A019279 -id:A019279 - cp's OEIS frontend

A019280 Let sigma_m(n) be result of applying the sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m(n) = k*n; sequence gives log_2 of the (2,2)-perfect numbers.

1, 2, 4, 6, 12, 16, 18, 30, 60

Offset: 1

Cohen and te Riele prove that any even (2,2)-perfect number (a "superperfect" number) must be of the form 2^(p-1) with 2^p-1 prime (Suryanarayana) and the converse also holds. Any odd superperfect number must be a perfect square (Kanold). Searches up to > 10^20 did not find any odd examples. - Ralf Stephan, Jan 16 2003

See also the Cohen-te Riele links under A019276.

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Cf. A019278, A019279.

Coincides with A000043(n) - 1 unless odd superperfect numbers exist.

a(8)-a(9) from Jud McCranie, Jun 01 2000

A067709 Numbers k such that phi(2sigma(k)) = 2sigma(phi(k)).

Original entry on oeis.org

2, 4, 16, 18, 64, 100, 450, 516, 1458, 4096, 4624, 13932, 14406, 20124, 21780, 28900, 29262, 29616, 36450, 45996, 62500, 65536, 92778, 95916, 106092, 106308, 114630, 114930, 121872, 125652, 130050, 140130, 145794, 149124, 160986, 179562, 185100

Offset: 1

Benoit Cloitre, Feb 05 2002

nonn

Every even superperfect number (A019279) is a term of the sequence. - Vladeta Jovovic, Feb 11 2002

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000010, A000203, A019279.

with(numtheory); A067709:=n->`if`( phi(2*sigma(n)) = 2*sigma(phi(n)), n, NULL); seq(A067709(n), n=1..200000); # Wesley Ivan Hurt, Apr 07 2014

Select[Range[200000], EulerPhi[2*DivisorSigma[1, #]] == 2*DivisorSigma[1, EulerPhi[#]] &] (* Amiram Eldar, May 13 2022 *)

isok(k) = eulerphi(2*sigma(k)) == 2*sigma(eulerphi(k)); \\ Michel Marcus, May 13 2022

More terms from Vladeta Jovovic, Feb 11 2002

A134709 Even superperfect numbers divided by 2, minus 1.

Original entry on oeis.org

0, 1, 7, 31, 2047, 32767, 131071, 536870911, 576460752303423487, 154742504910672534362390527, 40564819207303340847894502572031, 42535295865117307932921825928971026431

Offset: 1

Omar E. Pol, Nov 07 2007

nonn

			a(5)=2047 because the 5th even superperfect number is 4096 and (4096/2)-1=2047.

Cf. A019279, A061652, A133028.

a(n) = (A061652(n)/2) - 1.

More terms from Jinyuan Wang, Mar 14 2020

A134710 a(n) = n-th even superperfect number divided by 2^n.

Original entry on oeis.org

1, 1, 2, 4, 128, 1024, 2048, 4194304, 2251799813685248, 302231454903657293676544, 39614081257132168796771975168, 20769187434139310514121985316880384

Offset: 1

Omar E. Pol, Nov 07 2007

nonn

a(13) and a(14) have 153 and 179 digits respectively and are too large to include here. - R. J. Mathar, Jan 07 2008

			a(5) = 128 because the 5th even superperfect number is 4096 and 2^5 = 32 and 4096/32 = 128.

Amiram Eldar, Table of n, a(n) for n = 1..18
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000043, A000396, A000668, A019279, A061652 (even superperfect numbers), A133028.

A000043 := proc(n) op(n,[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213]) ; end: A061652 := proc(n) 2^(A000043(n)-1) ; end: A134710 := proc(n) A061652(n)/2^n ; end: seq(A134710(n),n=1..14) ; # R. J. Mathar, Jan 07 2008

With[{max = 12}, 2^(MersennePrimeExponent[Range[max]] - Range[max] - 1)] (* Amiram Eldar, Oct 21 2024 *)

a(n) = A061652(n)/(2^n).

a(n) = 2^(A000043(n)-n-1). - Amiram Eldar, Oct 21 2024

More terms from R. J. Mathar, Jan 07 2008

A134712 Base-2 logarithm of (n-th even superperfect number divided by 2^n).

Original entry on oeis.org

0, 0, 1, 2, 7, 10, 11, 22, 51, 78, 95, 114, 507, 592, 1263, 2186, 2263, 3198, 4233, 4402, 9667, 9918, 11189, 19912, 21675, 23182, 44469, 86214, 110473, 132018, 216059, 756806, 859399, 1257752, 1398233, 2976184, 3021339, 6972554, 13466877

Offset: 1

Omar E. Pol, Nov 07 2007

nonn

			a(5) = 7 because the 5th even superperfect number is 4096, 2^5 = 32, 4096/32 = 128 and log_2(128) = 7 (because 2^7 = 128).

Amiram Eldar, Table of n, a(n) for n = 1..48
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000043, A000396, A000668, A019279, A061652, A090748, A133028.

With[{max = 48}, MersennePrimeExponent[Range[max]] - Range[max] - 1] (* Amiram Eldar, Oct 21 2024 *)

a(n) = log_2(A061652(n)/(2^n)) = A000043(n) - n - 1 = A090748(n) - n.

A134713 Base-2 logarithm of (n-th even superperfect number divided by 2^n), plus 1.

Original entry on oeis.org

1, 1, 2, 3, 8, 11, 12, 23, 52, 79, 96, 115, 508, 593, 1264, 2187, 2264, 3199, 4234, 4403, 9668, 9919, 11190, 19913, 21676, 23183, 44470, 86215, 110474, 132019, 216060, 756807, 859400, 1257753, 1398234, 2976185, 3021340, 6972555, 13466878

Offset: 1

Omar E. Pol, Nov 07 2007

nonn

			a(5) = 8 because the 5th even superperfect number is 4096, 2^5 = 32, 4096/32 = 128, log_2(128) = 7 (because 2^7 = 128) and 7+1 = 8.

Amiram Eldar, Table of n, a(n) for n = 1..48
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000043, A000396, A000668, A019279, A061652, A090748, A133028.

With[{max = 48}, MersennePrimeExponent[Range[max]] - Range[max]] (* Amiram Eldar, Oct 21 2024 *)

a(n) = 1 + log_2(A061652(n)/(2^n)) = A000043(n) - n = A090748(n) - n + 1.

A135651 Even superperfect numbers written in base 2.

Original entry on oeis.org

10, 100, 10000, 1000000, 1000000000000, 10000000000000000, 1000000000000000000, 1000000000000000000000000000000

Offset: 1

Omar E. Pol, Feb 23 2008

Also, superperfect numbers (A019279) written in base 2 (If there are no odd perfect numbers).
Also, concatenation of "1" and A090748(n) digits "0".
The number of digits of a(n) is equal to A000043(n) and also equal to the number of digits of n-th Mersenne prime written in base 2 (see A117293, A135650).

			a(3)=10000 because the 3rd even superperfect number A061652(3)=16 and 16 written in base 2 is equal to 10000.

Cf. A000043, A000668, A019279, A061652, A090748, A117293, A135650.

A135656 Perfect numbers divided by 2, written in base 2.

Original entry on oeis.org

11, 1110, 11111000, 111111100000, 111111111111100000000000, 11111111111111111000000000000000, 111111111111111111100000000000000000, 111111111111111111111111111111100000000000000000000000000000

Offset: 1

Omar E. Pol, Feb 28 2008

The number of divisors of a(n) is equal to the number of its digits. This number is equal to 2*A000043(n)-2. The number of divisors of a(n) that are powers of 2 is equal to the number of divisors that are multiples of n-th Mersenne prime A000668(n) and this number of divisors is equal to A090748(n). The first digits of a(n) are "1". For n>1 the last digits are "0". The number of digits "1" is equal to A000043(n). The number of digits "0" is equal to A000043(n)-2. The concatenation of digits "1" gives the n-th Mersenne prime written in binary (see A117293(n)). The structure of divisors of a(n) represent a triangle (see example).

			a(4)=111111100000 because the 4th. perfect number is 8128 and 8128/2=4064 and 4064 written in base 2 is 111111100000. Note that 1111111 is the 4th. Mersenne prime A000668(4)=127, written in base 2.
The structure of divisors of a(4)=111111100000

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Perfect numbers divided by 2: A133028. Cf. A000396, A000668, A019279, A090748, A117293, A135650.

a(n)=A133028(n) written in base 2.

A138838 Concatenation of initial and final digits of n-th even superperfect number A061652(n), divided by 2.

Original entry on oeis.org

11, 22, 8, 32, 23, 33, 12, 7, 8, 18, 42, 42, 18, 12, 27, 37, 13, 8, 48, 7, 13, 8, 8, 13, 13, 13, 23, 12, 12, 13, 17, 42, 33, 12, 23, 18, 33, 13, 23, 32, 7, 32, 8, 33, 7, 43, 8

Offset: 1

Omar E. Pol, Apr 02 2008

Also, concatenation of initial and final digits of n-th superperfect number A019279(n), divided by 2, if there are no odd superperfect numbers.

Also, concatenation of A138124(n) and A138125(n), divided by 2.

			a(5)=23 because the 5th even superperfect number A061652(5) is 4096 and the concatenation of initial and final digits of 4096 is 46 and 46/2 = 23.

Cf. A019279, A061652, A073729, A138124, A138125, A138842.

a(n) = A138842(n)/2. - Jinyuan Wang, Mar 14 2020

More terms from Jinyuan Wang, Mar 14 2020

A139248 Triangle read by rows: row n lists the proper divisors of n-th even superperfect number A061652(n).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 8, 1, 2, 4, 8, 16, 32, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 1, 2, 4, 8, 16

Offset: 1

Omar E. Pol, Apr 22 2008

Also, row n list the proper divisors of n-th superperfect number A019279(n), if there are no odd superperfect numbers.

Row n has A000043(n) - 1 = A090748(n) terms.

			Triangle begins:
  1
  1, 2
  1, 2, 4, 8
  1, 2, 4, 8, 16, 32
  1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048
  1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768
  ...

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000043, A000668, A018254, A018487, A019279, A061652, A090748, A133024, A133025, A133031, A133049, A135652, A135653, A135654, A135655, A139246.

cp's OEIS Frontend

A019280 Let sigma_m(n) be result of applying the sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m(n) = k*n; sequence gives log_2 of the (2,2)-perfect numbers.

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A067709 Numbers k such that phi(2*sigma(k)) = 2*sigma(phi(k)).

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A134709 Even superperfect numbers divided by 2, minus 1.

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A134710 a(n) = n-th even superperfect number divided by 2^n.

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A134712 Base-2 logarithm of (n-th even superperfect number divided by 2^n).

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A134713 Base-2 logarithm of (n-th even superperfect number divided by 2^n), plus 1.

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Mathematica

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A135651 Even superperfect numbers written in base 2.

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A135656 Perfect numbers divided by 2, written in base 2.

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A138838 Concatenation of initial and final digits of n-th even superperfect number A061652(n), divided by 2.

A067709 Numbers k such that phi(2sigma(k)) = 2sigma(phi(k)).