A072995 -id:A072995 - cp's OEIS frontend

A253560 Multiply n by its largest prime factor: a(n) = A006530(n) * n.

1, 4, 9, 8, 25, 18, 49, 16, 27, 50, 121, 36, 169, 98, 75, 32, 289, 54, 361, 100, 147, 242, 529, 72, 125, 338, 81, 196, 841, 150, 961, 64, 363, 578, 245, 108, 1369, 722, 507, 200, 1681, 294, 1849, 484, 225, 1058, 2209, 144, 343, 250, 867, 676, 2809, 162, 605, 392, 1083, 1682, 3481, 300, 3721, 1922, 441

Offset: 1

Antti Karttunen, Jan 03 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Essentially the same as A129598, except that here we have a(1) = 1.
Cf. A070003 (same sequence without 1, sorted into ascending order).
Cf. A000040, A006530, A052126, A061395, A122111, A253550, A253561, A253563, A253565.
Differs from A072995 for the first time at n=15, where a(15) = 75, while A072995(15) = 225.

A253560 := proc(n)
    n*A006530(n) # code re-use
end proc:
seq( A253560(n),n=1..80) ; # R. J. Mathar, Jun 27 2024

a[n_] := n FactorInteger[n][[-1, 1]];
Array[a, 100] (* Jean-François Alcover, Feb 15 2021 *)

a(n) = if (n==1, 1, n*vecmax(factor(n)[,1])); \\ Michel Marcus, Feb 15 2021

Scheme
```
(define (A253560 n) (* (A006530 n) n))
```

a(1) = 1; for n > 1, a(n) = A006530(n) * n = A000040(A061395(n)) * n.
Other identities:
a(n) >= A253550(n) for all n >= 1.
a(n) = A129598(n) for all n >= 2.
A052126(a(n)) = n. [A052126 works as an inverse function for this injection.]

A050399 Least k such that n = A009195(k) (= gcd(phi(k), k)).

Original entry on oeis.org

1, 4, 9, 8, 25, 18, 49, 16, 27, 50, 121, 36, 169, 98, 225, 32, 289, 54, 361, 100, 147, 242, 529, 72, 125, 338, 81, 196, 841, 450, 961, 64, 1089, 578, 1225, 108, 1369, 722, 507, 200, 1681, 294, 1849, 484, 675, 1058, 2209, 144, 343, 250, 2601, 676, 2809

Offset: 1

Christian G. Bower, Nov 15 1999

nonn

Coincides with A072995 for many terms, but differs, e.g., in n = 20, 40, 52, ... in addition to the zeros in A072995. See also the comments in A072994. - M. F. Hasler, Feb 23 2014

a(n) <= n^2. - Robert G. Wilson v, Feb 27 2014

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

t = Table[0, {10000}]; k = 1; While[k < 100000001, a = GCD[k, EulerPhi@ k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++]; t (* Robert G. Wilson v, Feb 27 2014 *)

A050399=n->for(k=1,oo,gcd(eulerphi(k),k)==n&&return(k)) \\ M. F. Hasler, Feb 23 2014

A072994 Number of solutions to x^n==1 (mod n), 1<=x<=n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 1, 8, 1, 6, 1, 8, 3, 2, 1, 8, 5, 2, 9, 4, 1, 4, 1, 16, 1, 2, 1, 12, 1, 2, 3, 16, 1, 12, 1, 4, 3, 2, 1, 16, 7, 10, 1, 8, 1, 18, 5, 8, 3, 2, 1, 16, 1, 2, 9, 32, 1, 4, 1, 8, 1, 4, 1, 24, 1, 2, 5, 4, 1, 12, 1, 32, 27, 2, 1, 24, 1, 2, 1, 8, 1, 12, 1, 4, 3

Offset: 1

Benoit Cloitre, Aug 21 2002

More generally, if the equation a(x)*m=x has solutions, solutions are congruent to m: a(x)*7=x for x=7, 14, 21, 28, 49, 56, 63, 98, 112, ... . There are some composite values of m such that a(x)*m=x has solutions, as m=15. a(n) coincides with A009195(n) at many values of n, but not at n = 20, 30, 40, 42, 52, 60, 66, 68, 70, 78, 80, 84, 90, 100, ... . It seems also that for n large enough sum_{k=1..n} a(k) > n*log(n)*log(log(n)).

Similar (if not the same) coincidences and differences occur between A072995 and A050399. Sequence A072989 lists these indices. - M. F. Hasler, Feb 23 2014

T. D. Noe, Table of n, a(n) for n=1..10000

1, seq(nops(select(t -> t^n mod n = 1, [$1..n-1])),n=2..100); # Robert Israel, Dec 07 2014

f[n_] := (d = If[ OddQ@ n, 1, 2]; d*Length@ Select[ Range[ n/d], PowerMod[#, n, n] == 1 &]); f[1] = f[2] = 1; Array[f, 93] (* or *)
f[n_] := Length@ Select[ Range@ n, PowerMod[#, n, n] == 1 &]; f[n_] := 1 /; n<2; Array[f, 93] (* Robert G. Wilson v, Dec 06 2014 *)

A072994=n->sum(k=1,n,Mod(k,n)^n==1) \\ M. F. Hasler, Feb 23 2014

For n>0, a(A003277(n)) = 1, a(2^n) = 2^(n-1), a(A065119(n)) = A065119(n)/3.

For n>1, a(A026383(n)) = A026383(n)/5.

Corrected by T. D. Noe, May 19 2007

A129598 a(n) = n * A111089(n).

Original entry on oeis.org

2, 4, 9, 8, 25, 18, 49, 16, 27, 50, 121, 36, 169, 98, 75, 32, 289, 54, 361, 100, 147, 242, 529, 72, 125, 338, 81, 196, 841, 150, 961, 64, 363, 578, 245, 108, 1369, 722, 507, 200, 1681, 294, 1849, 484, 225, 1058, 2209, 144, 343, 250, 867, 676, 2809, 162, 605

Offset: 1

Antti Karttunen, May 01 2007

nonn

Conjecture: differs from A050399 at the positions given by A089966. E.g., a(15)=75, instead of A050399(15)=225, a(30)=150, instead of A050399(30)=450, a(33)=363, instead of A050399(33)=1089, a(45)=225, instead of A050399(45)=1225. Conjecture 2: for all n > 1, a(n) divides A050399(n).

Antti Karttunen, Table of n, a(n) for n = 1..10000
G. Guninski, List of automatically found conjectural relations between this and other sequences

Row 2 of A129595.
Essentially the same as A253560, except that here we have a(1) = 2.
Cf. also A050399, A089966, A111089, A072995.

Join[{2},Table[n*Last@(First/@FactorInteger[n]),{n,2,200}]] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2011 *)

a(1) = 2; for n >= 2, a(n) = A253560(n).

A238367 Even numbers n such that n is not divisible by 2^d where d is the number of distinct odd prime factors of n.

Original entry on oeis.org

30, 42, 66, 70, 78, 90, 102, 110, 114, 126, 130, 138, 150, 154, 170, 174, 182, 186, 190, 198, 210, 222, 230, 234, 238, 246, 258, 266, 270, 282, 286, 290, 294, 306, 310, 318, 322, 330, 342, 350, 354, 366, 370, 374, 378, 390, 402, 406, 410, 414, 418, 420, 426, 430, 434, 438, 442, 450, 462, 470, 474, 490, 494, 498

Offset: 1

Robert G. Wilson v, Feb 25 2014

nonn

Inspired by comment by Don Reble in A072995.

2*A061346 = A238367 with the exceptions of 420, 660, 780, 924, 1020, 1092, 1140, 1260, ... .

			30 is in the sequence because the odd primes that divide 30 are 3 and 5, but 2^2 does not divide 30.

Cf. A072995.

Select[2 Range[250], Mod[#, 2^(PrimeNu[#] - 1)] != 0 &]

cp's OEIS Frontend

A253560 Multiply n by its largest prime factor: a(n) = A006530(n) * n.

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A050399 Least k such that n = A009195(k) (= gcd(phi(k), k)).

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A072994 Number of solutions to x^n==1 (mod n), 1<=x<=n.

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A129598 a(n) = n * A111089(n).

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A238367 Even numbers n such that n is not divisible by 2^d where d is the number of distinct odd prime factors of n.

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Mathematica