A005277 -id:A005277 - cp's OEIS frontend

A347771 Unitary nontotient numbers: values not in range of unitary totient function uphi(n).

5, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 33, 34, 35, 37, 38, 39, 41, 43, 45, 47, 49, 50, 51, 53, 55, 57, 59, 61, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 81, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 99, 101, 103, 105, 107, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121, 122, 123, 125, 129, 131, 133, 134, 135

Offset: 1

Eric Chen, Sep 13 2021

nonn

Numbers not appearing in A047994.
Indices of -1 in A135347.
Unitary version of A007617.
This sequence to A047994 is A007617 to A000010.
This sequence to A135347 is A007617 to A049283 (for the case that no such numbers exist, A135347 uses -1 and A049283 uses 0).
All odd numbers not of the form 2^k-1 (i.e. not in A000225) are in this sequence, since uphi(n) = A047994(n) is an even number unless n is a power of 2 (A000079), in this case uphi(n) = n-1.
The intersection of this sequence and A049225 is empty, since for squarefree numbers, all divisors are unitary divisors, note that the intersection of this sequence and A002202 is not empty, the number 110 is in both sequences.

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

Cf. A000010, A002202, A005277, A007617, A047994, A049225, A049283, A135347, A361966, A361967.

Select[Range[135], Length[invUPhi[#]] == 0 &] (* Amiram Eldar, Apr 01 2023, using the function invUPhi from A361966 *)

A047994(n)=my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1)
is(n)=for(k=1,n^2,if(A047994(k)==n,return(0)));1 \\ after A047994

A361967(a(n)) = 0. - Amiram Eldar, Apr 01 2023

A058763 Integers which are neither totient nor cototient.

Original entry on oeis.org

26, 34, 50, 86, 122, 134, 146, 154, 170, 186, 202, 206, 218, 244, 266, 274, 290, 298, 326, 340, 362, 386, 394, 404, 412, 436, 470, 474, 482, 518, 532, 534, 554, 566, 596, 626, 634, 650, 666, 680, 686, 698, 706, 722, 724, 730, 746, 778, 794, 818, 834, 842

Offset: 1

Labos Elemer, Jan 02 2001

nonn

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A000010, A051953, A005277, A005278.

Intersection(A005277, A005278).

More terms from David Wasserman, May 14 2002

Offset corrected by Donovan Johnson, Sep 07 2013

A058887 Smallest prime p such that (2^n)*p is a nontotient number.

Original entry on oeis.org

3, 7, 17, 19, 19, 19, 31, 31, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47

Offset: 0

Labos Elemer, Jan 08 2001

nonn

For n=8,9,...,582, a(n) = 47. Note that A040076(47)=583.
For n=583,584,...,6392, a(n) = 383. Note that A040076(383)=6393.
Subsequent primes are 2897, 3061, 5297, and 7013 (cf. A057192 and A071628). [These are primes p such that the least e such that 2^e*p + 1 is prime sets a new record. - Jianing Song, Dec 14 2021]
Starting with some large N, a(n)=p for all n >= N. This prime p will likely be the first prime Sierpiński number, which is conjectured to be 271129.
In particular, a(n) <= 271129 for all n.
From Jianing Song, Dec 14 2021: (Start)
a(n) is the smallest prime p such that 2^e*p + 1 is composite for all 0 <= e <= n. A proof is given in the a-file below.
a(n) is also the smallest number k such that 2^n*k is a nontotient number (see A181662). (End)

			For n=1, the initial segment of {2p} sequence is nops(invphi({2p}))={4, 4, 2, 0, 2, 0, 0, 0, 2, 2, ...}, where the position of the first 0 is 4, corresponding to p(4)=7, so a(1)=7.
For n=8 the same initial segment is: {11, 32, 23, 18, 24, 10, 11, 4, 9, 21, 2, 16, 9, 12, 0, 14, 5, 6, 12, ...}, where the first 0 is the 15th, corresponding to p(15)=47, thus a(8)=47.

David Harden, Posting to Sequence Fans Mailing List, Sep 19 2010.
J. L. Selfridge, Solution to Problem 4995, Amer. Math. Monthly, 70:1 (1963), page 101.

D. Bressoud, CNT.m Computational Number Theory Mathematica package.
Jianing Song, Proof that a(n) is the smallest prime p such that 2^e*p + 1 is composite for all 0 <= e <= n.

Cf. A005277, A007617, A057192, A071628, A076336 (Sierpiński numbers), A000010, A181662.

Needs["CNT`"]; Table[p=3; While[PhiInverse[p*2^n] != {}, p=NextPrime[p]]; p, {n,0,20}]

a(n) = my(p=2); while(istotient(2^n*p), p=nextprime(p+1)); p; \\ Michel Marcus, May 14 2020

Min{p|p is prime and card(invphi((2^n)*p))=0}.
From Jianing Song, Dec 14 2021: (Start)
a(0) = 3;
a(1) = 7;
a(2) = 17;
a(3..5) = 19;
a(6..7) = 31;
a(8..582) = 47;
a(583..6392) = 383;
a(6393..9714) = 2897;
a(9715..33287) = 3061;
a(33288..50010) = 5297;
a(50011..126112) = 7013;
a(126113..31172164) = 10223.
a(n) = A181662(n) / 2^n. (End)

Edited by T. D. Noe, Nov 15 2010

Edited by Max Alekseyev, Nov 19 2010

A083534 First difference sequence of A007617. Difference between consecutive values not being in the range of phi (A000010).

Original entry on oeis.org

2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2

Offset: 1

Labos Elemer, May 20 2003

nonn

a(n) is either 2 or 1 since odd numbers are in A007619.

If a(n) = 1 then A007619(n+1) is an even number not in the range of phi.

			{11,13,14,15,17} are not in the range of phi and the corresponding differences are {2,1,1,2}.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000010, A002202, A005277, A007617, A007619, A083531, A083532, A083533, A083535, A083536.

a083534 n = a083534_list !! (n-1)
a083534_list = zipWith (-) (tail a007617_list) a007617_list
-- Reinhard Zumkeller, Nov 26 2015

t0[x_] := Table[j, {j, 1, x}]; t=Table[EulerPhi[w], {w, 1, 10000}]; u=Union[%]; c=Complement[t0[10000], u]; Delete[c-RotateRight[c], 1]

list(lim) = {my(k1 = 3, k2 = 3); while(k1 < lim, until(!istotient(k2), k2++); print1(k2 - k1, ", "); k1 = k2); } \\ Amiram Eldar, Feb 22 2025

a(n) = A007617(n+1) - A007617(n).

A333019 Numbers k such that both k and k + 2 are totient numbers (A002202).

Original entry on oeis.org

2, 4, 6, 8, 10, 16, 18, 20, 22, 28, 30, 40, 42, 44, 46, 52, 54, 56, 58, 64, 70, 78, 80, 82, 100, 102, 104, 106, 108, 110, 126, 128, 130, 136, 138, 148, 160, 162, 164, 166, 176, 178, 190, 196, 198, 208, 210, 220, 222, 224, 226, 238, 250, 260, 262, 268, 270, 280

Offset: 1

Amiram Eldar, Mar 05 2020

nonn

			2 is a term since both 2 and 4 are totient numbers.

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

Cf. A000010, A002202, A005277, A063512, A083533, A333020, A333021, A333022, A333023, A333024.

for(k = 1, 150, if(istotient(2*k) && istotient(2*k+2), print1(2*k,", ")))

A333020 Starts of runs of 3 consecutive even numbers that are all totient numbers (A002202).

Original entry on oeis.org

2, 4, 6, 8, 16, 18, 20, 28, 40, 42, 44, 52, 54, 56, 78, 80, 100, 102, 104, 106, 108, 126, 128, 136, 160, 162, 164, 176, 196, 208, 220, 222, 224, 260, 268, 292, 328, 342, 344, 356, 378, 380, 416, 438, 440, 460, 462, 464, 476, 498, 500, 502, 504, 520, 560, 584

Offset: 1

Amiram Eldar, Mar 05 2020

nonn

			2 is a term since 2, 4 and 6 are all totient numbers.

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

Cf. A000010, A002202, A005277, A063512, A083533, A333019, A333021, A333022, A333023, A333024.

m = 3; v = vector(m); for(k=1, m, v[k] = istotient(2*k)); for(k = m+1, 300, if(Set(v) == [1], print1(2*(k-m),", ")); v = concat(v[2..m], istotient(2*k)))

A333021 Starts of runs of 4 consecutive even numbers that are all totient numbers (A002202).

Original entry on oeis.org

2, 4, 6, 16, 18, 40, 42, 52, 54, 78, 100, 102, 104, 106, 126, 160, 162, 220, 222, 342, 378, 438, 460, 462, 498, 500, 502, 856, 858, 880, 882, 1086, 1276, 1278, 1300, 1422, 1480, 1482, 1566, 1660, 1662, 1804, 1806, 1996, 2058, 2200, 2202, 2236, 2238, 3016, 3018

Offset: 1

Amiram Eldar, Mar 05 2020

nonn

			2 is a term since 2, 4, 6 and 8 are all totient numbers.

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

Cf. A000010, A002202, A005277, A063512, A083533, A333019, A333020, A333022, A333023, A333024.

m = 4; v = vector(m); for(k=1, m, v[k] = istotient(2*k)); for(k = m+1, 1500, if(Set(v) == [1], print1(2*(k-m),", ")); v = concat(v[2..m], istotient(2*k)))

A333022 Starts of runs of 5 consecutive even numbers that are all totient numbers (A002202).

Original entry on oeis.org

2, 4, 16, 40, 52, 100, 102, 104, 160, 220, 460, 498, 500, 856, 880, 1276, 1480, 1660, 1804, 2200, 2236, 3016, 3160, 3460, 4516, 4780, 5500, 5920, 6040, 6196, 6820, 7240, 7636, 7696, 7720, 8536, 8620, 9196, 9460, 9880, 10456, 12916, 13756, 13960, 14416, 15640

Offset: 1

Amiram Eldar, Mar 05 2020

nonn

			2 is a term since 2, 4, 6, 8 and 10 are all totient numbers.

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

Cf. A000010, A002202, A005277, A063512, A083533, A333019, A333020, A333021, A333023, A333024.

m = 5; v = vector(m); for(k=1, m, v[k] = istotient(2*k)); for(k = m+1, 7500, if(Set(v) == [1], print1(2*(k-m),", ")); v = concat(v[2..m], istotient(2*k)))

A333023 Starts of runs of 6 consecutive even numbers that are all totient numbers (A002202).

Original entry on oeis.org

2, 100, 102, 498, 267670, 26734060, 26734062, 31253680, 65974998, 70938496, 118428800, 1232747200, 2764919296, 3149734998, 3149735000, 3413655896, 3415058276, 3755544796, 4446555802, 5727840798, 6156991616, 10080661998, 10464983096, 11054945296, 11953158220

Offset: 1

Amiram Eldar, Mar 05 2020

nonn

			2 is a term since 2, 4, 6, 8, 10 and 12 are all totient numbers.

Amiram Eldar, Table of n, a(n) for n = 1..46
Wikipedia, Totient numbers.

Cf. A000010, A002202, A005277, A063512, A083533, A333019, A333020, A333021, A333022, A333024.

m = 6; v = vector(m); for(k=1, m, v[k] = istotient(2*k)); for(k = m+1, 1.5e5, if(Set(v) == [1], print1(2*(k-m),", ")); v = concat(v[2..m], istotient(2*k)))

A350316 Totient numbers k such that 3*k is a nontotient.

Original entry on oeis.org

1, 30, 58, 78, 82, 106, 130, 138, 150, 172, 178, 198, 222, 226, 238, 268, 282, 316, 342, 358, 366, 378, 382, 388, 418, 438, 462, 478, 498, 502, 506, 508, 546, 562, 570, 598, 606, 618, 630, 642, 646, 652, 658, 682, 690, 718, 738, 742, 772, 786, 810, 826, 838

Offset: 1

Jianing Song, Dec 24 2021

nonn

			30 is a term since 30 = phi(31) = phi(62), but phi(n) = 3*30 = 90 has no solution.
58 is a term since 58 = phi(59) = phi(118), but phi(n) = 3*58 = 174 has no solution.

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

Totient numbers k such that m*k is a nontotient: this sequence (m=3), A350317 (m=5), A350318 (m=7), A350319 (m=9), A350320 (m=10), A350321 (m=14).

Cf. A002202, A005277.

isA350316(n) = istotient(n) && !istotient(3*n)

cp's OEIS Frontend

A347771 Unitary nontotient numbers: values not in range of unitary totient function uphi(n).

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A058763 Integers which are neither totient nor cototient.

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A058887 Smallest prime p such that (2^n)*p is a nontotient number.

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A083534 First difference sequence of A007617. Difference between consecutive values not being in the range of phi (A000010).

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A333019 Numbers k such that both k and k + 2 are totient numbers (A002202).

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A333020 Starts of runs of 3 consecutive even numbers that are all totient numbers (A002202).

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A333021 Starts of runs of 4 consecutive even numbers that are all totient numbers (A002202).

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A333022 Starts of runs of 5 consecutive even numbers that are all totient numbers (A002202).

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