A124223 -id:A124223 - cp's OEIS frontend

A328617 Multiplicative with a(p^e) = p^e, if e = 0 mod p, otherwise a(p^e) = p^((p*floor(e/p)) + A124223(A000720(p),e mod p)).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 125, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 2401, 250, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 375, 76, 77, 78, 79, 80, 81

Offset: 1

Antti Karttunen, Oct 23 2019

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Cf. A124223, A327938, A289234.

Cf. also A328618, A328619.

A328617(n) = { my(f = factor(n), m, q); for(k=1, #f~, q = (f[k, 2]\f[k, 1]); m = (f[k, 2]%f[k, 1]); if(m, f[k, 2] = q*f[k, 1] + lift(1/Mod(m,f[k, 1])))); factorback(f); };

For all n >= 0, A276085(a(A276086(n))) = A289234(n).

A006093 a(n) = prime(n) - 1.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270

Offset: 1

N. J. A. Sloane

These are also the numbers that cannot be written as i*j + i + j (i,j >= 1). - Rainer Rosenthal, Jun 24 2001; Henry Bottomley, Jul 06 2002
The values of k for which Sum_{j=0..n} (-1)^j*binomial(k, j)*binomial(k-1-j, n-j)/(j+1) produces an integer for all n such that n < k. Setting k=10 yields [0, 1, 4, 11, 19, 23, 19, 11, 4, 1, 0] for n = [-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], so 10 is in the sequence. Setting k=3 yields [0, 1, 1/2, 1/2] for n = [-1, 0, 1, 2], so 3 is not in the sequence. - Dug Eichelberger (dug(AT)mit.edu), May 14 2001
n such that x^n + x^(n-1) + x^(n-2) + ... + x + 1 is irreducible. - Robert G. Wilson v, Jun 22 2002
Records for Euler totient function phi.
Together with 0, n such that (n+1) divides (n!+1). - Benoit Cloitre, Aug 20 2002; corrected by Charles R Greathouse IV, Apr 20 2010
n such that phi(n^2) = phi(n^2 + n). - Jon Perry, Feb 19 2004
Numbers having only the trivial perfect partition consisting of a(n) 1's. - Lekraj Beedassy, Jul 23 2006
Numbers n such that the sequence {binomial coefficient C(k,n), k >= n } contains exactly one prime. - Artur Jasinski, Dec 02 2007
Record values of A143201: a(n) = A143201(A001747(n+1)) for n > 1. - Reinhard Zumkeller, Aug 12 2008
From Reinhard Zumkeller, Jul 10 2009: (Start)
The first N terms can be generated by the following sieving process:
  start with {1, 2, 3, 4, ..., N - 1, N};
  for i := 1 until SQRT(N) do
  (if (i is not striked out) then
  (for j := 2 * i + 1 step i + 1 until N do
  (strike j from the list)));
  remaining numbers = {a(n): a(n) <= N}. (End)
a(n) = partial sums of A075526(n-1) = Sum_{1..n} A075526(n-1) = Sum_{1..n} (A008578(n+1) - A008578(n)) = Sum_{1..n} (A158611(n+2) - A158611(n+1)) for n >= 1. - Jaroslav Krizek, Aug 04 2009
A006093 U A072668 = A000027. - Juri-Stepan Gerasimov, Oct 22 2009
A171400(a(n)) = 1 for n <> 2: subsequence of A171401, except for a(2) = 2. - Reinhard Zumkeller, Dec 08 2009
Numerator of (1 - 1/prime(n)). - Juri-Stepan Gerasimov, Jun 05 2010
Numbers n such that A002322(n+1) = n. This statement is stronger than repeating the property of the entries in A002322, because it also says in reciprocity that this sequence here contains no numbers beyond the Carmichael numbers with that property. - Michel Lagneau, Dec 12 2010
a(n) = A192134(A095874(A000040(n))); subsequence of A192133. - Reinhard Zumkeller, Jun 26 2011
prime(a(n)) + prime(k) < prime(a(k) + k) for at least one k <= a(n): A212210(a(n),k) < 0. - Reinhard Zumkeller, May 05 2012
Except for the first term, numbers n such that the sum of first n natural numbers does not divide the product of first n natural numbers; that is, n*(n + 1)/2 does not divide n!. - Jayanta Basu, Apr 24 2013
BigOmega(a(n)) equals BigOmega(a(n)*(a(n) + 1)/2), where BigOmega = A001222. Rationale: BigOmega of the product on the right hand side factorizes as BigOmega(a/2) + Bigomega(a+1) = BigOmega(a/2) + 1 because a/2 and a + 1 are coprime, because BigOmega is additive, and because a + 1 is prime. Furthermore Bigomega(a/2) = Bigomega(a) - 1 because essentially all 'a' are even. - Irina Gerasimova, Jun 06 2013
Record values of A060681. - Omar E. Pol, Oct 26 2013
Deficiency of n-th prime. - Omar E. Pol, Jan 30 2014
Conjecture: All the sums Sum_{k=s..t} 1/a(k) with 1 <= s <= t are pairwise distinct. In general, for any integers d >= -1 and m > 0, if Sum_{k=i..j} 1/(prime(k)+d)^m = Sum_{k=s..t} 1/(prime(k)+d)^m with 0 < i <= j and 0 < s <= t then we must have (i,j) = (s,t), unless d = m = 1 and {(i,j),(s,t)} = {(4,4),(8,10)} or {(4,7),(5,10)}. (Note that 1/(prime(8)+1)+1/(prime(9)+1)+1/(prime(10)+1) = 1/(prime(4)+1) and Sum_{k=5..10} 1/(prime(k)+1) = 1/(prime(4)+1) + Sum_{k=5..7} 1/(prime(k)+1).) - Zhi-Wei Sun, Sep 09 2015
Numbers n such that (prime(i)^n + n) is divisible by (n+1), for all i >= 1, except when prime(i) = n+1. - Richard R. Forberg, Aug 11 2016
a(n) is the period of Fubini numbers (A000670) over the n-th prime. - Federico Provvedi, Nov 28 2020

Archimedeans Problems Drive, Eureka, 40 (1979), 28.
Harvey Dubner, Generalized Fermat primes, J. Recreational Math., 18 (1985): 279-280.
M. Gardner, The Colossal Book of Mathematics, pp. 31, W. W. Norton & Co., NY, 2001.
M. Gardner, Mathematical Circus, pp. 251-2, Alfred A. Knopf, NY, 1979.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
Thomas F. Bloom, Unit fractions with shifted prime denominators, arXiv:2305.02689 [math.NT], 2023.
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
Harvey Dubner, Generalized Fermat primes, J. Recreational Math. 18.4 (1985-1986), 279. (Annotated scanned copy)
Armel Mercier, Problem E 3065, American Mathematical Monthly, 1984, p. 649.
Armel Mercier, S. K. Rangarajan, J. C. Binz and Dan Marcus, Problem E 3065, American Mathematical Monthly, No. 4, 1987, pp. 378.
Poo-Sung Park, Additive uniqueness of PRIMES-1 for multiplicative functions, arXiv:1708.03037 [math.NT], 2017.
J. R. Rickard and J. J. Hitchcock, Problem Drive 4, Archimedeans Problems Drive, Eureka, 40 (1979), 28-29, 40. (Annotated scanned copy)
Index entries for sequences generated by sieves

a(n) = K(n, 1) and A034693(K(n, 1)) = 1 for all n. The subscript n refers to this sequence and K(n, 1) is the index in A034693. - Labos Elemer
Cf. A000040, A034694. Different from A075728.
Complement of A072668 (composite numbers minus 1), A072670(a(n))=0.
Essentially the same as A039915.
Cf. A084920, A006093, A050997, A008864, A060800, A131991, A131992, A131993.
Cf. A101301 (partial sums), A005867 (partial products).
Column 1 of the following arrays/triangles: A087738, A249741, A352707, A378979, A379010.
The last diagonal of A162619, and of A174996, the first diagonal in A131424.
Row lengths of irregular triangles A086145, A124223, A212157.

Filtered([1..280],IsPrime)-1; # Muniru A Asiru, Nov 25 2018

a006093 = (subtract 1) . a000040  -- Reinhard Zumkeller, Mar 06 2012

[NthPrime(n)-1: n in [1..100]]; // Vincenzo Librandi, Nov 17 2015

for n from 2 to 271 do if (n! mod n^2 = n*(n-1) and (n<>4) then print(n-1) fi od; # Gary Detlefs, Sep 10 2010
# alternative
A006093 := proc(n)
    ithprime(n)-1 ;
end proc:
seq(A006093(n),n=1..100) ; # R. J. Mathar, Feb 06 2019

Table[Prime[n] - 1, {n, 1, 30}] (* Vladimir Joseph Stephan Orlovsky, Apr 27 2008 *)
a[ n_] := If[ n < 1, 0, -1 + Prime @ n] (* Michael Somos, Jul 17 2011 *)
Prime[Range[60]] - 1 (* Alonso del Arte, Oct 26 2013 *)

isA006093(n) = isprime(n+1) \\ Michael B. Porter, Apr 09 2010

A006093(n) = prime(n)-1 \\ Michael B. Porter, Apr 09 2010

\\ Sieve as described in Rainer Rosenthal's comment.
m=270;s=vector(m);for(i=1,m,for(j=i,m,k=i*j+i+j;if(k<=m,s[k]=1)));for(k=1,m,if(s[k]==0,print1(k,", "))); \\ Hugo Pfoertner, May 14 2019

from sympy import prime
for n in range(1,100): print(prime(n)-1, end=', ') # Stefano Spezia, Nov 30 2018

a(n) = (p-1)! mod p where p is the n-th prime, by Wilson's theorem. - Jonathan Sondow, Jul 13 2010
a(n) = A000010(prime(n)) = A000010(A006005(n)). - Antti Karttunen, Dec 16 2012
a(n) = A005867(n+1)/A005867(n). - Eric Desbiaux, May 07 2013
a(n) = A000040(n) - 1. - Omar E. Pol, Oct 26 2013
a(n) = A033879(A000040(n)). - Omar E. Pol, Jan 30 2014

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010
Obfuscating comments removed by Joerg Arndt, Mar 11 2010
Edited by Charles R Greathouse IV, Apr 20 2010

A122585 Reciprocal of n modulo smallest prime greater than n.

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 8, 7, 5, 10, 6, 12, 4, 11, 8, 16, 9, 18, 17, 15, 11, 22, 24, 23, 7, 19, 14, 28, 15, 30, 6, 22, 9, 12, 18, 36, 10, 27, 20, 40, 21, 42, 35, 31, 23, 46, 44, 21, 13, 35, 26, 52, 49, 47, 44, 39, 29, 58, 30, 60, 11, 40, 50, 22, 33, 66, 53, 47, 35, 70, 36, 72, 13, 63, 59

Offset: 1

Franklin T. Adams-Watters, Oct 20 2006

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
Eric Weisstein's World of Mathematics, Modular Inverse

Cf. A006093, A040976, A124223, A151800, A362254.

a:= n-> n&^(-1) mod nextprime(n):
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 13 2023

Table[PowerMod[n,-1,NextPrime[n]],{n,80}] (* Harvey P. Dale, Apr 13 2023 *)

from sympy import nextprime
def A122585(n): return pow(n,-1,nextprime(n)) # Chai Wah Wu, Apr 13 2023

From Alois P. Heinz, Apr 13 2023: (Start)
a(n) = n <=> n in { A006093 }.
a(n) = (n+1)/2 <=> n in { A040976 } \ { 0 }. (End)

A124224 Table T(n,k) = reciprocal of k-th number prime to n, modulo n, for 1 <= k <= phi(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 4, 5, 2, 3, 6, 1, 3, 5, 7, 1, 5, 7, 2, 4, 8, 1, 7, 3, 9, 1, 6, 4, 3, 9, 2, 8, 7, 5, 10, 1, 5, 7, 11, 1, 7, 9, 10, 8, 11, 2, 5, 3, 4, 6, 12, 1, 5, 3, 11, 9, 13, 1, 8, 4, 13, 2, 11, 7, 14, 1, 11, 13, 7, 9, 3, 5, 15, 1, 9, 6, 13, 7, 3, 5, 15, 2, 12, 14, 10, 4, 11, 8, 16

Offset: 1

Franklin T. Adams-Watters, Oct 20 2006

T(n,k) = smallest m such that A038566(n,k) * m = 1 (mod n).

For n>1 every row begins with 1 and ends with n-1. T(n,k) = A038566(n,k)^(phi(n) - 1) (mod n). - Geoffrey Critzer, Jan 03 2015

			The table T(n,k) starts:
n\k 1  2  2  3 4  5 6  7 8  9 10 11
1:  0
2:  1
3:  1  2
4:  1  3
5:  1  3  2  4
6:  1  5
7:  1  4  5  2 3  6
8:  1  3  5  7
9:  1  5  7  2 4  8
10: 1  7  3  9
11: 1  6  4  3 9  2 8  7 5 10
12: 1  5  7 11
13: 1  7  9 10 8 11 2  5 3  4  6 12
14: 1  5  3 11 9 13
15: 1  8  4 13 2 11 7 14
16: 1 11 13  7 9  3 5 15
...
n = 17: 1  9  6 13 7  3  5 15 2 12 14 10 4 11 8 16,
n = 18: 1 11 13  5 7 17,
n = 19: 1 10 13  5 4 16 11 12 17 2 7 8 3 15 14 6 9 18,
n = 20: 1 7 3 9 11 17 13 19.
... reformatted (extended and corrected), - _Wolfdieter Lang_, Oct 06 2016

Robert Israel, Table of n, a(n) for n = 1..10060
Eric Weisstein's World of Mathematics, Modular Inverse

Cf. A124223, A102057, A038566, A000010 (row lengths), A023896 (row sums after first)

0,seq(seq(i^(-1) mod m, i = select(t->igcd(t,m)=1, [$1..m-1])),m=1..100); # Robert Israel, May 18 2014

Table[nn = n; a = Select[Range[nn], CoprimeQ[#, nn] &];
PowerMod[a, -1, nn], {n, 1, 20}] // Grid (* Geoffrey Critzer, Jan 03 2015 *)

T(n,k) * A038566(n,k) = 1 (mod n), for n >=1 and k=1..A000010(n). - Wolfdieter Lang, Oct 06 2016

A289251 Triangle T(n, k), n > 0 and 0 <= k < n, read by rows; if gcd(n, k) = 1, then T(n, k) = modular inverse of k (mod n), otherwise T(n, k) = k.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 3, 2, 4, 0, 1, 2, 3, 4, 5, 0, 1, 4, 5, 2, 3, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 5, 3, 7, 2, 6, 4, 8, 0, 1, 2, 7, 4, 5, 6, 3, 8, 9, 0, 1, 6, 4, 3, 9, 2, 8, 7, 5, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 7, 9, 10, 8, 11

Offset: 1

Rémy Sigrist, Jun 29 2017

The n-th row has n terms, and is a self-inverse permutation of the first n nonnegative numbers.
T(n, 0) = 0 for any n > 0.
T(n, 1) = 1 for any n > 1.
T(n, n-1) = n-1 for any n > 0.
If n > 0 and gcd(n, k) = 1 then T(n, k) = A102057(n, k).
T(prime(n), k) = A124223(n, k) for any n > 0 and k in 1..prime(n)-1.

			The first rows are:
n\k  0 1 2 3 4 5 6 7 8 9
1    0
2    0 1
3    0 1 2
4    0 1 2 3
5    0 1 3 2 4
6    0 1 2 3 4 5
7    0 1 4 5 2 3 6
8    0 1 2 3 4 5 6 7
9    0 1 5 3 7 2 6 4 8
10   0 1 2 7 4 5 6 3 8 9

Rémy Sigrist, Rows n=1..100 of triangle, flattened

Cf. A102057, A124223.

T[n_, k_] := If[GCD[n, k] == 1, PowerMod[k, -1, n], k];
Table[T[n, k], {n, 1, 13}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Oct 31 2017 *)

T(n, k) = if (gcd(n, k)==1, lift(1/Mod(k, n)), k)

cp's OEIS Frontend

A328617 Multiplicative with a(p^e) = p^e, if e = 0 mod p, otherwise a(p^e) = p^((p*floor(e/p)) + A124223(A000720(p),e mod p)).

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A006093 a(n) = prime(n) - 1.

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A122585 Reciprocal of n modulo smallest prime greater than n.

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A124224 Table T(n,k) = reciprocal of k-th number prime to n, modulo n, for 1 <= k <= phi(n).

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A289251 Triangle T(n, k), n > 0 and 0 <= k < n, read by rows; if gcd(n, k) = 1, then T(n, k) = modular inverse of k (mod n), otherwise T(n, k) = k.

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