A108951 -id:A108951 - cp's OEIS frontend

A329614 Smallest prime factor of the number of divisors of A108951(n).

1, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2

Offset: 1

Antti Karttunen, Nov 17 2019

nonn

Differs from A071187 for the first time at n=324, where a(324) = 5, while A071187(324) = 3. The positions of the differences are listed at A329613.

			324 = 18^2 = 2^2 * 3^4, thus A108951(324) = 2^2 * (2*3)^4 = 2^6 * 3^4 = 5184, which has (6+1)*(4+1) = 7 * 5 = 35 divisors, thus a(324) = A020639(35) = 5.

Cf. A020639, A034386, A071187, A108951, A329605, A329608, A329612, A329613.

Array[FactorInteger[DivisorSigma[0, #]][[1, 1]] &@ Apply[Times, Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]] &, 105] (* Michael De Vlieger, Nov 18 2019 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A071187(n) = if(1==n, n, my(f = factor(numdiv(n))); vecmin(f[, 1]));
A329614(n) = A071187(A108951(n));

a(n) = A071187(A108951(n)).

a(n) = A020639(A329605(n)).

A342920 a(n) = A342002(A108951(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 8, 12, 2, 1, 4, 1, 2, 6, 16, 1, 24, 1, 4, 6, 2, 1, 26, 50, 2, 16, 4, 1, 62, 1, 10, 6, 2, 126, 48, 1, 2, 6, 18, 1, 24, 1, 4, 46, 2, 1, 22, 1486, 100, 6, 4, 1, 32, 94, 8, 6, 2, 1, 54, 1, 2, 72, 20, 264, 12, 1, 4, 6, 120, 1, 376, 1, 2, 1142, 4, 242, 12, 1, 36, 342, 2, 1, 48, 272, 2, 6, 8, 1, 92, 318

Offset: 1

Antti Karttunen, Apr 06 2021

nonn

Cf. A108951, A324886, A327860, A328572, A342002, A342463.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };
A342920(n) = A342002(A108951(n)); \\ Needs also code from A342002.

a(n) = A342002(A108951(n)) = A327860(A108951(n)) / A328572(A108951(n)).

A346096 Numerator of the primorial deflation of A276086(A108951(n)): a(n) = A319626(A324886(n)).

Original entry on oeis.org

2, 3, 5, 9, 7, 25, 11, 5, 7, 49, 13, 625, 17, 121, 117649, 25, 19, 49, 23, 2401, 1771561, 169, 29, 175, 14641, 289, 55, 14641, 31, 26411, 37, 21, 4826809, 361, 299393809, 2401, 41, 529, 24137569, 11, 43, 13, 47, 28561, 161051, 841, 53, 343, 6311981, 214358881, 47045881, 83521, 59, 3025, 48841, 214358881, 148035889, 961

Offset: 1

Antti Karttunen, Jul 07 2021

Numerator of ratio A324886(n) / A329044(n).

Cf. A346097 (denominators).
Cf. A003961, A108951, A319626, A324886, A329044, A346095, A346098, A346106, A346108.
Cf. also A337376, A345941.

A064989(n) = { my(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); };
A319626(n) = (n / gcd(n, A064989(n)));
A346096(n) = A319626(A324886(n)); \\ Rest of program in A324886.

a(n) = A319626(A324886(n)).
a(n) = A324886(n) / A346095(n) = A324886(n) / gcd(A324886(n), A329044(n)).
For n >= 1, A108951(A346096(n)) / A108951(A346097(n)) = A324886(n).
For n > 1, a(n) = A003961(A346098(n)).

A346097 Denominator of the primorial deflation of A276086(A108951(n)): a(n) = A319627(A324886(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 2, 3, 25, 11, 81, 13, 49, 15625, 4, 17, 9, 19, 625, 117649, 121, 23, 27, 1225, 169, 21, 2401, 29, 3125, 31, 10, 1771561, 289, 5764801, 81, 37, 361, 4826809, 5, 41, 7, 43, 14641, 12005, 529, 47, 75, 1127357, 1500625, 24137569, 28561, 53, 441, 14641, 5764801, 47045881, 841, 59, 125, 61, 961, 343, 100, 302875106592253

Offset: 1

Antti Karttunen, Jul 07 2021

Denominator of ratio A324886(n) / A329044(n).

Cf. A346096 (numerators).
Cf. A006530, A020639, A108951, A276086, A319627, A324886, A329044, A345941 [= gcd(n, a(n))], A346095, A346098, A346107, A346109.
Cf. also A337377.

A064989(n) = { my(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)); \\ Rest of program given in A324886.

a(n) = A319627(A324886(n)).
a(n) = A329044(n) / A346095(n) = A329044(n) / gcd(A324886(n), A329044(n)).
A020639(a(n)) = A006530(n).
A108951(a(n)) = A346107(n).
A346105(a(n)) = A346109(n).

A337474 Number of prime shifts (x -> A003961(x)) needed before the result is deficient, when starting from x = A108951(n), the primorial inflation of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 2, 2, 2, 0, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 1, 2, 3, 2, 3, 0, 2, 2, 2, 1, 3, 3, 2, 2, 3, 2, 3, 2, 2, 3, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 3, 3, 2, 4, 3, 2, 0, 2, 2, 4, 2, 3, 2, 4, 1, 4, 3, 2, 3, 2, 2, 4, 2, 1, 3, 4, 2, 2, 3, 3, 2, 4, 2, 2, 3, 3, 3, 3, 1, 4, 2, 2, 2, 4, 2, 4, 2, 2

Offset: 1

Antti Karttunen, Aug 28 2020

nonn

a(n) is the least k for which A337473(k, n) = 1.

Cf. A003961, A108951, A181815, A336835, A337473, A337475.

Cf. A337476 (position of the first occurrence of each n), A337478.

A337473sq(n, k) = if(1==k,k, my(f=factor(k), h = #f~, prevpid=primepi(f[h,1]), e=f[h,2], p, s=1); forstep(i=h-1,0,-1, if(!i,pid=0,pid=primepi(f[i,1])); forstep(j=prevpid,(1+pid),-1, p=prime(j+n); s *= ((p^(1+e)-1)/((p-1)*(p^e)))); if(!pid,return(floor(s))); prevpid = pid; e += f[i,2]); floor(s));
A337474(n) = for(i=0,oo,if(1==A337473sq(i,n),return(i)));

\\ This version uses binary search, which is faster in certain cases:
isA337473sq1(n, k) = if(1==k,k, my(f=factor(k), h = #f~, prevpid=primepi(f[h,1]), e=f[h,2], p, s=1); forstep(i=h-1,0,-1, if(!i,pid=0,pid=primepi(f[i,1])); forstep(j=prevpid,(1+pid),-1, p=prime(j+n); s *= ((p^(1+e)-1)/((p-1)*(p^e)))); if(!pid,return(s<2)); prevpid = pid; e += f[i,2]); (s<2));
A337474(n) = if(!bitand(n,n-1),0,my(imin=0,imax=n,imid); for(i=0,oo, imid=(imax+imin)\2; if(1!=isA337473sq1(imid,n), imin = imid+1, if(1!=isA337473sq1(imid-1,n),return(imid), imax = imid-1))));

a(n) = A336835(A108951(n)).
a(A181815(n)) = A337475(n).
For all n >= 0, a(A337476(n)) = n.
For all n >= 0, a(A337478(n)) >= n.

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

Original entry on oeis.org

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

A329600 Smallest number with the same set of distinct prime exponents as A108951(n).

Original entry on oeis.org

1, 2, 2, 4, 2, 12, 2, 8, 4, 12, 2, 24, 2, 12, 12, 16, 2, 72, 2, 24, 12, 12, 2, 48, 4, 12, 8, 24, 2, 360, 2, 32, 12, 12, 12, 144, 2, 12, 12, 48, 2, 360, 2, 24, 24, 12, 2, 96, 4, 72, 12, 24, 2, 432, 12, 48, 12, 12, 2, 720, 2, 12, 24, 64, 12, 360, 2, 24, 12, 360, 2, 288, 2, 12, 72, 24, 12, 360, 2, 96, 16, 12, 2, 720, 12, 12, 12, 48, 2, 2160, 12, 24, 12, 12, 12

Offset: 1

Antti Karttunen, Nov 17 2019

nonn

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

Cf. A007947, A034386, A071364, A108951, A122111, A181819, A181821, A328400, A329607.

Cf. A077462 (rgs-transform, from its term a(1)=1 onward).

Array[Times @@ MapIndexed[Prime[#2[[1]]]^#1 &, Reverse[Flatten[Cases[FactorInteger[#], {p_, k_} :> Table[PrimePi[p], {k}]]]]] &[Times @@ FactorInteger[#][[All, 1]]] &@ If[# == 1, 1, Times @@ Prime@ FactorInteger[#][[All, -1]]] &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]] &, 105] (* Michael De Vlieger, Nov 18 2019, after Gus Wiseman at A181821 *)

A007947(n) = factorback(factorint(n)[, 1]);
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A181821(n) = { my(f=factor(n),p=0,m=1); forstep(i=#f~,1,-1,while(f[i,2], f[i,2]--; m *= (p=nextprime(p+1))^primepi(f[i,1]))); (m); };
A328400(n) = A181821(A007947(A181819(n)));
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A329600(n) = A328400(A108951(n));

a(n) = A328400(A108951(n)) = A181821(A007947(A122111(n)))

a(n) = A181821(A329607(n)) = A181821(A122111(A071364(n))).

A342012 Primorial deflation of the n-th colossally abundant number: the unique integer k such that A108951(k) = A004490(n).

Original entry on oeis.org

2, 3, 6, 10, 20, 30, 42, 84, 132, 156, 312, 468, 780, 1020, 1140, 1380, 2760, 3480, 3720, 5208, 7812, 9324, 10332, 10836, 21672, 23688, 26712, 29736, 49560, 51240, 56280, 59640, 61320, 96360, 104280, 208560, 219120, 328680, 352440, 384120, 453960, 472680, 482040, 500760, 510120, 528840, 594360, 613080, 641160, 650520, 1301040

Offset: 1

Antti Karttunen, Mar 08 2021

nonn

In contrast to A329902, this sequence is monotonic, because each term is obtained from the previous, either by multiplying it by 2, or by "bumping" one [or hypothetically: two] of its prime factors one step up (i.e., replacing it with the next larger prime), and both operations are guaranteed to make the number larger.

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

Cf. A004490, A005940, A073751, A108951, A319626, A329900, A342010 [= A001222(a(n))], A342011, A342013.

Cf. also A217867, A329902.

v073751 = readvec("b073751_to.txt");
A073751(n) = v073751[n];
A004490list(v073751) = { my(v=vector(#v073751)); v[1] = 2; for(n=2,#v,v[n] = v073751[n]*v[n-1]); (v); };
v004490 = A004490list(v073751);
A004490(n) = v004490[n];
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)};
A319626(n) = (n / gcd(n, A064989(n)));
A342012(n) = A319626(A004490(n));

a(n) = A319626(A004490(n)) = A329900(A004490(n)).

a(n) = A005940(1+A342013(n)).

A346091 a(n) = A328571(A108951(n)).

Original entry on oeis.org

2, 3, 5, 3, 7, 5, 11, 15, 35, 7, 13, 5, 17, 11, 7, 15, 19, 35, 23, 7, 11, 13, 29, 35, 77, 17, 55, 11, 31, 77, 37, 21, 13, 19, 143, 35, 41, 23, 17, 77, 43, 143, 47, 13, 77, 29, 53, 35, 2431, 77, 19, 17, 59, 55, 221, 11, 23, 31, 61, 77, 67, 37, 143, 21, 323, 13, 71, 19, 29, 143, 73, 385, 79, 41, 1001, 23, 221, 17, 83, 77, 385

Offset: 1

Antti Karttunen, Jul 09 2021

nonn

All terms are squarefree (in A005117). - Antti Karttunen, Apr 03 2022

Cf. A000040, A002110, A005117, A007947, A108951, A276086, A324886, A328571, A344592, A346093 [= A276085(a(n))], A351955 (rgs-transform).

A002110(n) = prod(i=1,n,prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A002110(primepi(f[i, 1]))^f[i, 2]) };
A328571(n) = { my(m=1, p=2); while(n, m *= (p^!!(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A346091(n) = A328571(A108951(n));

a(n) = A328571(A108951(n)) = A007947(A276086(A108951(n))).
a(n) = A324886(n) / A344592(n).
For all n >= 1, a(A000040(n)) = A000040(1+n).

A346108 a(n) = A276085(A108951(A346096(n))), where A346096(n) gives the numerator of the primorial deflation of A276086(A108951(n)).

Original entry on oeis.org

1, 3, 9, 6, 39, 18, 249, 9, 39, 78, 2559, 36, 32589, 498, 234, 18, 543099, 78, 10242789, 156, 1494, 5118, 233335659, 57, 996, 65178, 258, 996, 6703028889, 405, 207263519019, 42, 15354, 1086198, 6612, 156, 7628001653829, 20485578, 195534, 249, 311878265181039, 2559, 13394639596851069, 10236, 1245, 466671318, 628284422185342479

Offset: 1

Antti Karttunen, Jul 08 2021

nonn

Cf. A002110, A064989, A108951, A276085, A276086, A324886, A319626, A346096, A346105, 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)};
A319626(n) = (n / gcd(n, A064989(n)));
A346096(n) = A319626(A324886(n));
A346106(n) = A108951(A346096(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)); };
A346108(n) = A276085(A346106(n)); \\ Rest of program given in A324886.

a(n) = A346105(A346096(n)) = A276085(A346106(n)) = A276085(A108951(A346096(n))).

a(n) = A108951(n) + A346109(n).

cp's OEIS Frontend

A329614 Smallest prime factor of the number of divisors of A108951(n).

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A342920 a(n) = A342002(A108951(n)).

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A346096 Numerator of the primorial deflation of A276086(A108951(n)): a(n) = A319626(A324886(n)).

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A346097 Denominator of the primorial deflation of A276086(A108951(n)): a(n) = A319627(A324886(n)).

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A337474 Number of prime shifts (x -> A003961(x)) needed before the result is deficient, when starting from x = A108951(n), the primorial inflation of n.

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

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A329600 Smallest number with the same set of distinct prime exponents as A108951(n).

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A342012 Primorial deflation of the n-th colossally abundant number: the unique integer k such that A108951(k) = A004490(n).

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