A039650 -id:A039650 - cp's OEIS frontend

A221741 a(n) = -4(((n+1)^(n+1)-(n+1))/((n+1)-1)^2-1)/((-3+(-1)^n)n).

1, 5, 9, 97, 373, 7625, 48913, 1513361, 13717421, 570623341, 6698798233, 350549891889, 5057809205989, 319164643134737, 5465701947765793, 403925909124187873, 8008631808527689309, 678470389458269406421, 15287592943577781017641

Offset: 1

Alexander R. Povolotsky, Jan 23 2013

Per exhaustive program, written for bases from 2 to 10, the number of permutations pairs, which have the same ratio, equal to A221740(n)/a(n) = (n^2 (n+1)^n-(n+1)^n+1) / (-n^2+n (n+1)^n+(n+1)^n-n-1), is: {2,2,3,3,5,3,7,5,7,...} for n>=1 where n=r-1 and r is the base radix. Judging by above sequence it appears that the number of such permutations pairs is related to phi, which is the Euler totient function - according to A039649, A039650, A214288 (see bullet 1 of the analysis in the answer section of the StackExchange link). Alexander R. Povolotsky, Jan 26 2013

G. C. Greubel, Table of n, a(n) for n = 1..385
NMBRTHRY, Number of specific permutations pairs relates to Euler Phi totient function ?
StackExchange, Permutations (with no duplicates) of decimal base digits 1,2,...,8,9,0.

Cf. A221740, A023811.

Table[-4*(((n + 1)^(n + 1) - (n + 1))/((n + 1) - 1)^2 - 1)/((-3 + (-1)^n)*n), {n,1,50}] (* G. C. Greubel, Feb 19 2017 *)

makelist(-4*(((n+1)^(n+1)-(n+1))/((n+1)-1)^2-1)/((-3+(-1)^n)*n), n, 1, 20); /* Martin Ettl, Jan 25 2013 */

for(n=1,25, print1(-4*(((n + 1)^(n + 1) - (n + 1))/((n + 1) - 1)^2 - 1)/((-3 + (-1)^n)*n), ", ")) \\ G. C. Greubel, Feb 19 2017

a(n) = -4*A023811(n+1)/((-3 + (-1)^n)*n).

A323075 The fixed point reached when map x -> 1+(x-(largest divisor d < x)) is iterated, starting from x = n.

Original entry on oeis.org

1, 2, 3, 3, 5, 3, 7, 5, 7, 3, 11, 7, 13, 5, 11, 7, 17, 3, 19, 11, 11, 7, 23, 13, 11, 5, 19, 11, 29, 7, 31, 17, 23, 3, 29, 19, 37, 11, 19, 11, 41, 7, 43, 23, 31, 13, 47, 11, 43, 5, 29, 19, 53, 11, 31, 29, 19, 7, 59, 31, 61, 17, 43, 23, 53, 3, 67, 29, 47, 19, 71, 37, 73, 11, 29, 19, 67, 11, 79, 41, 31, 7, 83, 43, 47, 23, 59, 31, 89, 13

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

After a(1) = 1, the fixed point reached is always a prime. Question: Do all odd primes occur infinitely often?

Yes. All odd primes occur infinitely often. A060681(2*k) = k + 1 and so for each k > 1 there exists an integer m such that a(m) = p where p is an odd prime. - David A. Corneth, Jan 07 2019

Antti Karttunen, Table of n, a(n) for n = 1..20669

Cf. A000040, A000079, A000918, A060681, A323076, A323079, A323164, A323165 (ordinal transform).

Cf. also A039650, A039654.

{1}~Join~Array[FixedPoint[1 + (# - Divisors[#][[-2]]) &, #] &, 89, 2] (* Michael De Vlieger, Jan 04 2019 *)

A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
A323075(n) = { my(nn = 1+A060681(n)); if(nn==n,n,A323075(nn)); };

If n == (1+A060681(n)), then a(n) = n, otherwise a(n) = a(1+A060681(n)).
a(2^k * p - 2^(k+1) + 2) = a(A000079(k) * p - A000918(k+1)) = p for k >= 0. - David A. Corneth, Jan 08 2019
a(1) = 1, and for n > 1, a(n) = A000040(A323164(n)). - Antti Karttunen, Jan 08 2019

A296080 Restricted growth sequence transform of A289625(1+phi(n)), where phi = A000010, Euler totient function.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 05 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A039649, A289625, A289626, A296078.

Cf. A039650.

allocatemem(2^30);
up_to = 65537;
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A289625(n) = { my(m=1,p=2,v=znstar(n)[2]); for(i=1,length(v),m *= p^v[i]; p = nextprime(p+1)); (m); };
write_to_bfile(1,rgs_transform(vector(up_to,n,A289625(1+eulerphi(n)))),"b296080.txt");

A229487 Conjectured greatest number that converges to prime(n) under the iteration x -> phi(x) + 1, where phi is Euler's totient function.

Original entry on oeis.org

2, 6, 12, 30, 22, 138, 60, 54, 46, 58, 62, 174, 498, 510, 94, 106, 118, 4314, 134, 142, 1038, 158, 166, 276, 420, 250, 206, 214, 750, 1758, 254, 262, 274, 278, 298, 302, 1182, 486, 334, 346, 358, 6360, 382, 1614, 394, 398, 422, 446, 454, 458, 5898, 478, 54582

Offset: 1

T. D. Noe, Oct 16 2013

nonn

Many terms are just twice a prime: 6, 22, 46, 58, 62, 94,....

			The number 138 has trajectory {138, 45, 25, 21, 13}, which is conjecturally the last number that terminates with 13 = prime(6). Hence a(6) = 138.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A039650, A039651, A039652.

t = Table[NestWhileList[EulerPhi[#] + 1 &, n, UnsameQ, All][[-1]], {n, 100000}]; Table[Position[t, p][[-1]], {p, Prime[Range[53]]}]

cp's OEIS Frontend

A221741 a(n) = -4*(((n+1)^(n+1)-(n+1))/((n+1)-1)^2-1)/((-3+(-1)^n)*n).

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A323075 The fixed point reached when map x -> 1+(x-(largest divisor d < x)) is iterated, starting from x = n.

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A296080 Restricted growth sequence transform of A289625(1+phi(n)), where phi = A000010, Euler totient function.

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A229487 Conjectured greatest number that converges to prime(n) under the iteration x -> phi(x) + 1, where phi is Euler's totient function.

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A221741 a(n) = -4(((n+1)^(n+1)-(n+1))/((n+1)-1)^2-1)/((-3+(-1)^n)n).