A000188 -id:A000188 - cp's OEIS frontend

A294895 a(n) = Product_{d|n, gcd(d,n/d)>1} prime(gcd(d,n/d)-1).

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 4, 1, 1, 1, 20, 1, 9, 1, 4, 1, 1, 1, 16, 7, 1, 9, 4, 1, 1, 1, 100, 1, 1, 1, 396, 1, 1, 1, 16, 1, 1, 1, 4, 9, 1, 1, 400, 13, 49, 1, 4, 1, 81, 1, 16, 1, 1, 1, 16, 1, 1, 9, 1700, 1, 1, 1, 4, 1, 1, 1, 17424, 1, 1, 49, 4, 1, 1, 1, 400, 171, 1, 1, 16, 1, 1, 1, 16, 1, 81, 1, 4, 1, 1, 1, 10000, 1, 169, 9, 4508, 1, 1, 1, 16, 1

Offset: 1

Antti Karttunen, Nov 21 2017

nonn

For all i, j: a(i) = a(j) => A294897(i) = A294897(j).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005117 (the positions of ones).

Cf. A294876, A294897.

A294895[n_] := Times @@ Prime[Select[Map[GCD[#, n/#] &, Divisors[n]], #>1 &] - 1];
Array[A294895, 100] (* Paolo Xausa, Feb 22 2024 *)

A294895(n) = { my(m=1); fordiv(n,d,if(gcd(d,n/d)>1, m *= prime(gcd(d,n/d)-1))); m; };

a(n) = Product_{d|n} A008578(gcd(d,n/d)).
a(n) = A064989(A294876(n)).
For n >= 1, A001222(a(n)) = A048105(n).
For n > 1, 1+A061395(a(n)) = A000188(n).

A295658 Multiplicative with a(p^e) = p^max(0,(floor(e/2)-1)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 28 2017

a(n) differs from A053164(n) = A000188(A000188(n)) for the first time at n=64, where a(64) = 4, while A053164(64) = 2.

			For n = 64 = 2^6, a(64) = 2^(floor(6/2)-1) = 2^2 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000188, A003557, A004526, A020639, A028234, A053164, A067029, A295657.

Cf. A046100 (positions of ones), A157289.

f[p_, e_] := p^Max[0, Floor[e/2-1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 30 2022 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^max(0, floor(f[i,2]/2-1)));} \\ Amiram Eldar, Nov 30 2022

a(1) = 1; for n > 1, a(n) = A020639(n)^max(0,A004526(A067029(n))-1) * a(A028234(n)).
a(n) = A003557(A000188(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3)/zeta(6) = 1.181564... (A157289). - Amiram Eldar, Nov 30 2022

A329602 Square root of largest square dividing A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 6, 2, 1, 2, 1, 2, 6, 4, 1, 6, 1, 2, 6, 2, 1, 4, 30, 2, 6, 2, 1, 6, 1, 4, 6, 2, 30, 12, 1, 2, 6, 4, 1, 6, 1, 2, 6, 2, 1, 4, 210, 30, 6, 2, 1, 12, 30, 4, 6, 2, 1, 12, 1, 2, 6, 8, 30, 6, 1, 2, 6, 30, 1, 12, 1, 2, 30, 2, 210, 6, 1, 4, 36, 2, 1, 12, 30, 2, 6, 4, 1, 12, 210, 2, 6, 2, 30, 8, 1, 210, 6, 60, 1, 6, 1, 4, 30

Offset: 1

Antti Karttunen, Nov 19 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10201

Cf. A000188, A108951.

A000188(n) = core(n, 1)[2]; \\ From A000188
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
A329602(n) = A000188(A108951(n));

a(n) = A000188(A108951(n)).

A337110 Number of length three 1..n vectors that contain their geometric mean.

Original entry on oeis.org

1, 2, 3, 10, 11, 12, 13, 20, 33, 34, 35, 42, 43, 44, 45, 64, 65, 78, 79, 86, 87, 88, 89, 96, 121, 122, 135, 142, 143, 144, 145, 164, 165, 166, 167, 198, 199, 200, 201, 208, 209, 210, 211, 218, 231, 232, 233, 252, 289, 314, 315, 322, 323, 336, 337, 344, 345, 346

Offset: 1

Hywel Normington, Aug 16 2020

From David A. Corneth, Aug 26 2020: (Start)
If x^2 == 0 (mod n) has only 1 solution then a(n) = a(n-1) + 1. Proof:
Let (a, b, n) be such a tuple. Let without loss of generality b be the geometric mean of the tuple. Then a*b*n = b^3 and as b is not 0 we have b^2 = a*n. So then b^2 == 0 (mod n). If b^2 == 0 (mod n) has only 1 solution then b = n. This gives the tuple (n, n, n) which has 1 permutation. So giving a(n) = a(n-1) + 1. (End)

			For n = 2, the a(2) = 2 solutions are: (1,1,1) and (2,2,2).
For n = 4, the a(4) = 10 solutions are: (1,1,1),(2,2,2),(3,3,3),(4,4,4) and the 6 permutations of (1,2,4).

Hywel Normington, Python code, 2020.
Hywel Normington, Julia code, 2023.

Cf. A120486, A248434, A337111, A013929, A057918, A000188.

first(n) = {my(s = 0, res = vector(n)); for(i = 1, n, s+=b(i); res[i] = s ); res }
b(n) = { my(s = factorback(factor(n)[, 1]), res = 1); for(i = 1, n \ s - 1, c = (s*i)^2/n; if(denominator(c) == 1 && c <= n, res+=6; ) ); res } \\ David A. Corneth, Aug 26 2020

a(n) = a(n-1) + 1 + 6*A057918(n).

A382890 The square root of the largest square dividing the n-th cubefree number.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 5, 1, 2, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 7, 5, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 7, 3, 10, 1, 1

Offset: 1

Amiram Eldar, Apr 07 2025

The product of the non-unitary prime divisors of the n-th cubefree number.
Also, the square root of the powerful part of the n-th cubefree number.
All the terms are squarefree.

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

Cf. A000188, A004709, A005117, A057521, A371188 (positions of 1's).

Similar sequences: A382888, A382889, A382891.

f[p_, e_] := p^If[e == 1, 0, 1]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]

list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 3, print1(prod(i = 1, #f~, f[i, 1]^if(f[i, 2] == 1, 0, 1)), ", ")));}

a(n) = A000188(A004709(n)).
a(n) = sqrt(A382889(n)).
a(n) = A004709(n)/A382888(n).
a(n) = sqrt(A004709(n)/A382891(n)).
a(A371188(n)) = 1.

A055654 Difference between n and the result of "Phi-summation" over unitary divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 2, 0, 0, 3, 0, 0, 0, 7, 0, 4, 0, 5, 0, 0, 0, 9, 4, 0, 8, 7, 0, 0, 0, 15, 0, 0, 0, 15, 0, 0, 0, 15, 0, 0, 0, 11, 10, 0, 0, 21, 6, 8, 0, 13, 0, 16, 0, 21, 0, 0, 0, 15, 0, 0, 14, 31, 0, 0, 0, 17, 0, 0, 0, 37, 0, 0, 12, 19, 0, 0, 0, 35, 26, 0, 0, 21, 0, 0, 0, 33, 0, 20, 0, 23

Offset: 1

Labos Elemer, Jun 07 2000

Squarefree numbers are roots of a(n)=0 equation, while Min n for which a(n)=k is k^2. See also A000188, A008833.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000188, A008833, A055653, A053570, A053571, A055631, A055632, A055653, A065465.

a055654 n = a055654_list !! (n-1)
a055654_list = zipWith (-) [1..] a055653_list
-- Reinhard Zumkeller, Mar 11 2012

Table[n - DivisorSum[n, EulerPhi[#] &, CoprimeQ[#, n/#] &], {n, 92}] (* Michael De Vlieger, Oct 26 2017 *)
f[p_, e_] := p^e - p^(e-1) + 1; a[1] = 0; a[n_] := n - Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 04 2024 *)

a(n) = n - sumdiv(n, d, if (gcd(d, n/d)==1, eulerphi(d))); \\ Michel Marcus, Oct 27 2017

a(n) = {my(f = factor(n)); n - prod(k = 1, #f~, f[k,1]^f[k,2] - f[k,1]^(f[k,2] - 1) + 1);} \\ Amiram Eldar, Oct 04 2024

a(n) = n - Sum_{u|n, gcd(u,n/u) = 1} phi(u), i.e. when u is a unitary divisor of n.
a(n) = n - A055653(n). - Sean A. Irvine, Mar 30 2022
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 - A065465 = 0.11848616... . - Amiram Eldar, Oct 04 2024

A056044 Let k be the largest number such that k^2 divides n! and let m be the largest number such that m!^2 divides n!; a(n) = k/m!.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 1, 2, 2, 2, 6, 3, 3, 2, 2, 2, 2, 2, 2, 2, 10, 10, 30, 2, 2, 12, 12, 3, 3, 6, 30, 10, 10, 10, 30, 6, 6, 2, 2, 2, 30, 60, 60, 30, 210, 42, 42, 42, 42, 1, 1, 2, 2, 4, 4, 4, 4, 4, 84, 21, 21, 14, 14, 14, 42, 6, 6, 2, 2, 2, 10, 10, 70, 140, 140, 14, 126, 3, 3, 6, 30

Offset: 1

Labos Elemer, Jul 25 2000

nonn

			For n = 11, 11! = 6! * 6! * 77, so A000188(11!) = A056038(11) = 6! and a(11) = 6!/6! = 1.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000188, A055772, A001405, A001057, A056038.

f[p_, e_] := p^Floor[e/2]; b[1] = 1; b[n_] := Times @@ f @@@ FactorInteger[n!];
c[n_] := Module[{k = 1}, NestWhile[#/(++k)^2 &, n!, IntegerQ]; (k-1)!];
a[n_] := b[n] / c[n]; Array[a, 100] (* Amiram Eldar, May 24 2024 *)

a(n) = A000188(n!)/A056038(n).

Name corrected by Amiram Eldar, May 24 2024

A056627 a(n) = A056622(n!).

Original entry on oeis.org

1, 1, 1, 1, 1, 12, 12, 12, 36, 720, 720, 480, 480, 1680, 3024, 12096, 12096, 145152, 145152, 7257600, 345600, 1900800, 1900800, 136857600, 684288000, 4447872000, 4447872000, 435891456000, 435891456000, 3138418483200, 3138418483200, 6276836966400, 190207180800

Offset: 1

Labos Elemer, Aug 08 2000

nonn

Previous name "Square root of largest unitary square divisor of n!" was incorrect. See A374989 for the correct sequence with this name. - Amiram Eldar, Jul 26 2024

			a(12) = A056622(12!) = A000188(12!)/A055229(12!) = 1440/3 = 480.

Cf. A055772, A055230, A000188, A055229, A056622, A056628, A374989.

f[p_, 1] := 1; f[p_, e_] := If[EvenQ[e], p^(e/2), p^((e-3)/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 33] (* Amiram Eldar, Jul 08 2024 *)

A055229(n) = { my(c=core(n)); gcd(c, n/c); }; \\ Charles R Greathouse IV, Nov 20 2012
A008833(n) = n/core(n) \\ Michael B. Porter, Oct 17 2009
A056623(n) = (A008833(n)/(A055229(n)^2)); \\ Antti Karttunen, Nov 19 2017
a(n) = sqrtint(A056623(n!)); \\ Michel Marcus, Aug 16 2020

a(n) = A055772(n)/A055230(n) = A000188(n!)/A055229(n!).
a(n) = A056622(n!). - Michel Marcus, Aug 16 2020
a(n) = sqrt(A056628(n)). - Amiram Eldar, Jul 08 2024

More terms from Michel Marcus, Aug 16 2020

Incorrect name replaced with a formula by Amiram Eldar, Jul 26 2024

A069292 Sum of square roots of square divisors of n <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 3, 1

Offset: 1

Reinhard Zumkeller, Mar 14 2002

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors

Cf. A035316, A046951, A000188, A069290, A069291, A069293.

Table[DivisorSum[n, Sqrt@ # &, And[IntegerQ@ Sqrt@ #, # <= Sqrt@ n] &], {n, 105}] (* Michael De Vlieger, Nov 20 2017 *)

A069292(n) = { my(r="NA"); sumdiv(n, d, (issquare(d,&r)&&((d^2)<=n))*r); } \\ Antti Karttunen, Nov 20 2017

G.f.: Sum_{k>=1} k * x^(k^4) / (1 - x^(k^2)). - Ilya Gutkovskiy, Aug 19 2021

More terms from Antti Karttunen, Nov 20 2017

A283989 Largest square dividing prime factorization representation of the n-th Stern polynomial: a(n) = A008833(A260443(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 9, 1, 1, 1, 9, 25, 9, 1, 225, 1, 1, 1, 9, 25, 225, 49, 2025, 25, 225, 1, 225, 1225, 225, 1, 11025, 1, 1, 1, 9, 25, 11025, 49, 50625, 1225, 275625, 121, 2480625, 30625, 1265625, 49, 2480625, 1225, 11025, 1, 11025, 1225, 275625, 5929, 2480625, 1225, 275625

Offset: 0

Antti Karttunen, Mar 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A003961, A008833, A260443, A277330, A283983.

Cf. A023758 (positions of ones).

A003961[p_?PrimeQ] := A003961[p] = Prime[ PrimePi[p] + 1]; A003961[1] = 1; A003961[n_] := A003961[n] = Times @@ ( A003961[First[#]] ^ Last[#] & ) /@ FactorInteger[n]  (* after Jean-François Alcover, Dec 01 2011 *); A260443[n_]:= If[n<2, n + 1, If[EvenQ[n], A003961[A260443[n/2]], A260443[(n - 1)/2] * A260443[(n + 1)/2]]]; A000188[n_]:= Sum[Boole[Mod[i^2, n] == 0], {i, n}]; Table[A000188[A260443[n]]^2, {n, 0, 20}] (* Indranil Ghosh, Mar 28 2017 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2)))); \\ Cf. Charles R Greathouse IV's code for "ps" in A186891 and A277013.
A008833(n) = n/core(n); \\ This function from Michael B. Porter, Oct 17 2009
A283989(n) = A008833(A260443(n));

(define (A283989 n) (A008833 (A260443 n)))
(define (A283989 n) (/ (A260443 n) (A277330 n)))

a(n) = A008833(A260443(n)).
a(n) = A260443(n) / A277330(n).
a(n) = A283983(n)^2.
a(2n) = A003961(a(n)).

cp's OEIS Frontend

A294895 a(n) = Product_{d|n, gcd(d,n/d)>1} prime(gcd(d,n/d)-1).

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A295658 Multiplicative with a(p^e) = p^max(0,(floor(e/2)-1)).

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A329602 Square root of largest square dividing A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

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A337110 Number of length three 1..n vectors that contain their geometric mean.

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A382890 The square root of the largest square dividing the n-th cubefree number.

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A055654 Difference between n and the result of "Phi-summation" over unitary divisors of n.

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A056044 Let k be the largest number such that k^2 divides n! and let m be the largest number such that m!^2 divides n!; a(n) = k/m!.

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A056627 a(n) = A056622(n!).