A039649 -id:A039649 - cp's OEIS frontend

A039689 Numbers k such that phi(k) + 1 is not a prime.

15, 16, 20, 24, 25, 30, 33, 35, 39, 44, 45, 50, 51, 52, 56, 64, 65, 66, 68, 69, 70, 72, 78, 80, 81, 84, 85, 87, 90, 92, 96, 102, 104, 105, 112, 116, 120, 121, 123, 128, 129, 130, 136, 138, 140, 141, 143, 144, 147, 155, 156, 159, 160, 161, 162, 164, 165, 168, 170

Offset: 1

Olivier Gérard

			phi(20)+1 = 8+1 = 9 is not prime.

Antti Karttunen, Table of n, a(n) for n = 1..39340

Cf. A000010, A007614, A039649, A039698 (complement).
Positions of zeros in A296079.
Cf. also A263029.

Select[Range[200],!PrimeQ[EulerPhi[#]+1]&] (* Harvey P. Dale, Aug 31 2018 *)

isok(k) = !isprime(eulerphi(k)+1); \\ Michel Marcus, Jun 28 2021

Name edited by Antti Karttunen, Dec 05 2017

A063530 Numbers k such that phi(k)+1 is a square.

Original entry on oeis.org

15, 16, 20, 24, 30, 35, 39, 45, 52, 56, 65, 70, 72, 78, 84, 90, 104, 105, 112, 123, 130, 140, 143, 144, 155, 156, 164, 165, 168, 175, 176, 180, 183, 200, 203, 210, 215, 220, 225, 231, 244, 245, 246, 248, 261, 264, 286, 300, 308, 310, 323, 330, 339, 344, 350

Offset: 1

Labos Elemer, Aug 02 2001

nonn

Numbers k such that A000010(k) = -1 + m^2 for some m.

			If k = p*(p+2), a product of twin primes (from A037074), then k is in the sequence. The corresponding square is p^2. Other solutions are k = {56,72,78,84}, since phi(k) + 1 = 25 for all. Also phi(123) + 1 = 9^2, the square of a composite.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000010, A037074, A039649.

Select[Range[400],IntegerQ[Sqrt[1+EulerPhi[#]]]&] (* Harvey P. Dale, Jul 31 2020 *)

{ n=0; for (a=1, 10^9, if (issquare(eulerphi(a) + 1), write("b063530.txt", n++, " ", a); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 25 2009

A138539 Primes p_n for which A140141(n) < 2p_n, where p_n = n-th prime (A000040).

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 37, 41, 43, 61, 73, 89, 97, 101, 109, 113, 157, 163, 181, 193, 233, 241, 257, 277, 281, 313, 337, 349, 353, 397, 401, 409, 421, 433, 449, 457, 461, 487, 521, 541, 577, 593, 601, 613, 617, 641, 661, 673, 701, 733, 757, 761, 769, 821, 829

Offset: 1

Vladimir Shevelev, May 10 2008

nonn

Apparently this is the union of {2} and A058341. - R. J. Mathar, Jul 03 2009

This sequence is the complement of A058340. - Torlach Rush, Jun 29 2018

Cf. A140141, A039649, A000010, A000040, A058340.

Corrected and extended by Ray Chandler, May 20 2008

A138786 "Left" odd composite numbers n for which n < A140607((n-1)/2).

Original entry on oeis.org

9, 21, 25, 27, 35, 45, 49, 55, 69, 75, 77, 81, 93, 95, 99, 105, 115, 119, 121, 125, 133, 135, 141, 143, 147, 153, 155, 161, 165, 169, 175, 187, 189, 203, 207, 209, 213, 215, 217, 219, 221, 225, 231, 235, 237, 243, 245, 247, 253, 259, 261, 267, 279, 285, 287

Offset: 1

Vladimir Shevelev, May 18 2008

nonn

There are odd composite numbers which are neither in this sequence nor in A140608. The first such number is 91, see A140667.

Ray Chandler, Table of n, a(n) for n=1..5713

Cf. A140607, A039649, A140140, A138193, A138217, A138227, A138535, A138536, A138537, A138539, A140141, A000010, A000040, A140608, A140667.

Extended by Ray Chandler, May 20 2008

A140608 "Right" odd composite numbers n for which n > A140607((n-1)/2).

Original entry on oeis.org

15, 33, 39, 51, 57, 63, 65, 85, 87, 111, 117, 123, 129, 145, 159, 171, 177, 183, 185, 195, 201, 205, 249, 255, 265, 273, 275, 291, 303, 305, 315, 321, 327, 333, 339, 341, 393, 399, 411, 417, 435, 447, 451, 455, 465, 471, 481, 485, 489, 505, 511, 513, 519, 537

Offset: 1

Vladimir Shevelev, May 18 2008

nonn

Conjecture. The sequence is infinite.

Ray Chandler, Table of n, a(n) for n=1..2022

Cf. A140607, A039649, A140140, A138193, A138217, A138227, A138535, A138536, A138537, A138539, A140141, A000010, A000040, A138786, A140667.

Extended by Ray Chandler, May 20 2008

A186987 Number of subsets of {1, 2, ..., n} containing n and having <=3 pairwise coprime elements.

Original entry on oeis.org

1, 2, 4, 4, 10, 4, 18, 11, 19, 10, 42, 11, 58, 21, 30, 33, 96, 22, 120, 36, 62, 48, 172, 37, 147, 69, 128, 70, 270, 37, 308, 123, 158, 117, 208, 75, 432, 147, 218, 119, 530, 78, 584, 186, 228, 212, 696, 133, 594, 191, 380, 256, 882, 166, 547

Offset: 1

Alois P. Heinz, Mar 03 2011

			a(6) = 4 because there are 4 subsets of {1,2,3,4,5,6} containing 6 and having <=3 pairwise coprime elements: {6}, {1,6}, {5,6}, {1,5,6}.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Column 3 of triangle A186975. Sum of A039649 and A185953 for n>1.

A214288 Primes of the form phi(n)+1 sorted by increasing n, where phi is the Euler totient function.

Original entry on oeis.org

2, 2, 3, 3, 5, 3, 7, 5, 7, 5, 11, 5, 13, 7, 17, 7, 19, 13, 11, 23, 13, 19, 13, 29, 31, 17, 17, 13, 37, 19, 17, 41, 13, 43, 23, 47, 17, 43, 53, 19, 41, 37, 29, 59, 17, 61, 31, 37, 67, 71, 73, 37, 41, 37, 61, 79, 41, 83, 43, 41, 89, 73, 61, 47, 73, 97, 43, 61, 41, 101, 103, 53, 107, 37

Offset: 1

Vincenzo Librandi, Jul 13 2012

Primes in A039649.

All primes are in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000010, A039698 (associated n), A214287.

Select[Table[EulerPhi[n]+1,{n,1,1000}],PrimeQ]

A140667 Odd composite numbers k for which k = A140607((k-1)/2).

Original entry on oeis.org

91, 1581, 2465, 8481, 25761, 31609, 33355, 34945, 118405, 146611, 319507, 736291, 994507, 3270403, 3375487, 5176153, 6186403, 6228685, 8650951, 10679131, 22028203, 26017291, 31470211, 33796531, 41710411, 42149971, 42474547, 46672291, 48316969, 49019851, 58986091, 68182003, 69885649

Offset: 1

Ray Chandler, May 20 2008

nonn

Cf. A140607, A039649, A140140, A138193, A138217, A138227, A138535, A138536, A138537, A138539, A140141, A000010, A000040, A140608, A138786.

f(n) = (eulerphi(2*n+1) + 1 + g(n))/2; \\ A140607
g(n) = sumdiv(2*n+1, d, eulerphi(d)/(t=znorder(Mod(2, d))))*t-t+1; \\ A137576
isok(c) = if (!isprime(c) && (c%2), f((c-1)/2) == c); \\ Michel Marcus, Jan 31 2023

More terms from Michel Marcus, Jan 31 2023

A296080 Restricted growth sequence transform of A289625(1+phi(n)), where phi = A000010, Euler totient function.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 5, 3, 6, 4, 4, 4, 7, 4, 8, 4, 6, 5, 9, 4, 10, 6, 8, 6, 11, 4, 12, 7, 10, 7, 13, 6, 14, 8, 13, 7, 15, 6, 16, 10, 13, 9, 17, 7, 16, 10, 18, 13, 19, 8, 15, 13, 14, 11, 20, 7, 21, 12, 14, 18, 16, 10, 22, 18, 23, 13, 24, 13, 25, 14, 15, 14, 21, 13, 26, 18, 27, 15, 28, 13, 29, 16, 30, 15, 31, 13, 25, 23, 21, 17, 25, 18, 32, 16, 21, 15

Offset: 1

Antti Karttunen, Dec 05 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A039649, A289625, A289626, A296078.

Cf. A039650.

allocatemem(2^30);
up_to = 65537;
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A289625(n) = { my(m=1,p=2,v=znstar(n)[2]); for(i=1,length(v),m *= p^v[i]; p = nextprime(p+1)); (m); };
write_to_bfile(1,rgs_transform(vector(up_to,n,A289625(1+eulerphi(n)))),"b296080.txt");

A187263 Number of nonempty subsets of {1, 2, ..., n} with <=2 pairwise coprime elements.

Original entry on oeis.org

1, 3, 6, 9, 14, 17, 24, 29, 36, 41, 52, 57, 70, 77, 86, 95, 112, 119, 138, 147, 160, 171, 194, 203, 224, 237, 256, 269, 298, 307, 338, 355, 376, 393, 418, 431, 468, 487, 512, 529, 570, 583, 626, 647, 672, 695, 742, 759, 802, 823

Offset: 1

Alois P. Heinz, Mar 07 2011

nonn

			a(4) = 9 because there are 9 nonempty subsets of {1,2,3,4} with <=2 pairwise coprime elements: {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {3,4}.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Column 2 of triangle A187262. First differences are A039649 for n>1.

a(n) = A187262(n,2).

cp's OEIS Frontend

A039689 Numbers k such that phi(k) + 1 is not a prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A063530 Numbers k such that phi(k)+1 is a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A138539 Primes p_n for which A140141(n) < 2p_n, where p_n = n-th prime (A000040).

Views

Author

Keywords

Comments

Crossrefs

Extensions

A138786 "Left" odd composite numbers n for which n < A140607((n-1)/2).

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A140608 "Right" odd composite numbers n for which n > A140607((n-1)/2).

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A186987 Number of subsets of {1, 2, ..., n} containing n and having <=3 pairwise coprime elements.

Views

Author

Keywords

Examples

Links

Crossrefs

A214288 Primes of the form phi(n)+1 sorted by increasing n, where phi is the Euler totient function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A140667 Odd composite numbers k for which k = A140607((k-1)/2).

Views

Author

Keywords

Crossrefs

Programs

PARI

Extensions

A296080 Restricted growth sequence transform of A289625(1+phi(n)), where phi = A000010, Euler totient function.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A187263 Number of nonempty subsets of {1, 2, ..., n} with <=2 pairwise coprime elements.

Views