A007913 -id:A007913 - cp's OEIS frontend

A069087 Numbers m such that (1/m)Sum_{k=1..m} core(k) > phi(m) where core(n) = A007913(n) is the squarefree part of n: the smallest number such that na(n) is a square and phi(n) = A000010(n) is the Euler totient function.

2, 6, 12, 18, 24, 30, 36, 42, 48, 60, 66, 72, 78, 84, 90, 96, 102, 114, 120, 126, 132, 138, 144, 150, 156, 168, 174, 180, 186, 198, 204, 210, 222, 228, 234, 240, 246, 252, 258, 264, 270, 276, 282, 294, 300, 306, 312, 318, 330, 336, 342, 348, 360, 372, 378, 390

Offset: 1

Benoit Cloitre, Apr 05 2002

nonn

Equivalently, numbers m such that A069891(m) > m*phi(m).

The listed terms are all even, but there are some odd terms, including m = 111546435 = 3*5*7*11*13*17*19*23, for which A069891(m) = 4093453424286382 and m*phi(m) = 4070927302041600.

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

f[p_, e_] := If[OddQ[e], p, 1]; sqf[n_] := Times @@ (f @@@ FactorInteger[n]); seq = {}; s = 0; Do[s += sqf[n]; If[s > n*EulerPhi[n], AppendTo[seq, n]], {n, 1, 400}]; seq (* Amiram Eldar, Apr 02 2020 *)

is(n)=sum(k=1,n,core(k)) > n*eulerphi(n) \\ Charles R Greathouse IV, Feb 21 2013

Edited by Dean Hickerson, Apr 09 2002

A225882 Numbers k such that core(k) is equal to the sum of the proper square divisors of k, where core(k) = A007913(k).

Original entry on oeis.org

20, 90, 336, 650, 5440, 7371, 13000, 14762, 28730, 30240, 83810, 87296, 130682, 147420, 218400, 280370, 295240, 406875, 708122, 924482, 1397760, 1875530, 2613640, 3536000, 4881890, 4960032, 5884851, 7856640, 7893290, 8137500

Offset: 1

Antonio Roldán, May 19 2013

nonn

If p is prime and p^2 + 1 squarefree, then p^2*(p^2 + 1) is in the sequence.

			13000 is a term because core(13000) = 130 = 100 + 25 + 4 + 1.

Giovanni Resta, Table of n, a(n) for n = 1..282 (terms < 5*10^11; first 80 terms from Charles R Greathouse IV)

Cf. A007913, A035316, A225880, A225881.

for(n=2,10^8,if(core(n)==sumdiv(n,d,d*issquare(d)),print(n)))

ssd(f)=prod(i=1,#f[,1],(f[i,1]^(f[i,2]+2-f[i,2]%2)-1)/(f[i,1]^2-1))
is(n)=my(f=factor(n));prod(i=1,#f[,1],f[i,1]^(f[i,2]%2))==ssd(f) && n>1 \\ Charles R Greathouse IV, May 20 2013

A249388 Put a [+] b = A(A(a) + A(b)), where A=A007913. The sequence lists consecutive row "sums" of triangle A248473, using [+].

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 2, 94, 1, 1, 1, 2, 6, 2, 17, 2, 2, 1187, 6, 2, 1, 62, 2, 56883, 14, 3, 14471, 2, 14, 3018, 34, 6, 3, 29, 67, 19, 1, 38, 528846, 9758, 14, 18278015, 163506530, 767014, 7, 2, 2611563, 2081053770, 3, 2, 2, 53654, 94, 17175330570, 2, 1612685866

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Oct 27 2014

nonn

Cf. A007913, A248470, A248473.

a7913[n_]:=a7913[n]=Times@@(#[[1]]^Mod[#[[2]],2])&[Transpose[FactorInteger[n]]];
ab[x_,y_]:=ab[x,y]=a7913[a7913[x]+a7913[y]];
Map[Fold[ab,First[#],Rest[#]]&,Table[a7913[Binomial[a7913[m],a7913[k]]],{m,0,50},{k,0,m}]] (* Peter J. C. Moses, Oct 27 2014 *)

A336641 Numbers k such that A007913(k) divides sigma(k) and A008833(k)-1 either divides A326127(k) (= sigma(k)-core(k)-k), or both are zero.

Original entry on oeis.org

6, 24, 28, 96, 120, 150, 294, 384, 496, 1014, 1536, 3276, 3750, 3780, 6144, 8128, 14406, 20328, 24576, 32760, 93750, 98304, 171366, 306180, 393216, 705894, 1241460, 1572864, 2343750, 6291456, 16380000, 24800580, 25165824, 28960854, 30387840, 33550336, 34588806, 58593750, 100663296, 165143160, 332226048, 402653184

Offset: 1

Antti Karttunen, Jul 28 2020

nonn

Numbers k such that A326128(k) = A326129(k) form a subsequence of this sequence. So far it is not known whether it contains any other terms apart from those of A000396. See comments in A326129.
Sequence is infinite because all numbers of the form 6*4^n (A002023) are present.
Question: Are there any odd terms?

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000203, A007913, A228058, A326126, A326127, A326128, A326129, A336642.
Cf. A000396, A002023 (subsequences).
Cf. also A336550 for a similar construction.

isA336641(n) = { my(c=core(n), s=sigma(n), u=((n/c)-1)); (!(s%c) && (gcd(u,(s-c-n))==u)); };

A347119 Squarefree part of A005940(1+(3*A156552(n))): a(n) = A007913(A332449(n)).

Original entry on oeis.org

1, 1, 1, 10, 1, 1, 1, 30, 21, 1, 1, 22, 1, 1, 1, 10, 1, 10, 1, 10, 1, 1, 1, 66, 55, 1, 105, 10, 1, 1, 1, 30, 1, 1, 1, 154, 1, 1, 1, 30, 1, 1, 1, 10, 39, 1, 1, 22, 91, 21, 1, 10, 1, 30, 1, 30, 1, 1, 1, 34, 1, 1, 21, 10, 1, 1, 1, 10, 1, 1, 1, 462, 1, 1, 21, 10, 1, 1, 1, 10, 21, 1, 1, 22, 1, 1, 1, 30, 1, 22, 1, 10

Offset: 1

Antti Karttunen, Aug 22 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000290, A005117 (positions of 1's), A005940, A007913, A156552, A332449, A347120.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A332449(n) = A005940(1+(3*A156552(n)));
A347119(n) = core(A332449(n));

a(n) = A007913(A332449(n)).

a(n) = A332449(n) / A000290(A347120(n)).

A248473 Triangle of numbers b(i,j) = A(binomial(A(i), A(j))), where A = A007913, with the convention that A(0)=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 1, 0, 0, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 5, 6, 6, 1, 1, 7, 21, 35, 7, 21, 7, 1, 1, 2, 1, 0, 2, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 10, 5, 30, 10, 7, 210, 30, 5, 10, 1, 1, 11, 55, 165, 11, 462, 462, 330, 55, 11, 11, 1

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Oct 27 2014

By definition, all terms are squarefree (A005117).

			For i=8, j=4, we have A(8)=2, A(4)=1, hence b(8,4) = A(binomial(2,1)) = 2.
Triangle begins
1
1   1
1   2   1
1   3   3   1
1   1   0   0   1
1   5  10  10   5   1
1   6  15   5   6   6   1
1   7  21  35   7  21   7   1
1   2   1   0   2   0   0   0   1
1   1   0   0   1   0   0   0   0   1
1  10   5  30  10   7 210  30   5  10  1
..........................................

Cf. A005117, A007318, A007913, A248470.

a7913[n_]:=a7913[n]=Times@@(#[[1]]^Mod[#[[2]],2])&[Transpose[FactorInteger[n]]];
Flatten[Table[a7913[Binomial[a7913[m],a7913[k]]],{m,0,10},{k,0,m}]] (* Peter J. C. Moses, Oct 27 2014 *)

A069186 Numbers n such that core(n)=floor(sqrt(n)), where core(x)=A007913(x) is the squarefree part of x and floor(sqrt(x))=A000196(x).

Original entry on oeis.org

1, 8, 12, 63, 224, 240, 575, 1224, 1260, 2303, 3968, 6399, 14399, 20448, 20592, 28223, 38024, 38220, 50175, 65024, 65280, 82943, 104328, 104652, 129599, 159200, 159600, 193599, 233288, 233772, 278783, 330624, 389375, 455624

Offset: 1

Benoit Cloitre, Apr 14 2002

Apart from 1, numbers of the form x*y^2 for y >= 2, where x is squarefree and is either y^2-2 or y^2-1. - Robert Israel, Apr 11 2019

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000196, A007913.

select(numtheory:-issqrfree, [1, seq(seq(b^2+j,j=-2..-1),b=2..100)]); # Robert Israel, Apr 11 2019

isok(n) = core(n) == sqrtint(n); \\ Michel Marcus, Apr 12 2019

Name edited by Robert Israel, Apr 12 2019

A069189 Numbers k such that A007913(k) < sqrt(k).

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 25, 27, 32, 36, 45, 48, 49, 50, 54, 63, 64, 72, 75, 80, 81, 96, 98, 100, 108, 112, 121, 125, 128, 144, 147, 150, 160, 162, 169, 175, 176, 180, 192, 196, 200, 208, 216, 224, 225, 240, 242, 243, 245, 250, 252, 256, 275, 288, 289, 294, 300, 320

Offset: 1

Benoit Cloitre, Apr 10 2002

Equivalently, numbers k whose squarefree part, A007913(k), is smaller than their square part, A008833(k). - Peter Munn, Mar 26 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square Part.
Eric Weisstein's World of Mathematics, Squarefree Part.

Cf. A007913, A008833.
Subsequence of A048098.
A116882 is a near equivalent with respect to a number's odd part.

f[p_, e_] := If[OddQ[e], p, 1]; sqf[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[320], sqf[#] < Sqrt[#] &] (* Amiram Eldar, Apr 02 2020 *)

PARI
```
for(n=1,200,if(core(n)
				
```

A069260 a(n) = core(1)core(2)...*core(n) where core(n) is the squarefree part of n (A007913).

Original entry on oeis.org

1, 2, 6, 6, 30, 180, 1260, 2520, 2520, 25200, 277200, 831600, 10810800, 151351200, 2270268000, 2270268000, 38594556000, 77189112000, 1466593128000, 7332965640000, 153992278440000, 3387830125680000, 77920092890640000, 467520557343840000, 467520557343840000

Offset: 1

Benoit Cloitre, Apr 14 2002

A "core" analog of n! (A000142) - might be called a "c-factorial" (see formula). - Vladimir Shevelev, Oct 22 2014

Amiram Eldar, Table of n, a(n) for n = 1..538

Cf. A007913, A000142.

core[n_] := Times @@ (First[#]^Mod[Last[#], 2] & /@ FactorInteger[n]); FoldList[Times, 1, core /@ Range[2, 23]] (* Amiram Eldar, Sep 05 2020 *)

a(n) = prod(i=1, n, core(i)); \\ Michel Marcus, Aug 09 2013

Let p_n = prime(n). a(n) = n!^(c) = p_1^b_1*p_2^b_2*...*p_k^b_k, where p_k is maximal prime <= n and b_i = floor(n/p_i)- floor(n/p_i^2) + floor(n/p_i^3)-..., i.e., for exponents of primes of c-factorial we have an alternating sum, instead of the similar sum for exponents of primes for n! - Vladimir Shevelev, Oct 22 2014

More terms from Amiram Eldar, Sep 05 2020

A074786 Numbers k such that phi(k) = Sum_{d|k} core(d) where core(x) is the squarefree part of x (A007913).

Original entry on oeis.org

1, 99, 1080, 1836, 4743, 5670, 7644, 8307, 14384, 14784, 15225, 15824, 16065, 20300, 21584, 25704, 29760, 34544, 46816, 71568, 94240, 128412, 169290, 264160, 266266, 312480, 331731, 364941, 404550, 445050, 454575, 458052, 479655, 497781

Offset: 1

Benoit Cloitre, Sep 07 2002

nonn

Numbers k such that A000010(k) = A069088(k). - Amiram Eldar, Apr 28 2025

Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..500 from Donovan Johnson)

Cf. A000010, A007913, A069088.

f[p_, e_] := If[OddQ[e], (p+1)*(e+1)/2, (p+1)*e/2 + 1] / ((p-1)*p^(e-1)); q[1] = True; q[n_] := Times @@ f @@@ FactorInteger[n] == 1; Select[Range[500000], q] (* Amiram Eldar, Apr 28 2025 *)

isok(n) = eulerphi(n) == sumdiv(n, d, core(d)); \\ Michel Marcus, Aug 09 2013

isok(k) = {my(f = factor(k)); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(e % 2, (p+1)*(e+1)/2, (p+1)*e/2 + 1) / ((p-1)*p^(e-1))) == 1;} \\ Amiram Eldar, Apr 28 2025

More terms from Michel Marcus, Aug 09 2013

cp's OEIS Frontend

A069087 Numbers m such that (1/m)*Sum_{k=1..m} core(k) > phi(m) where core(n) = A007913(n) is the squarefree part of n: the smallest number such that n*a(n) is a square and phi(n) = A000010(n) is the Euler totient function.

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A225882 Numbers k such that core(k) is equal to the sum of the proper square divisors of k, where core(k) = A007913(k).

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A249388 Put a [+] b = A(A(a) + A(b)), where A=A007913. The sequence lists consecutive row "sums" of triangle A248473, using [+].

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Mathematica

A336641 Numbers k such that A007913(k) divides sigma(k) and A008833(k)-1 either divides A326127(k) (= sigma(k)-core(k)-k), or both are zero.

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A347119 Squarefree part of A005940(1+(3*A156552(n))): a(n) = A007913(A332449(n)).

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A248473 Triangle of numbers b(i,j) = A(binomial(A(i), A(j))), where A = A007913, with the convention that A(0)=0.

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A069186 Numbers n such that core(n)=floor(sqrt(n)), where core(x)=A007913(x) is the squarefree part of x and floor(sqrt(x))=A000196(x).

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A069189 Numbers k such that A007913(k) < sqrt(k).

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A069260 a(n) = core(1)*core(2)*...*core(n) where core(n) is the squarefree part of n (A007913).

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A069087 Numbers m such that (1/m)Sum_{k=1..m} core(k) > phi(m) where core(n) = A007913(n) is the squarefree part of n: the smallest number such that na(n) is a square and phi(n) = A000010(n) is the Euler totient function.

A069260 a(n) = core(1)core(2)...*core(n) where core(n) is the squarefree part of n (A007913).