A005114 -id:A005114 - cp's OEIS frontend

A048187 Palindromic untouchable numbers.

2, 5, 88, 262, 292, 474, 626, 818, 848, 898, 2992, 4224, 4884, 6006, 6776, 6996, 8008, 8228, 8558, 8778, 8998, 20502, 20802, 21112, 21512, 21712, 21812, 22022, 22122, 22322, 22422, 22822, 23232, 23732, 24042, 24142, 24642, 24742, 24942, 25252, 25552, 25652

Offset: 1

Naohiro Nomoto

Amiram Eldar, Table of n, a(n) for n = 1..1554

Intersection of A002113 and A005114.

a(22)-a(42) from Donovan Johnson, Aug 12 2011

A057978 Number of untouchable numbers <= 10^n.

Original entry on oeis.org

2, 5, 89, 1212, 13863, 150232, 1574973, 16246940, 165826606, 1681871718, 16988116409, 171128671374

Offset: 1

Naohiro Nomoto, Nov 13 2000

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 Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, Transactions of the American Mathematical Society, Series B Vol. 3, No. 1 (2016), pp. 1-26.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Mathematics of Computation, Vol. 83, No. 288 (2014), pp. 1903-1913.
Index entries for sequences related to numbers of primes in various ranges

Cf. A005114.

a(5)-a(6) from Donovan Johnson, Jul 19 2008
a(7) from Donovan Johnson, Aug 13 2011
a(8)-a(10) from Amiram Eldar; a(8) was taken from the paper by Pomerance and Yang, a(9)-a(10) were taken from the paper by Pollack and Pomerance
a(11)-a(12) added by Amiram Eldar, Sep 29 2018 from the paper by Chum et al.

A120534 Untouchable 3-almost primes.

Original entry on oeis.org

52, 124, 188, 238, 246, 268, 290, 292, 322, 406, 426, 430, 474, 498, 518, 530, 556, 628, 658, 668, 670, 782, 892, 894, 902, 964, 1002, 1028, 1074, 1102, 1146, 1162, 1222, 1246, 1258, 1266, 1342, 1348, 1388, 1398, 1406, 1506, 1510, 1542, 1578, 1588, 1682, 1732, 1758, 1842, 1844, 1852

Offset: 1

Jonathan Vos Post, Aug 06 2006

nonn

Numbers having 3 prime factors (with multiplicity) and which are not the sum of the aliquot parts of any number.

2 and 5 are almost surely the only untouchable primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Giovanni Resta)

Intersection of A005114 and A014612.

Cf. A119379.

A121683 Untouchable cubes.

Original entry on oeis.org

216, 1728, 5832, 8000, 13824, 21952, 27000, 74088, 110592, 157464, 175616, 195112, 216000, 238328, 287496, 314432, 343000, 373248, 729000, 884736, 941192, 1061208, 1124864, 1259712, 1481544, 1728000, 2000376, 2097152, 2628072, 2744000, 2985984, 3241792, 3511808

Offset: 1

Klaus Brockhaus, Aug 15 2006

nonn

Untouchable numbers (cf. A005114) that are third powers (cf. A000578); a(1) to a(7) are the third powers of 6, 12, 18, 20, 24, 28, 30.

			216 = 6^3 = A000578(6) = A005114(13).

Amiram Eldar, Table of n, a(n) for n = 1..45

Intersection of A000578 and A005114.

Cf. A111278 (untouchable squares).

a(8)-a(30) from Donovan Johnson, Jul 19 2008

Terms a(26)-a(30) corrected and a(31)-a(33) added by Amiram Eldar, Sep 23 2022

A191749 Numbers not the sum of a smaller number and its prime factors (with multiplicity).

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 12, 13, 16, 18, 20, 21, 25, 27, 28, 30, 32, 37, 43, 44, 45, 48, 49, 50, 52, 57, 60, 61, 64, 66, 67, 68, 70, 73, 75, 77, 78, 80, 81, 85, 87, 90, 91, 92, 97, 100, 101, 102, 104, 108, 110, 112, 115, 117, 126, 129, 130, 132, 133, 135, 137, 139, 144, 145

Offset: 1

Alonso del Arte, Jul 13 2011

nonn

If a number is not squarefree, then its repeated prime factors are added as many times as the exponent indicates (e.g., the sum of prime factors of 8 is 6 since 8 = 2 * 2 * 2 and 2 + 2 + 2 = 6).

No even semiprime (A100484) can be in this sequence, since, if nothing else, it is the sum of a prime number and that prime number's only prime factor (itself).

			3 is in the sequence since neither 1 + sopfr(1) nor 2 + sopfr(2) add up to 3 (instead these equal 2 and 4 respectively).
Because 2 + sopfr(2) = 4, the number 4 is not in this sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A096461, A192896 (only a(1) of those sequences can be in this sequence). Cf. also A001414. Analogous to A005114.

pfAddSeq[start_, max_] := NestWhileList[# + Plus@@Times@@@FactorInteger@# &, start, # < max &]; Complement[Range[200], Flatten[Table[Drop[pfAddSeq[n, 200], 1], {n, 2, 200}]]] (* corrected by Amiram Eldar, Aug 14 2025 *)

upto(n) = {
	v = vector(n);
	for(i = 2, n,
		c = i + sopfr(i);
		if(c <= n,
			v[c] = 1));
	select(x -> x == 0, v, 1)}
sopfr(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i,1] * f[i,2])} \\ David A. Corneth, Aug 14 2025

2 inserted by and more terms from David A. Corneth, Aug 14 2025

A192277 Impossible values for both sum of proper divisors of k and sum of anti-divisors of k.

Original entry on oeis.org

120, 146, 206, 246, 268, 288, 290, 326, 342, 430, 448, 472, 530, 626, 658, 718, 726, 766, 804, 818, 836, 926, 982, 996, 1002, 1028, 1200, 1212, 1246, 1256, 1312, 1332, 1360, 1438, 1506, 1508, 1538, 1542, 1692, 1718, 1728, 1746, 1806, 1840, 2078, 2148, 2388

Offset: 1

Paolo P. Lava, Jul 06 2011

nonn

Amiram Eldar, Table of n, a(n) for n = 1..639

Intersection of A005114 and A074898.

Cf. A001065, A066272, A066417, A078923.

A209249 Even numbers that are not the sum of two untouchable numbers.

Original entry on oeis.org

2, 6, 8, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 92, 94, 96, 100, 102, 106, 108, 110, 112, 114, 116, 118, 120, 124, 128, 130, 132, 134, 136, 138

Offset: 1

Donovan Johnson, Jan 13 2013

nonn

It appears that this sequence has 118 terms, the last of which is 490. All even numbers from 492 to 10^7 are the sum of two untouchable numbers.

			90 is not a term because it is the sum of two untouchable numbers (2 + 88). 92 is a term because no two untouchable numbers sum to 92.

Donovan Johnson, Table of n, a(n) for n = 1..118
Eric Weisstein's World of Mathematics, Untouchable Number.
Wikipedia, Untouchable number

Cf. A005114.

mx=10000; v=vector(mx); w=vector(mx); un=vector(1212); for(i=669, 1229, v[prime(i)+1]=1); for(i=2, 24989857, al=sigma(i)-i; if(al<=mx, v[al]=1)); c=0; for(i=2, mx, if(v[i]==0, c++; un[c]=i)); for(i=1, 1212, for(j=i, 1212, s=un[i]+un[j]; if(s<=mx, w[s]=1, next(2)))); n=0; forstep(i=2, mx, 2, if(w[i]==0, n++; write("b209249.txt", n " " i)))

A291109 Numbers that are not the sum of the squarefree divisors of some natural number.

Original entry on oeis.org

2, 5, 7, 9, 10, 11, 13, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 39, 40, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 100, 101, 103, 105, 106, 107, 109, 111

Offset: 1

Ilya Gutkovskiy, Aug 17 2017

nonn

Impossible values for A048250 (numbers k in increasing order such that A048250(m) = k has no solution).
Numbers that are not of the form Product (p_i + 1), p is a prime, so all odd numbers (except 1 and 3) are in this sequence.
Also numbers that are not the sum of the divisors of some squarefree number.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors

Cf. A005114, A005117, A007369, A048250, A048995, A062822, A063948, A064000, A203444, A206778, A274793.

sort(convert({$1..1000} minus map(numtheory:-sigma, select(numtheory:-issqrfree, {$1..1000})),list)); # Robert Israel, Jun 26 2018

TakeWhile[Complement[Range@ #, Union@ Table[Total@ Select[Divisors@ n, SquareFreeQ], {n, 2 #}]], Function[k, k <= #]] &@ 111

A300325 Nonaliquot nontotient numbers.

Original entry on oeis.org

124, 146, 188, 206, 246, 248, 290, 304, 322, 326, 406, 426, 472, 474, 516, 518, 530, 626, 628, 668, 670, 714, 748, 766, 782, 802, 804, 818, 872, 892, 894, 898, 902, 926, 934, 964, 1002, 1028, 1046, 1074, 1078, 1134, 1146, 1148, 1246, 1254, 1256, 1266, 1314, 1342, 1346, 1348, 1388, 1406, 1418, 1420

Offset: 1

Robert G. Wilson v, Mar 03 2018

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2326 from Robert G. Wilson v)

Intersection of A005114 and A005277.

A325449 Psi-untouchable numbers: impossible values for A306927(n) = A001615(n) - n.

Original entry on oeis.org

30, 38, 58, 60, 66, 94, 98, 102, 118, 120, 132, 138, 146, 158, 174, 178, 188, 190, 204, 206, 222, 238, 240, 246, 262, 264, 276, 278, 282, 290, 292, 298, 306, 318, 322, 326, 338, 348, 354, 374, 380, 390, 398, 402, 406, 408, 426, 430, 444, 458, 462, 474, 476, 478

Offset: 1

Amiram Eldar, Sep 06 2019

nonn

Analogous to untouchable numbers (A005114) with Dedekind psi function (A001615) instead of the sum of divisors function, sigma (A000203).

te Riele named these numbers psi_1-untouchable. He calculated the first 2896 terms (terms below 20000). He proved that this sequence is infinite by showing that all the numbers of the form 2^k*3*5 (k >= 1, A110286(k) except for k = 0) are psi-untouchables.

Amiram Eldar, Table of n, a(n) for n = 1..10000
H. J. J. te Riele, A theoretical and computational study of generalized aliquot sequences (Doctoral thesis), MCT-74, Mathematisch Centrum, Amsterdam, 1976, Chapter 9.

Cf. A001615, A005114, A110286, A306927.

f[1] = 0; f[n_] := n*(Times @@ (1 + 1/FactorInteger[n][[;; , 1]]) - 1); m = 300; v = Table[0, {m}]; Do[j = f[k]; If[2 <= j <= m, v[[j]]++], {k, 1, m^2}]; Rest[Position[v, _?(# == 0 &)] // Flatten]

cp's OEIS Frontend

A048187 Palindromic untouchable numbers.

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A057978 Number of untouchable numbers <= 10^n.

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A120534 Untouchable 3-almost primes.

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A121683 Untouchable cubes.

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A191749 Numbers not the sum of a smaller number and its prime factors (with multiplicity).

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Mathematica

PARI

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A192277 Impossible values for both sum of proper divisors of k and sum of anti-divisors of k.

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A209249 Even numbers that are not the sum of two untouchable numbers.

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PARI

A291109 Numbers that are not the sum of the squarefree divisors of some natural number.

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Maple

Mathematica

A300325 Nonaliquot nontotient numbers.

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A325449 Psi-untouchable numbers: impossible values for A306927(n) = A001615(n) - n.

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