A039649 -id:A039649 - cp's OEIS frontend

A336192 Numbers of the form ab such that phi(ab) = a*b - 1 where ab is the concatenation of a and b.

1385, 1397, 15663, 19835, 37037, 238903, 719719, 1983035, 4337785, 5946445, 8099989, 15276063, 64438507, 97919791, 238639687, 325776657, 1926629941, 3228792383, 4387457627, 4652069941, 9801019901, 44898935609, 68135795923, 115563539473, 129898064149, 390084197561

Offset: 1

M. Farrokhi D. G., Jul 11 2020

Is the sequence infinite?

If phi(ab) = a*b - 1 then ab is a composite number.

			phi(1385) = 13 * 85 - 1
phi(1397) = 13 * 97 - 1
phi(15663) = 15 * 663 - 1
phi(19835) = 19 * 835 - 1
phi(37037) = 3703 * 7 - 1
phi(238903) = 23 * 8903 - 1
phi(719719) = 719 * 719 - 1
phi(1983035) = 19 * 83035 - 1
phi(4337785) = 4337 * 785 - 1
phi(5946445) = 5 * 946445 - 1
phi(8099989) = 809 * 9989 - 1
phi(15276063) = 1527 * 6063 - 1
phi(64438507) = 6443 * 8507 - 1
phi(97919791) = 9791 * 9791 - 1
phi(238639687) = 23 * 8639687 - 1
phi(325776657) = 32577 * 6657 - 1

Cf. A000010, A039649, A336191.

seqQ[n_] := Module[{d = IntegerDigits[n]}, MemberQ[Times @@@ Table[FromDigits /@ {Take[d, k], Take[d, -Length[d] + k]}, {k, 1, Length[d] - 1}], EulerPhi[n] + 1]]; Select[Range[10, 10^5], seqQ] (* Amiram Eldar, Jul 11 2020 *)

isok(m) = {my(tm=eulerphi(m)+1, d=digits(m)); for (i=1, #d-1, if (fromdigits(vector(i, k, d[k]))*fromdigits(vector(#d-i, k, d[i+k])) == tm, return(1)););} \\ Michel Marcus, Jul 11 2020

Missing terms 1983035 & 5946445 from Amiram Eldar, Jul 11 2020

More terms from Giovanni Resta, Jul 13 2020

A062789 a(n) = gcd(n, phi(n) * (phi(n) + 1)).

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 4, 3, 10, 11, 4, 13, 14, 3, 8, 17, 6, 19, 4, 3, 22, 23, 24, 5, 26, 9, 4, 29, 6, 31, 16, 3, 34, 5, 12, 37, 38, 3, 8, 41, 6, 43, 4, 15, 46, 47, 16, 7, 10, 3, 4, 53, 18, 5, 8, 3, 58, 59, 4, 61, 62, 9, 32, 1, 6, 67, 4, 3, 10, 71, 24, 73, 74, 5, 4, 1, 6, 79, 16, 27, 82, 83, 12

Offset: 1

Reinhard Zumkeller, Jul 19 2001

Harry J. Smith, Table of n, a(n) for n = 1..2000

Cf. A009195, A000010, A039649.

a062789 n = gcd n (phi * (phi + 1)) where phi = a000010 n
-- Reinhard Zumkeller, Oct 07 2015

Table[GCD[n, EulerPhi[n]*(EulerPhi[n] + 1)], {n, 75}] (* Stefan Steinerberger, Apr 03 2006 *)
ep[n_]:=Module[{epn=EulerPhi[n]},GCD[n,epn(epn+1)]]; Array[ep,90] (* Harvey P. Dale, Jun 23 2011 *)

for (n=1, 2000, p=eulerphi(n); write("b062789.txt", n, " ", gcd(n, p*(p + 1)))) \\ Harry J. Smith, Aug 11 2009

More terms from Stefan Steinerberger, Apr 03 2006

More terms from Harry J. Smith, Aug 11 2009

A212958 Array with a variable number of columns, where terms in the n-th row are the differences (computed in decimal base and divided by 9) between equal ratio permutations, found in the base n>=2, and the first (in ascending order of digits) minimal value permutation of {0,1,...,n}.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 12, 22, 0, 21, 22, 123, 131, 343, 0, 342, 343, 1234, 2531, 4664, 0, 1421, 3242, 4663, 12345, 58985, 0, 58984, 58985, 23456, 497531, 713306, 0, 137421, 276842, 436463, 575884, 713305, 713306, 1234567, 1810675, 2907844, 4002993, 6197531, 8367727

Offset: 1

Alexander R. Povolotsky, Feb 14 2013

It is conjectured (conjecture #1 - based on the observation of the programmatical exhaustive computation results) that among all possible distinct {0,1,...,n} permutations for each base n>=2, there are at least two pairs of such permutations that yield the same ratio, equal to A221740(p)/A221741(p) = (p^2*(p+1)^p -(p+1)^p+1)/(-p^2+p*(p+1)^p+(p+1)^p-p-1) where p=n-1. Additionally it is conjectured (conjecture #2) that the number of such pairs for each n is equal to A039649(p).
Construction of the terms for the n-th row of this sequence (where n is the base, n>=2) requires pre-calculation of the (mentioned in the above conjecture #1) permutation pairs (see the StackExchange link) that have the same ratio. Given the known (as defined above) ratio, actual values of the permutation pairs for each n (there are A039649(p) such pairs for each n>=2 - as conjectured above) are findable either by trial and error, or via executing an exhaustive computer program (see the link, featuring PARI/GP program - detection script for finding the equal-ratio permutation pairs for each base in the range from 2 to 10).
The terms of this sequence should be viewed in the table layout. The above mentioned table is an array with a variable number of columns. The number of terms (columns)in n-th row is, as conjectured above, equal to 2*A039649(p). Each n-th row (rows are counted from n=2) starts with a zero term because (as conjectured - conjecture #3) the 1st in ascending order minimal value permutation of {0,1,...,n} is always present among elements of the pairs, and, as it appears, always ends with the a(n=sum(i=2...n,2*A039649(i))) term, which has a nonzero value, and always seems to be equal to A215940((n)!) - conjecture #4.
Denoting the decimal value of the number made by the concatenation of the digits of the first (identical) base-n permutation of {0,1,...,n} as Pn, and considering terms of this sequence a(n) as members of a two-dimensional array b(n,k), where n is the row number and k is the position number in the n-th row (n>1, 1 <= k <=2*A039649(p)), then (per conjecture #1) it is true that in each array row "n" there are A039649(p) pairs of decimal integer values j and l such that (b(n,j)+ Pn)/(b(n,l)+ Pn) = A221740(p)/A221741(p) for n>=2, 1 <= j <= A039649(p), 1 <= l <= j.
It also appears that in the "next" (n+1)-th row there is always present a term that is less by one in decimal value as compared with the ending term in the "previous" n-th row (conjecture #5).

			For the fourth (n=4) row, which relates to base-4 four-digit {0, 1, 2, 3} distinct permutations, there are A039649(p) pairs where p = n-1 and thus for n=4, p=3, A039649(3)=3 - so there are three pairs in the fourth row.
Those pairs are supposed to have the same ratio, which can be calculated using the expression A221740(p)/A221741(p) = (p^2*(p+1)^p - (p+1)^p+1)/(-p^2 + p*(p+1)^p + (p+1)^p - p - 1), which for n=4 (p = n-1 = 3) yields 19/9 = 2.111.
The exhaustive computer program featured in the link finds that in decimal notation those three pairs with the ratio 19/9 = 2.111... are:
  (1) {114,54}; (2) {57,27}; (3) {228,108}
In base-4 notation, those 3 pairs of distinct permutations are:
  (1) {1302, 0312}; (2) {0321,0123}; (3) {3210,1230};
Now we calculate the fourth row terms per the sequence's definition:
  (1302-0123)/9 = 131; (0312-0123)/9 = 21; (0321-0123)/9 = 22; (0123-0123)/9 = 0; (3210-0123)/9 = 343; (1230-0123)/9 = 123;
Thus, for the fourth row (n=4), which corresponds to base 4 (note that rows in the table are counted starting with n=2, which corresponds to base 2) we get the following 6 (three pairs) sequence terms, presented as sorted in ascending order: 0, 21, 22, 123, 131, 343, ...

R. J. Cano, PARI/GP program - detection script for finding the equal ratio permutation pairs for each base in the range from 2 to 10
A. Povolotsky, Number of specific permutations pairs relates to Euler Phi totient function ?, NMBRTHRYY 27 Jan 2013
StackExchange, Permutations (with no duplicates) of decimal base digits 1,2,...,8,9,0.

Cf. A215940, A221740, A221741.

cp's OEIS Frontend

A336192 Numbers of the form ab such that phi(ab) = a*b - 1 where ab is the concatenation of a and b.

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A062789 a(n) = gcd(n, phi(n) * (phi(n) + 1)).

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Haskell

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Extensions

A212958 Array with a variable number of columns, where terms in the n-th row are the differences (computed in decimal base and divided by 9) between equal ratio permutations, found in the base n>=2, and the first (in ascending order of digits) minimal value permutation of {0,1,...,n}.

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Comments

Examples

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Crossrefs