A005114 -id:A005114 - cp's OEIS frontend

A048191 Let (u1,u2) be successive untouchable numbers such that phi(u1) = phi(u2) = k; sequence gives values of k.

32, 144, 192, 312, 240, 592, 672, 760, 648, 936, 1600, 864, 1560, 1224, 1872, 2160, 2688, 1920, 2088, 1920, 3312, 3732, 3760, 2640, 3984, 4240, 4272, 2880, 4248, 6432, 5976, 5112, 7760, 5280, 7932, 4320, 8968, 9000, 6176, 7680, 10224, 7560, 12072, 12432, 12552

Offset: 1

Naohiro Nomoto

nonn

			32 is a term since 96 and 120 are successive untouchable numbers and phi(96) = phi(120) = 32.
144 is a term since 292 and 304 are successive untouchable numbers and phi(292) = phi(304) = 144.

Amiram Eldar, Table of n, a(n) for n = 1..4309 (terms 1..81 from Michel Marcus)

Cf. A000010, A005114, A048188, A048189, A048190.

a(n) = A000010(A048189(n)) = A000010(A048190(n)). - Amiram Eldar, Sep 23 2022

Offset changed to 1 and more terms from Michel Marcus, Jun 08 2015

A135244 Largest m such that the sum of the aliquot parts of m (A001065) equals n, or 0 if no such number exists.

Original entry on oeis.org

0, 4, 9, 0, 25, 8, 49, 15, 14, 21, 121, 35, 169, 33, 26, 55, 289, 77, 361, 91, 38, 85, 529, 143, 46, 133, 28, 187, 841, 221, 961, 247, 62, 253, 24, 323, 1369, 217, 81, 391, 1681, 437, 1849, 403, 86, 493, 2209, 551, 94, 589, 0, 667, 2809, 713, 106, 703, 68, 697, 3481

Offset: 2

Ophir Spector (ospectoro(AT)yahoo.com), Nov 25 2007

nonn

Previous name: Aliquot predecessors with the largest values.
Find each node's predecessors in aliquot sequences and choose the largest predecessor.
Climb the aliquot trees on shortest paths (see A135245 = Climb the aliquot trees on thickest branches).
The sequence starts at offset 2, since all primes satisfy sigma(n)-n = 1. - Michel Marcus, Nov 11 2014

			a(25) = 143 since 25 has 3 predecessors (95,119,143), 143 being the largest.
a(5) = 0 since it has no predecessors (see Untouchables - A005114).

Amiram Eldar, Table of n, a(n) for n = 2..10000 (terms 2..150 from Ophir Spector)
Wolfgang Creyaufmueller, Aliquot sequences.
J. O. M. Pedersen, Tables of Aliquot Cycles. [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]
Eric Weisstein's World of Mathematics, Aliquot sequence.

Cf. A001065, A005114, A125601, A135245, A057709, A057710, A063769, A080907, A121507, A037020, A126016.

seq[max_] := Module[{s = Table[0, {n, 1, max}], i}, Do[If[(i = DivisorSigma[1, n] - n) <= max, s[[i]] = Max[s[[i]], n]], {n, 2, (max - 1)^2}]; Rest @ s]; seq[50]

lista(nn) = {for (n=2, nn, k = (n-1)^2; while(k && (sigma(k)-k != n), k--); print1(k, ", "););} \\ Michel Marcus, Nov 11 2014

a(1)=0 removed and offset set to 2 by Michel Marcus, Nov 11 2014

New name from Michel Marcus, Oct 31 2023

A135245 Aliquot predecessors with the largest degrees.

Original entry on oeis.org

0, 0, 4, 9, 0, 25, 8, 49, 15, 14, 21, 121, 35, 169, 33, 12, 55, 289, 65, 361, 91, 20, 85, 529, 143, 46, 133, 28, 187, 841, 161, 961, 247, 62, 253, 24, 323, 1369, 217, 81, 391, 1681, 341, 1849, 403, 86, 493, 2209, 551, 40, 481, 0, 667, 2809, 533, 106, 703, 68, 697, 3481

Offset: 1

Ophir Spector, ospectoro (AT) yahoo.com, Nov 25 2007

nonn

Find each node's predecessors in aliquot sequences and choose the node with largest number of predecessors.

Climb the aliquot trees on thickest branches (see A135244 = Climb the aliquot trees on shortest paths).

			a(25) = 143 since 25 has 3 predecessors (95,119,143) with degrees (4,5,7), 143 having the largest degree. a(5) = 0 since it has no predecessors (see Untouchables - A005114).

Ophir Spector, Table of n, a(n) for n = 1..150
W. Creyaufmueller, Aliquot sequences
J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]
Eric Weisstein's World of Mathematics, Aliquot sequence

Cf. A001065 A005114 A125601 A135244 A057709 A057710 A063769 A080907 A121507 A037020 A126016.

A372740 Coreful untouchable numbers: numbers that are not the sum of aliquot coreful divisors (A336563) of any number.

Original entry on oeis.org

1, 4, 8, 9, 16, 20, 25, 27, 28, 32, 40, 44, 45, 48, 49, 50, 52, 54, 63, 64, 68, 72, 75, 76, 81, 88, 92, 99, 100, 104, 108, 116, 117, 121, 124, 125, 128, 136, 144, 147, 148, 152, 153, 160, 162, 164, 169, 171, 172, 175, 176, 184, 188, 189, 192, 196, 200, 207, 208

Offset: 1

Amiram Eldar, May 12 2024

nonn

A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958).
Numbers k such that A372739(k) = 0.
Numbers that are not in the range of A336563.
Except for 1, all the terms are not squarefree (A013929), because if k is squarefree (A005117), and there is a prime p such that p|k, then A336563(p*k) = k.
Includes all the squares of primes (A001248).
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are , 4, 29, 281, 2762, 27690, ... . Apparently, the asymptotic density of this sequence exists and equals 0.27... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A013929, A307958, A336563, A372739.
A001248 is a subsequence.
Similar sequences: A005114, A063948 (unitary), A324276 (bi-unitary), A324277 (infinitary).

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; seq[max_] := Module[{v = Table[0, {max}], i}, Do[i = s[k]; If[0 < i <= max, v[[i]]++], {k, 1, max^2}]; Position[v, _?(# == 0 &)] // Flatten]; seq[200]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2] + 1) - 1)/(f[i, 1] - 1) - 1) - n;}
lista(nmax) = {my(v = vector(nmax), i); for(k = 1, nmax^2, i = s(k); if(i > 0 && i <= nmax, v[i]++)); for(k = 1, nmax, if(v[k] == 0, print1(k, ", ")));}

A070016 Least m such that Chowla's function value of m [A048050(m)] equals n or 0 if no such number exists.

Original entry on oeis.org

0, 4, 9, 0, 6, 8, 10, 15, 14, 21, 121, 27, 22, 16, 12, 39, 289, 65, 34, 18, 20, 57, 529, 95, 46, 69, 28, 115, 841, 32, 58, 45, 62, 93, 24, 155, 1369, 217, 44, 63, 30, 50, 82, 123, 52, 129, 2209, 75, 40, 141, 0, 235, 42, 36, 106, 99, 68, 265, 3481, 371, 118, 64, 56, 117

Offset: 1

Labos Elemer, Apr 12 2002

nonn

Remark that A070016(n)=A070015(n+1) in accordance with A048995(k)+1=A005114(k).

			n=127: a(n)=16129, divisors={1,127,16129}, 127=sigma[n]-n-1=127 and 16129 is the smallest.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A001065, A048050, A051444, A007369, A070016, A005114, A048995.

f1[x_] := DivisorSigma[1, x]-x-1; t=Table[0, {128}]; Do[b=f1[n]; If[b<129&&t[[b]]==0, t[[b]]=n], {n, 1, 1000000}]; t

a(n)=Min{x; A048050(x)=n} or a(n)=0 if n is from A048995.

A074898 Impossible values for sum of anti-divisors of n.

Original entry on oeis.org

1, 6, 7, 9, 11, 15, 17, 20, 21, 25, 26, 27, 29, 31, 33, 35, 37, 38, 43, 44, 45, 47, 49, 51, 53, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 75, 77, 79, 81, 82, 83, 85, 87, 89, 91, 93, 95, 97, 99, 100, 103, 105, 109, 111, 113, 115, 117, 119, 120, 121, 123, 125, 127, 128, 129, 131, 133, 134, 135, 137, 139, 141, 143, 145, 146, 149, 151, 153, 155, 157, 158, 159, 161, 163, 165, 167, 168, 169, 170, 171

Offset: 1

Jason Earls, Sep 14 2002

nonn

See A066272 for definition of anti-divisor.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A005114, A066417.

More terms from Paolo P. Lava, Jul 06 2011

A111278 Untouchable squares.

Original entry on oeis.org

324, 576, 784, 1296, 2304, 2500, 2704, 3136, 3600, 4356, 5184, 6084, 7056, 8100, 9216, 11236, 11664, 14400, 14884, 15876, 17424, 24336, 24964, 25600, 27556, 28224, 30276, 32400, 33856, 34596, 36864, 38416, 39204, 41616, 49284, 51076, 51984

Offset: 1

Tanya Khovanova, Aug 05 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..498 (terms 1..200 from Donovan Johnson)

Intersection of A000290 (squares) and A005114 (untouchable numbers).

a(16) to a(33) from Klaus Brockhaus, Aug 10 2006

More terms obtained from b-file of A005114 by David Wasserman, Jan 07 2009

A119379 Untouchable semiprimes: semiprimes which are not the sum of the aliquot parts of any number.

Original entry on oeis.org

146, 206, 262, 326, 562, 626, 718, 766, 802, 818, 898, 926, 934, 982, 1046, 1186, 1318, 1346, 1418, 1438, 1522, 1538, 1642, 1718, 1766, 1774, 1822, 1838, 1894, 2062, 2078, 2098, 2174, 2218, 2258, 2302, 2306, 2446, 2498, 2518, 2602, 2606, 2614, 2642, 2762

Offset: 1

Tanya Khovanova, Jul 24 2006

nonn

From Amiram Eldar, Feb 13 2021: (Start)
Assuming that 5 is the only odd untouchable number, all the terms are of the form 2*p, where p is a prime.
Alanen (1972) calculated the first 70 terms of this sequence (terms below 5000). (End).

Amiram Eldar, Table of n, a(n) for n = 1..9035 (terms below 10^6, calculated using data from Giovanni Resta; terms 1..1000 from Donovan Johnson)
Jack David Alanen, Empirical study of aliquot series, Ph.D Thesis, Yale University, 1972.
Giovanni Resta, Untouchable numbers (includes a list of the 150232 untouchable numbers below 10^6).

Intersection of A001358 and A005114.

seq[max_] := Module[{v = Table[0, {max}]}, Do[k = DivisorSigma[1, n] - n; If[2 <= k <= max, v[[k]]++], {n, 1, max^2}]; Select[Rest[Position[v, ?(# == 0 &)] // Flatten], PrimeOmega[#] == 2 &]]; seq[1000] (* _Amiram Eldar, Feb 13 2021 *)

More terms from Franklin T. Adams-Watters and Don Reble, Jul 28 2006

A324278 Exponential untouchable numbers: numbers that are not the sum of aliquot exponential divisors of any number.

Original entry on oeis.org

1, 4, 8, 9, 16, 20, 25, 27, 28, 32, 40, 44, 45, 49, 52, 54, 63, 64, 68, 75, 76, 81, 88, 92, 96, 99, 104, 108, 116, 117, 121, 124, 125, 128, 136, 144, 147, 148, 152, 153, 160, 164, 169, 171, 172, 175, 176, 184, 188, 189, 192, 196, 200, 207, 208, 212, 216, 224

Offset: 1

Amiram Eldar, Feb 20 2019

nonn

The terms are conjectural and based on a search for solutions to esigma(x) - x = k for k in the range of the data section and x < 10^12 (esigma(x) - x = A051377(x) - x = A126164(x) is the sum of aliquot exponential divisors of x). - Amiram Eldar, Jan 22 2020

Cf. A051377, A126164, A005114, A063948 (unitary), A324276 (bi-unitary), A324277 (infinitary).

fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := esigma[n] = Times @@ fun @@@ FactorInteger[n]; untouchableQ[n_] := Catch[ Do[ If[n == esigma[k]-k, Throw[True]], {k, 0, (n+1)^2}]] === Null; Reap[ Table[ If[ untouchableQ[n], Sow[n]], {n, 1, 130}]][[2, 1]] (* after Jean-François Alcover at A005114 *)

Data corrected by Amiram Eldar, Jan 22 2020

A366110 a(n) is the difference between the maximum and minimum number whose proper divisors sum to n, or 0 if there is no such number.

Original entry on oeis.org

0, 0, 0, 0, 19, 0, 39, 0, 0, 0, 0, 8, 147, 17, 14, 16, 0, 12, 327, 73, 18, 28, 0, 48, 0, 64, 0, 72, 0, 189, 903, 202, 0, 160, 0, 168, 0, 0, 37, 328, 1651, 387, 1767, 280, 34, 364, 0, 476, 54, 448, 0, 432, 2767, 677, 0, 604, 0, 432, 0, 528, 3603, 753, 66, 826, 0, 768, 0, 720, 0

Offset: 2

Michel Marcus, Oct 28 2023

nonn

A152454 is the irregular triangle in which row n lists the numbers whose proper divisors sum to n.

			A152454 begins as []; [4]; [9]; []; [6, 25]; [8]; [10, 49]...
so sequence begins 0, 0, 0, 0, 19, 0, 39, ...

Michel Marcus, Table of n, a(n) for n = 2..10001

Cf. A001065, A152454.
Cf. A070015, A135244.
Cf. A005114, A057709.

lista(nn) = my(v = vector(nn, k, [])); forcomposite (i=1, nn^2, my(x=sigma(i)-i); if (x <=  nn, v[x] = concat(v[x], i));); vector(nn-1, k, k++; if (#v[k], vecmax(v[k]) - vecmin(v[k])));

a(n) = A135244(n) - A070015(n).

a(A005114(n)) = a(A057709(n)) = 0.

cp's OEIS Frontend

A048191 Let (u1,u2) be successive untouchable numbers such that phi(u1) = phi(u2) = k; sequence gives values of k.

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A135244 Largest m such that the sum of the aliquot parts of m (A001065) equals n, or 0 if no such number exists.

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A135245 Aliquot predecessors with the largest degrees.

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A372740 Coreful untouchable numbers: numbers that are not the sum of aliquot coreful divisors (A336563) of any number.

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A070016 Least m such that Chowla's function value of m [A048050(m)] equals n or 0 if no such number exists.

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A074898 Impossible values for sum of anti-divisors of n.

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A111278 Untouchable squares.

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A119379 Untouchable semiprimes: semiprimes which are not the sum of the aliquot parts of any number.

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Mathematica

Extensions

A324278 Exponential untouchable numbers: numbers that are not the sum of aliquot exponential divisors of any number.

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