(n) = A056239(n) - A356159(n) = A056239(n) - A056239(A046523(n)).

For all n, a(n) >= A358218(n). - Antti Karttunen, Nov 05 2022

A358219 Indices k where A358217(k) != A358218(k).

Original entry on oeis.org

15, 35, 45, 70, 75, 77, 105, 135, 140, 143, 154, 165, 175, 195, 221, 225, 231, 245, 255, 280, 285, 286, 308, 315, 323, 345, 350, 375, 385, 405, 429, 435, 437, 442, 450, 455, 462, 465, 490, 495, 525, 539, 555, 560, 572, 585, 595, 615, 616, 645, 646, 663, 665, 667, 675, 693, 700, 705, 715, 735, 765, 770, 795, 805

Offset: 1

Antti Karttunen, Nov 04 2022

nonn

Index entries for sequences related to primorial numbers

Cf. A319627, A328478, A358217, A358218.

A366877 Lexicographically earliest infinite sequence such that a(i) = a(j) => A337377(i) = A337377(j) for all i, j >= 0, where A337377 is the primorial deflation (denominator) of Doudna sequence.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 26 2023

nonn

Restricted growth sequence transform of A337377.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A005940, A064989, A319627, A337377.

Cf. also A366876.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A319627(n) = (A064989(n) / gcd(n, A064989(n)));
A337377(n) = A319627(A005940(1+n));
v366877 = rgs_transform(vector(1+up_to,n,A337377(n-1)));
A366877(n) = v366877[1+n];

cp's OEIS Frontend

A307035 a(n) is the unique integer k such that A108951(k) = n!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A346109 a(n) = A276085(A108951(A346097(n))), where A346097(n) gives the denominator of the primorial deflation of A276086(A108951(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A348993 a(n) = A064989(sigma(n) / gcd(sigma(n), A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, while A064989 shifts it back towards smaller primes, and sigma is the sum of divisors function.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A346107 a(n) = A108951(A346097(n)), where A346097(n) gives the denominator of the primorial deflation of A276086(A108951(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A348994 a(n) = A003961(n) / gcd(n, A003961(n)), where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A348990 a(n) = n / gcd(n, A003961(n)), where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A355930 Sum of the prime indices of n minus the sum of the prime indices of the smallest number with same prime signature as n, when the sum is taken with multiplicity, as in A056239.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A358219 Indices k where A358217(k) != A358218(k).

Views