A231964 -id:A231964 - cp's OEIS frontend

A005114 Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function (A001065).

2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, 516, 518, 520, 530, 540, 552, 556, 562, 576, 584, 612, 624, 626, 628, 658

Offset: 1

N. J. A. Sloane

Complement of A078923. - Lekraj Beedassy, Jul 19 2005
Chen & Zhao show that the lower density of this sequence is at least 0.06, improving on te Riele. - Charles R Greathouse IV, Dec 28 2013
Numbers k such that A048138(k) = 0. A048138(k) measures how "touchable" k is. - Jeppe Stig Nielsen, Jan 12 2020
From Amiram Eldar, Feb 13 2021: (Start)
The term "untouchable number" was coined by Alanen (1972). He found the 570 terms below 5000.
Erdős (1973) proved that the lower asymptotic density of untouchable numbers is positive, te Riele (1976) proved that it is > 0.0324, and Banks and Luca (2004, 2005) proved that it is > 1/48.
Pollack and Pomerance (2016) conjectured that the asymptotic density is ~ 0.17. (End)
The upper asymptotic density is less than 1/2 by the 'almost all' binary Goldbach conjecture, independently proved by Nikolai Chudakov, Johannes van der Corput, and Theodor Estermann. (In this context, this shows that the density of the odd numbers of this form is 0 (consider A001065(p*q) for prime p, q); full Goldbach would prove that 5 is the only odd number in this sequence.) - Charles R Greathouse IV, Dec 05 2022

Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, section B10, pp. 100-101.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 65.
József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 93.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 125.

Giovanni Resta, Table of n, a(n) for n = 1..13863 (terms < 10^5; first 8153 terms from Klaus Brockhaus)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], p. 840.
Jack David Alanen, Empirical study of aliquot series, Ph.D Thesis, Yale University, 1972.
William D. Banks and Florian Luca, Noncototients and Nonaliquots, arXiv:math/0409231 [math.NT], 2004.
William D. Banks and Florian Luca, Nonaliquots and Robbins numbers, Colloq. Math., Vol. 103, No. 1 (2005), pp. 27-32.
Yong-Gao Chen and Qing-Qing Zhao, Nonaliquot numbers, Publ. Math. Debrecen, Vol. 78, No. 2 (2011), pp. 439-442.
K. Chum, R. K. Guy, M. J. Jacobson, Jr., and A. S. Mosunov, Numerical and statistical analysis of aliquot sequences, Experimental Mathematics (2018), pp. 1-12.
Paul Erdős, Über die Zahlen der Form sigma(n)-n und n-phi(n), Elemente der Math., Vol. 28 (1973), pp. 83-86; alternative link (in German).
Shyam Sunder Gupta, Beauty of Number 153, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 15, 399-410.
Victor Meally, Letter to N. J. A. Sloane, no date.
Paul Pollack, Not Always Buried Deep: A Second Course in Elementary Number Theory, AMS, 2009, p. 272.
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B, Vol. 3 (2016), pp. 1-26; Errata.
Carl Pomerance and Hee-Sung Yang, On untouchable numbers and related problems, 2012.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp., Vol. 83, No. 288 (2014), pp. 1903-1913; alternative link.
Giovanni Resta, Untouchable numbers (the 150232 terms up to 10^6).
H. J. J. te Riele, A theoretical and computational study of generalized aliquot sequences, Mathematisch Centrum, Amsterdam, 1976. See chapter 9.
Eric Weisstein's World of Mathematics, Untouchable Number.
Wikipedia, Untouchable number.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jul. 1992.

Cf. A001065, A048138, A057709, A064000, A078923, A152454, A231964, A283152, A284147.

untouchableQ[n_] := Catch[ Do[ If[n == DivisorSigma[1, k]-k, Throw[True]], {k, 0, (n-1)^2}]] === Null; Reap[ Table[ If[ untouchableQ[n], Print[n]; Sow[n]], {n, 2, 700}]][[2, 1]] (* Jean-François Alcover, Jun 29 2012, after Benoit Cloitre *)

isA078923(n)=if(n==0 || n==1, return(1)); for(m=1,(n-1)^2, if( sigma(m)-m == n, return(1))); 0
isA005114(n)=!isA078923(n)
for(n=1,700, if (isA005114(n), print(n))) \\ R. J. Mathar, Aug 10 2006

is(n)=if(n%2 && n<4e18, return(n==5)); forfactored(m=1,(n-1)^2, if(sigma(m)-m[1]==n, return(0))); 1 \\ Charles R Greathouse IV, Dec 05 2022

from sympy import divisor_sigma as sigma
from functools import cache
@cache
def f(m): return sigma(m)-m
def okA005114(n):
    if n < 2: return 0
    return not any(f(m) == n for m in range(1, (n-1)**2+1))
print([k for k in range(289) if okA005114(k)]) # Michael S. Branicky, Nov 16 2024

# faster for intial segment of sequence
from itertools import count, islice
from sympy import divisor_sigma as sigma
def agen(): # generator of terms
    n, touchable, t = 2, {0, 1}, 1
    for m in count(2):
        touchable.add(sigma(m)-m)
        while m > t:
            if n not in touchable:
                yield n
            else:
                touchable.discard(n)
            n += 1
            t = (n-1)**2
print(list(islice(agen(), 20))) # Michael S. Branicky, Nov 16 2024

More terms from David W. Wilson

A306952 Lesser member of twin weird numbers: weird numbers n (A006037) such that n+2 is also weird.

Original entry on oeis.org

512468, 540890, 688028, 1390268, 1565828, 1741388, 2268068, 3525410, 3848108, 4374788, 6481508, 6657068, 7534868, 7885988, 7914410, 8089970, 8838968, 9143330, 9290468, 10021130, 10343828, 10898930, 12654530, 12801668, 12872510, 13152788, 13181210, 14234570

Offset: 1

Amiram Eldar, Mar 17 2019

nonn

Number of terms below 10^k for k = 6, 7, ... 10: 19, 231, 2111, 22426.

The first occurrences of 2 consecutive pairs of twin weirds are (21607670, 21607672, 21608090, 21608092), (873951608, 873951610, 873951890, 873951892), ...

			512468 is in the sequence since both 512468 and 512470 are weird numbers.

Amiram Eldar, Table of n, a(n) for n = 1..22426 (terms below 10^10).

Cf. A006037, A125109, A231086 (supersequence), A231964.

A231965 Smallest integer starting a group of exactly n consecutive untouchable numbers (A005114) with term differences of 2.

Original entry on oeis.org

246, 288, 892, 9020, 11456, 23480, 52274, 33686, 190070, 1741856, 1668564, 7806762

Offset: 2

Michel Marcus, Nov 16 2013

Such n-tuplets from A005114 correspond to n+1 positive numbers interspersed with n zeros in A070015, and starting at a(n) - 1. For instance, a(4) = 892 is related to A070015(891) and consecutive values: 2661, 0, 4147, 0, 2945, 0, 1287, 0, 9757.

			a(5) = 9020, because 9020, 9022, 9024, 9026, 9028 are untouchable, while 9018 and 9030 are not so (A001065). For examples with smaller n, see A231964 comments.

Eric Weisstein's World of Mathematics, Untouchable Number.
Wikipedia, Untouchable number

Cf. A001065, A005114, A231964.

Cf. A110875 (analog for sigma(n)).

oksucc(v, vi, n) = {for (i = 1, n-1, if (! vecsearch(v, vi+2*i, ) , return (0));); return(! vecsearch(v, vi-2) && !vecsearch(v, vi+2*n));}
a(n) = {v = readvec("untouchable.log"); for (i=1, #v, vi = v[i]; if (oksucc(v, vi, n), return(vi)););} \\ readvec reads the file obtained by keeping the second column of b005114.txt seen as a csv file.

a(10)-a(13) from Donovan Johnson, Nov 16 2013

Definition, data, and Pari script corrected by Michel Marcus with Donovan Johnson, Nov 18 2013

A333100 Even numbers k such that both k and k + 2 are nontotients (A005277).

Original entry on oeis.org

74, 122, 152, 186, 234, 242, 244, 246, 284, 302, 338, 362, 374, 402, 404, 410, 412, 426, 434, 470, 472, 482, 494, 514, 516, 530, 532, 548, 572, 594, 602, 608, 626, 666, 668, 678, 722, 728, 746, 752, 788, 802, 804, 842, 844, 866, 868, 870, 872, 890, 892, 914, 942

Offset: 1

Amiram Eldar, Mar 07 2020

nonn

			74 is a term since both 74 and 76 are nontotients.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Nontotient.
Wikipedia, Nontotient.

Cf. A005277, A063512, A231964, A306952, A333101.

forstep(k=2, 100, 2, if(!istotient(k) && !istotient(k+2), print1(k,", ")))

A333101 Numbers k such that both k and k + 2 are noncototients (A005278).

Original entry on oeis.org

50, 170, 266, 290, 344, 518, 532, 534, 650, 686, 722, 730, 872, 962, 1036, 1158, 1166, 1332, 1394, 1462, 1464, 1586, 1634, 1682, 1804, 1864, 1922, 1946, 1970, 2034, 2072, 2074, 2116, 2134, 2262, 2314, 2316, 2318, 2330, 2420, 2534, 2598, 2666, 2668, 2772, 2822

Offset: 1

Amiram Eldar, Mar 07 2020

nonn

			50 is a term since both 50 and 52 are noncototients.

Eric Weisstein's World of Mathematics, Noncototient.
Wikipedia, Noncototient.

Cf. A005278, A072296, A231964, A306952, A333100.

nmax = 3000; cototientQ[n_?EvenQ] := (x = n; While[test = x - EulerPhi[x] == n ; Not[test || x > 2*nmax], x++]; test); cototientQ[n_?OddQ] = True; nonc = Select[Range[nmax], !cototientQ[#]&]; nonc[[Flatten[Position[Differences[nonc], 2]]]] (* after Jean-François Alcover at A005278 *)

A357323 Numbers k such that k and k+2 are both unitary untouchable numbers (A063948).

Original entry on oeis.org

2, 3, 5, 30756, 34182, 46128, 51816, 56352, 72522, 86640, 88896, 119796, 133062, 133618, 149682, 164290, 207282, 207642, 213636, 245708, 257820, 261156, 279730, 283050, 286356, 286858, 310842, 318060, 327300, 339402, 339612, 349030, 360390, 371820, 377940, 384576, 396090

Offset: 1

Amiram Eldar, Sep 24 2022

nonn

Except for k=3, are there any other numbers k such that k, k+2 and k+4 are all unitary untouchable numbers? There are no such numbers below 10^6.

Amiram Eldar, Table of n, a(n) for n = 1..101

The unitary version of A231964.

Cf. A063948.

u = Cases[Import["https://oeis.org/A063948/b063948.txt", "Table"], {, }][[;; , 2]]; Select[u, MemberQ[u, # + 2] &]

A370355 Highly touchable numbers sandwiched between untouchable twin pairs.

Original entry on oeis.org

1681, 5251, 7771, 36961, 39271, 170941, 196351, 360361, 510511, 1009471, 9699691

Offset: 1

Amiram Eldar, Feb 16 2024

Highly touchable numbers k have a record number of solutions x to A001065(x) = k, while untouchable numbers k have no solution to this equation.

Intersection of A238895 and {A231964(n) + 1};
Cf. A001065, A005114.
Similar sequences: A068507, A113839.

seq[nmax_] := Module[{v = Table[0, {nmax}], i, s = {}, vmax = -1}, Do[i = DivisorSigma[1, n] - n; If[0 < i <= nmax, v[[i]]++], {n, 1, nmax^2}]; Do[If[v[[n]] > vmax, vmax = v[[n]]; If[v[[n - 1]] == 0 && v[[n + 1]] == 0, AppendTo[s, n]]], {n, 2, nmax - 1}]; s]; seq[8000]

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A005114 Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function (A001065).

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A306952 Lesser member of twin weird numbers: weird numbers n (A006037) such that n+2 is also weird.

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A231965 Smallest integer starting a group of exactly n consecutive untouchable numbers (A005114) with term differences of 2.

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A333100 Even numbers k such that both k and k + 2 are nontotients (A005277).

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A333101 Numbers k such that both k and k + 2 are noncototients (A005278).

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A357323 Numbers k such that k and k+2 are both unitary untouchable numbers (A063948).

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Mathematica

A370355 Highly touchable numbers sandwiched between untouchable twin pairs.

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