A110084 -id:A110084 - cp's OEIS frontend

A112009 Numbers n with even length such that phi(n)=d_1^d_2d_3^d_4...* d_(k-1)^d_k where d_1 d_2 ... d_k is the decimal expansion of n.

113724, 116680, 126620, 176453, 236520, 12146841, 12514635, 13334445, 13469331, 13813728, 16473510, 18259344, 20116537, 20119347, 21324832, 23336066, 27923616, 30352728, 34425425, 35424571, 36311184, 37837170, 39171345, 43362816, 45429360, 45449216, 45916416, 46544032, 50713684, 50816880, 61642672, 62193744, 62226711, 62263890, 62288272, 64245272, 64808352, 64832560, 66707233, 66807126, 66827180, 81913446, 84943040

Offset: 1

Farideh Firoozbakht, Aug 26 2005

			27923616 is in the sequence because phi(27923616)=2^7*9^2*3^6*1^6.
11600069 and 23635500 are not members, since 0^0 is undefined.

Cf. A110084, A112010, A112011.

Do[h = IntegerDigits[n]; k = Length[h]; If[EvenQ[k] && Select[ Range[k/2], h[[2#-1]] == 0 &] == {} && EulerPhi[n]==Product[ h[[2j-1]]^h[[2j]], {j, k/2}], Print[n]], {n, 30000000}]

Edited by N. J. A. Sloane, Apr 02 2009

More terms from Max Alekseyev, Oct 16 2012

A112010 Numbers m with even length such that phi(m)=phi(d_1^d_2d_3^d_4...* d_(k-1)^d_k) where d_1 d_2 ... d_k is the decimal expansion of m.

Original entry on oeis.org

24, 1064, 2592, 6520, 106434, 145166, 237165, 262535, 372780, 491520, 531765, 546410, 566250, 636352, 12716544, 12806910, 13666320, 15116832, 15408692, 17473715, 21645616, 23473515, 23726640, 23728264, 26722436, 26757024, 27933192, 30537364, 30869280, 32118177, 33452293, 34114338, 39602752, 42262365, 44373490

Offset: 1

Farideh Firoozbakht, Aug 26 2005

			33452293 is in the sequence because phi(33452293)=phi(3^3*4^5*2^2*9^3).

Cf. A110084, A112009, A112011.

Do[h = IntegerDigits[n]; k = Length[h]; If[EvenQ[k] && Select[ Range[k/2], h[[2#-1]] == 0 &] == {} && EulerPhi[n]==EulerPhi [Product[h[[2j-1]]^h[[2j]], {j, k/2}]], Print[n]], {n, 31000000}]
epQ[n_]:=Module[{idn=IntegerDigits[n]},EvenQ[Length[idn]]&& FreeQ[ Take[ idn, {1,-1,2}],0] && EulerPhi[n] == EulerPhi[Times@@(#[[1]]^#[[2]]&/@ Partition[ idn,2])]]; Join[Select[Range[10,99],epQ],Select[Range[ 1000,9999], epQ], Select[Range[100000,999999],epQ], Select[Range[ 10000000, 44999999], epQ]] (* Harvey P. Dale, Feb 24 2016 *)

More terms from Max Alekseyev, Oct 16 2012

A112011 Numbers n with even length such that phi(n)=phi(d_1^d_2)phi(d_3^d_4) ...*phi(d_(k-1)^d_k) where d_1 d_2 ... d_k is the decimal expansion of n.

Original entry on oeis.org

24, 1064, 2592, 6520, 9234, 145166, 245344, 296480, 372780, 491520, 531765, 546410, 566250, 664062, 12806910, 12826710, 14466530, 15408692, 15621268, 17473715, 19946352, 22297520, 23256720, 30537364, 30869280, 32118177

Offset: 1

Farideh Firoozbakht, Aug 26 2005

For the third term we have the relation 2592=2^5*9^2. So phi(2592)=phi(2^5*9^2)=phi(2^5)*phi(9^2).

			39602752 is in the sequence because phi(39602752)=
phi(3^9)*phi(6^0)*phi(2^7)*phi(5^2).

Cf. A110084, A112009, A112010.

Do[h = IntegerDigits[n]; k = Length[h]; If[EvenQ[k] && Select[ Range[k/2], h[[2#-1]] == 0 &] == {} && EulerPhi[n]== Product[EulerPhi[h[[2j-1]]^h[[2j]]], {j, k/2}], Print[n]], {n, 35000000}]

A099650 Solutions to x+phi(x) = sigma(x)/2.

Original entry on oeis.org

456, 828, 7584, 33462, 1596048, 1964544, 19800384, 26211264, 31451136, 106805184, 156868224, 316113024, 502349274, 503291904, 1557940992, 8024671392, 8052965376, 11697091872, 22149447168, 87877745664, 443520605184, 626058783744

Offset: 1

Labos Elemer, Nov 05 2004

nonn

If 5*2^n-1 is prime then m=3*2^(n+1)*(5*2^n-1) is in the sequence because m+phi(m)=2^(n+1)*3*(5*2^n-1)+2^(n+1)*(5*2^n-2)=2^(n+1) *(20*2^n-5)=2^(n+1)*5*(2^(n+2)-1)=1/2*4*(2^(n+2)-1)*(5*2^n)= 1/2*sigma(3)*sigma(2^(n+1))*sigma(5*2^n-1)=1/2*sigma(3*2^(n+1) *(5*2^n-1))=1/2*sigma(m). So 3*2^(A001770+1)*(5*2^A001770-1) is a subsequence of this sequence. A110084 is this subsequence. Next term is greater than 10^8. - Farideh Firoozbakht, Aug 04 2005

a(23) > 10^12. - Donovan Johnson, Feb 29 2012

			n=456: phi(456) = 144, sigma(456) = 1200.

Cf. A000010, A000203, A001770, A110084.

Do[If[DivisorSigma[1, m] == 2m + 2 EulerPhi[m], Print[m]], {m, 100000000}] (Firoozbakht)

Two more terms from Farideh Firoozbakht, Aug 04 2005

a(10)-a(22) from Donovan Johnson, Feb 29 2012

cp's OEIS Frontend

A112009 Numbers n with even length such that phi(n)=d_1^d_2d_3^d_4...* d_(k-1)^d_k where d_1 d_2 ... d_k is the decimal expansion of n.

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A112010 Numbers m with even length such that phi(m)=phi(d_1^d_2d_3^d_4...* d_(k-1)^d_k) where d_1 d_2 ... d_k is the decimal expansion of m.

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A112011 Numbers n with even length such that phi(n)=phi(d_1^d_2)phi(d_3^d_4) ...*phi(d_(k-1)^d_k) where d_1 d_2 ... d_k is the decimal expansion of n.

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A099650 Solutions to x+phi(x) = sigma(x)/2.

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cp's OEIS Frontend

A112009 Numbers n with even length such that phi(n)=d_1^d_2*d_3^d_4*...* d_(k-1)^d_k where d_1 d_2 ... d_k is the decimal expansion of n.

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Extensions

A112010 Numbers m with even length such that phi(m)=phi(d_1^d_2*d_3^d_4*...* d_(k-1)^d_k) where d_1 d_2 ... d_k is the decimal expansion of m.

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A112011 Numbers n with even length such that phi(n)=phi(d_1^d_2)*phi(d_3^d_4) *...*phi(d_(k-1)^d_k) where d_1 d_2 ... d_k is the decimal expansion of n.

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A099650 Solutions to x+phi(x) = sigma(x)/2.

Views

Author

Keywords

Comments

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Programs

Mathematica

Extensions

A112009 Numbers n with even length such that phi(n)=d_1^d_2d_3^d_4...* d_(k-1)^d_k where d_1 d_2 ... d_k is the decimal expansion of n.

A112010 Numbers m with even length such that phi(m)=phi(d_1^d_2d_3^d_4...* d_(k-1)^d_k) where d_1 d_2 ... d_k is the decimal expansion of m.

A112011 Numbers n with even length such that phi(n)=phi(d_1^d_2)phi(d_3^d_4) ...*phi(d_(k-1)^d_k) where d_1 d_2 ... d_k is the decimal expansion of n.