A049084 -id:A049084 - cp's OEIS frontend

A039703 a(n) = n-th prime modulo 5.

2, 3, 0, 2, 1, 3, 2, 4, 3, 4, 1, 2, 1, 3, 2, 3, 4, 1, 2, 1, 3, 4, 3, 4, 2, 1, 3, 2, 4, 3, 2, 1, 2, 4, 4, 1, 2, 3, 2, 3, 4, 1, 1, 3, 2, 4, 1, 3, 2, 4, 3, 4, 1, 1, 2, 3, 4, 1, 2, 1, 3, 3, 2, 1, 3, 2, 1, 2, 2, 4, 3, 4, 2, 3, 4, 3, 4, 2, 1, 4, 4, 1, 1, 3, 4, 3, 4, 2, 1, 3, 2, 4, 2, 1, 4, 3, 4, 1, 3, 1, 2, 2, 3, 4, 1

Offset: 1

Clark Kimberling

a(A049084(A045356(n-1))) = even; a(A049084(A045429(n-1))) = odd. - Reinhard Zumkeller, Feb 25 2008

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000040, A010874, A039701, A039702, A039704-A039706, A038194, A007652, A039709-A039715.

[p mod 5: p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 5, n=1..105); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 5], {n, 105}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 5] (* Vincenzo Librandi, May 06 2014 *)

primes(105) % 5 \\ Zak Seidov, Apr 09 2013

Sum_k={1..n} a(k) ~ (5/2)*n. - Amiram Eldar, Dec 11 2024

A066246 a(n) = 0 unless n is a composite number A002808(k) then a(n) = k.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 4, 5, 0, 6, 0, 7, 8, 9, 0, 10, 0, 11, 12, 13, 0, 14, 15, 16, 17, 18, 0, 19, 0, 20, 21, 22, 23, 24, 0, 25, 26, 27, 0, 28, 0, 29, 30, 31, 0, 32, 33, 34, 35, 36, 0, 37, 38, 39, 40, 41, 0, 42, 0, 43, 44, 45, 46, 47, 0, 48, 49, 50, 0, 51, 0, 52, 53, 54, 55, 56, 0, 57

Offset: 1

Reinhard Zumkeller, Dec 09 2001

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A002808, A239968, A010051, A049084.

Cf. A026233, A026238, A065855, A005171.

import Data.List (unfoldr, genericIndex)
a066246 n = genericIndex a066246_list (n - 1)
a066246_list = unfoldr x (1, 1, a002808_list) where
   x (i, z, cs'@(c:cs)) | i == c = Just (z, (i + 1, z + 1, cs))
                        | i /= c = Just (0, (i + 1, z, cs'))
-- Reinhard Zumkeller, Jan 29 2014

Module[{k=1},Table[If[CompositeQ[n],k;k++,0],{n,100}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 03 2019 *)

a(n)=if(isprime(n),0,max(0,n-primepi(n)-1)) \\ Charles R Greathouse IV, Aug 21 2011

a(n) = A239968(n) + A010051(n) - 1. - Reinhard Zumkeller, Mar 30 2014

a(n) = A065855(n)*A005171(n). - Ridouane Oudra, Jul 29 2025

A241917 If n is a prime with index i, p_i, a(n) = i, (with a(1)=0), otherwise difference (i-j) of the indices of the two largest primes p_i, p_j, i >= j in the prime factorization of n: a(n) = A061395(n) - A061395(A052126(n)).

Original entry on oeis.org

0, 1, 2, 0, 3, 1, 4, 0, 0, 2, 5, 1, 6, 3, 1, 0, 7, 0, 8, 2, 2, 4, 9, 1, 0, 5, 0, 3, 10, 1, 11, 0, 3, 6, 1, 0, 12, 7, 4, 2, 13, 2, 14, 4, 1, 8, 15, 1, 0, 0, 5, 5, 16, 0, 2, 3, 6, 9, 17, 1, 18, 10, 2, 0, 3, 3, 19, 6, 7, 1, 20, 0, 21, 11, 0, 7, 1, 4, 22, 2, 0, 12, 23

Offset: 1

Antti Karttunen, May 13 2014

nonn

Note: the two largest primes in the multiset of prime divisors of n are equal for all numbers that are in A070003, thus, after a(1)=0, A070003 gives the positions of the other zeros in this sequence.

Cf. A052126, A061395, A070003, A122111, A241909.
Cf. A049084, A027746.
Cf. A241919, A242411, A243055 for other variants.

a241917 n = i - j where
            (i:j:_) = map a049084 $ reverse (1 : a027746_row n)
-- Reinhard Zumkeller, May 15 2014

A241917(n) = if(isprime(n), primepi(n), if(1>=omega(n), 0, my(f=factor(n)); if(f[#f~,2]>1, 0, primepi(f[#f~,1])-primepi(f[(#f~)-1,1])))); \\ Antti Karttunen, Jul 10 2024

from sympy import primefactors, primepi
def a061395(n): return 0 if n==1 else primepi(primefactors(n)[-1])
def a052126(n): return 1 if n==1 else n/primefactors(n)[-1]
def a(n): return 0 if n==1 else a061395(n) - a061395(a052126(n)) # Indranil Ghosh, May 19 2017

(define (A241917 n) (- (A061395 n) (A061395 (A052126 n))))

a(n) = A061395(n) - A061395(A052126(n)).

A065381 Primes not of the form p + 2^k, p prime and k >= 0.

Original entry on oeis.org

2, 127, 149, 251, 331, 337, 373, 509, 599, 701, 757, 809, 877, 907, 977, 997, 1019, 1087, 1259, 1549, 1597, 1619, 1657, 1759, 1777, 1783, 1867, 1973, 2203, 2213, 2293, 2377, 2503, 2579, 2683, 2789, 2843, 2879, 2909, 2999, 3119, 3163, 3181, 3187, 3299

Offset: 1

Reinhard Zumkeller, Nov 03 2001

nonn

Sequence is infinite. For example, Pollack shows that numbers which are 1260327937 mod 2863311360 are not of the form p + 2^k for any prime p and k >= 0, and there are infinitely many primes in this congruence class by Dirichlet's theorem. - Charles R Greathouse IV, Jul 20 2014

			127 is a prime, 127-2^0 through 127-2^6 are all nonprimes.

T. D. Noe, Table of n, a(n) for n = 1..1000
P. Pollack, Not Always Buried Deep: Selections from Analytic and Combinatorial Number Theory, p. 193, ex. 5.1.6, p. 216ff. [?Broken link]
P. Pollack, Not Always Buried Deep: Selections from Analytic and Combinatorial Number Theory, p. 193, ex. 5.1.6, p. 216ff.
Lei Zhou, Between 2^n and primes.

Equals A000040 minus A065380.
Cf. A010051, A006285, A102630, A094076, A156695.
Cf. A098237.

a065381 n = a065381_list !! (n-1)
a065381_list = filter f a000040_list where
   f p = all ((== 0) . a010051 . (p -)) $ takeWhile (<= p) a000079_list
-- Reinhard Zumkeller, Nov 24 2011

fQ[n_] := Block[{k = Floor[Log[2, n]], p = n}, While[k > -1 && ! PrimeQ[p - 2^k], k--]; If[k > 0, True, False]]; Drop[Select[Prime[Range[536]], ! fQ[#] &], {2}] (* Robert G. Wilson v, Feb 10 2005; corrected by Arkadiusz Wesolowski, May 05 2012 *)

is(p)=my(k=1);while(kp,return(isprime(p)));0 \\ Charles R Greathouse IV, Jul 20 2014

A078687(A049084(a(n))) = 0; subsequence of A118958. - Reinhard Zumkeller, May 07 2006

Link and cross-reference fixed by Charles R Greathouse IV, Nov 09 2008

A251239 Indices of prime numbers in A098550.

Original entry on oeis.org

2, 3, 9, 15, 22, 23, 30, 43, 51, 61, 62, 79, 87, 88, 101, 114, 127, 132, 142, 153, 158, 167, 175, 194, 204, 215, 222, 233, 238, 247, 274, 283, 296, 301, 324, 329, 338, 355, 364, 375, 386, 393, 414, 423, 430, 435, 452, 479, 490, 497, 506, 523, 528, 541, 550

Offset: 1

Reinhard Zumkeller, Dec 02 2014

nonn

It is conjectured that every prime appears in A098550, and if so then A098550(a(n)) = A000040(n). [Comment edited by N. J. A. Sloane, Dec 15 2014] [It is now known that every prime appears in A098550, although it is not known that they appear in their right order. - N. J. A. Sloane, Dec 25 2014]
A010051(A098550(a(n))) = 1; A049084(A098550(a(n))) > 0.
Conjecture: a(n) = A251541(n) + 2 for n > 4. - Reinhard Zumkeller, Dec 16 2014
A253049(n) = A098550(a(n)+1). - Reinhard Zumkeller, Dec 29 2014

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

Cf. A098550, A010051, A049084, A251392, A247253 (first differences), A251595, A251541, A253049, A253048.

This is a subsequence of A251391 and A251241,

a251239 n = a251239_list !! (n-1)
a251239_list = filter ((== 1) . a010051' . a098550) [1..]

a098550[lst_List] :=
Block[{k = 4},
  While[GCD[lst[[-2]], k] == 1 || GCD[lst[[-1]], k] > 1 ||
    MemberQ[lst, k], k++]; Append[lst, k]];
a251239[n_] :=
Flatten@Position[Nest[a098550, {1, 2, 3}, n], Integer?PrimeQ]; a251239[550] (* _Michael De Vlieger, Dec 23 2014, based on Robert G. Wilson v at A098550 *)

A276079 Numbers n such that prime(k)^(k+1) divides n for some k.

Original entry on oeis.org

4, 8, 12, 16, 20, 24, 27, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 135, 136, 140, 144, 148, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 240, 243, 244, 248, 252, 256, 260, 264, 268, 270, 272

Offset: 1

Antti Karttunen, Aug 18 2016

nonn

The asymptotic density of this sequence is 1 - Product_{i>=1} 1-prime(i)^(-1-i) = 0.2789766... - Amiram Eldar, Oct 21 2020

			625 = 5*5*5*5 = prime(3)^4 so it is divisible by prime(3)^(3+1), and thus 625 is included in the sequence.

Antti Karttunen, Table of n, a(n) for n = 1..5000

Positions of nonzeros in A276077.
Complement: A276078.
Cf. A000040, A000720, A008586 (a subsequence).
Differs from its subsequence A100716 for the first time at n=175, where a(175) = 625, while that value is missing from A100716.

from sympy import primepi, isprime, primefactors, factorint
def a028234(n):
    f=factorint(n)
    minf = min(f)
    return 1 if n==1 else n//(minf**f[minf])
def a067029(n):
    f=factorint(n)
    return 0 if n==1 else f[min(f)]
def a049084(n): return primepi(n) if isprime(n) else 0
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def a(n): return 0 if n==1 else a(a028234(n)) + (1 if a067029(n) > a055396(n) else 0)
print([n for n in range(1, 301) if a(n)!=0]) # Indranil Ghosh, Jun 21 2017

A028416 Primes p such that the decimal expansion of 1/p has a periodic part of even length.

Original entry on oeis.org

7, 11, 13, 17, 19, 23, 29, 47, 59, 61, 73, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 157, 167, 179, 181, 193, 197, 211, 223, 229, 233, 241, 251, 257, 263, 269, 281, 293, 313, 331, 337, 349, 353, 367, 373, 379, 383, 389, 401, 409, 419, 421, 433

Offset: 1

Mario Velucchi (mathchess(AT)velucchi.it)

Primes whose reciprocals have even period length.
Primes p such that the order of 10 mod p is even. - Joerg Arndt, Mar 04 2014
A002371(A049084(a(n))) mod 2 == 0.
Not the same as A040121: a(33)=241 is not in A040121.
Let (d(i): 1<=i<=2*K) be the period of the decimal expansion of 1/a(n), K=A002371(A049084(a(n)))/2, then d(i) + d(i+K) = 9 for i with 1<=i<=K, or, equivalently: u + v = 10^K - 1 with u = Sum_{i=1..K} d(i)*10^(K-i) and v = Sum_{i=1..K} d(i+K)*10^(K-i). - Reinhard Zumkeller, Oct 05 2008

			From _Reinhard Zumkeller_, Oct 05 2008: (Start)
(0,5,8,8,2,3,5,2,9,4,1,1,7,6,4,7) is the period of 1/17 (see A007450),
K = A002371(A049084(17))/2 = A002371(7)/2 = 16/2 = 8,
u = 5882352, v = 94117647: u + v = 99999999 = 10^8 - 1. (End)

H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, "Die periodischen Dezimalbrueche". [Reinhard Zumkeller, Oct 05 2008]

Cf. A087000, A186635.

A028416 := proc(n) local st:
st := ithprime(n):
if (modp(numtheory[order](10,st),2) = 0) then
   RETURN(st)
fi: end:  seq(A028416(n), n=1..100); # Jani Melik, Feb 24 2011

Select[Prime[Range[4,100]],EvenQ[Length[RealDigits[1/#][[1,1]]]]&] (* Harvey P. Dale, Jul 07 2011 *)

forprime(p=7,1e3,if(znorder(Mod(10,p))%2==0,print1(p", "))) \\ Charles R Greathouse IV, Feb 24 2011

from sympy import gcd, isprime, n_order
is_A028416 = lambda n: gcd(n,10)==1 and n>5 and n_order(10, n)%2==0 and isprime(n) # M. F. Hasler, Nov 19 2024

More terms from Reinhard Zumkeller, Jul 29 2003

A081092 Primes having a prime number of 1's in their binary representation.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 47, 59, 61, 67, 73, 79, 97, 103, 107, 109, 127, 131, 137, 151, 157, 167, 173, 179, 181, 191, 193, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 271, 283, 307, 313, 331, 367, 379, 397, 409, 419, 421, 431, 433, 439, 443

Offset: 1

Reinhard Zumkeller, Mar 05 2003

Same as primes with prime binary digit sum.
Primes with prime decimal digit sum are A046704.
Sum_{a(n) < x} 1/a(n) is asymptotic to log(log(log(x))) as x -> infinity; see Harman (2012). Thus the sequence is infinite. - Jonathan Sondow, Jun 09 2012
A049084(A000120(a(n))) > 0; A081091, A000215 and A081093 are subsequences.

			15th prime = 47 = '101111' with five 1's, therefore 47 is in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
G. Harman, Counting Primes whose Sum of Digits is Prime, J. Integer Seq., 15 (2012), Article 12.2.2.

Cf. A000040, A000120, A046704, A081093.

Subsequence of A052294.

a081092 n = a081092_list !! (n-1)
a081092_list = filter ((== 1) . a010051') a052294_list
-- Reinhard Zumkeller, Nov 16 2012

q:= n-> isprime(n) and isprime(add(i,i=Bits[Split](n))):
select(q, [$1..500])[];  # Alois P. Heinz, Sep 28 2023

Clear[BinSumOddQ];BinSumPrimeQ[a_]:=Module[{i,s=0},s=0;For[i=1,i<=Length[IntegerDigits[a,2]],s+=Extract[IntegerDigits[a,2],i];i++ ];PrimeQ[s]]; lst={};Do[p=Prime[n];If[BinSumPrimeQ[p],AppendTo[lst,p]],{n,4!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 06 2009 *)
Select[Prime[Range[100]], PrimeQ[Apply[Plus, IntegerDigits[#, 2]]] &] (* Jonathan Sondow, Jun 09 2012 *)

lista(nn) = {forprime(p=2, nn, if (isprime(hammingweight(p)), print1(p, ", ")););} \\ Michel Marcus, Jan 16 2015

from sympy import isprime
def ok(n): return isprime(n.bit_count()) and isprime(n)
print([k for k in range(444) if ok(k)]) # Michael S. Branicky, Dec 27 2023

A107740 Number of numbers m such that prime(n) = m + (digit sum of m).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 0, 1, 1, 0, 1

Offset: 1

Reinhard Zumkeller, May 23 2005

a(A049084(A006378(n))) = 0; a(A049084(A048521(n))) > 0. [Corrected by Reinhard Zumkeller, Sep 27 2014]
a(n) <= 2 for n <= 10^5. Conjecture: sequence is bounded.
I would rather conjecture the opposite. Of course a(n) >= m implies n >= A006064(m), having more than A230857(m) digits, i.e., 14, 25 and 1111111111125 digits of n, for a(n) = 3, 4, 5. - M. F. Hasler, Nov 09 2018

			A000040(26) = 101 = 91 + (9 + 1) = 100 + (1 + 0 + 0): a(26) = # {91, 100} = 2.

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

Cf. A007953, A000040, A062028, A047791, A107741.

Cf. A006378, A048521, A049084, A055642, A062028.

a107740 n = length [() | let p = a000040 n,
                         m <- [max 0 (p - 9 * a055642 p) .. p - 1],
                         a062028 m == p]
-- Reinhard Zumkeller, Sep 27 2014

Table[p=Prime[n];c=0;i=1;While[iJayanta Basu, May 03 2013 *)

apply( A107740(n)=A230093(prime(n)), [1..150]) \\ M. F. Hasler, Nov 08 2018

a(n) = A230093(prime(n)), i.e.: A107740 = A230093 o A000040. - M. F. Hasler, Nov 08 2018

A108548 Fully multiplicative with a(prime(j)) = A108546(j), where A108546 is the lexicographically earliest permutation of primes such that after 2 the forms 4k+1 and 4k+3 alternate, and prime(j) is the j-th prime in A000040.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 12, 11, 14, 15, 16, 17, 18, 19, 20, 21, 26, 29, 24, 25, 22, 27, 28, 23, 30, 37, 32, 39, 34, 35, 36, 31, 38, 33, 40, 41, 42, 43, 52, 45, 58, 53, 48, 49, 50, 51, 44, 47, 54, 65, 56, 57, 46, 61, 60, 59, 74, 63, 64, 55, 78, 73, 68, 87, 70, 67, 72

Offset: 1

Reinhard Zumkeller, Jun 10 2005

Multiplicative with a(2^e) = 2^e, else if p is the m-th prime then a(p^e) = q^e where q is the m/2-th prime of the form 4*k + 3 (A002145) for even m and a(p^e) = r^e where r is the (m-1)/2-th prime of the form 4*k + 1 (A002144) for odd m. - David A. Corneth, Apr 25 2022

Permutation of the natural numbers with fixed points A108549: a(A108549(n)) = A108549(n).

Antti Karttunen, Table of n, a(n) for n = 1..26927
Index entries for sequences that are permutations of the natural numbers

Cf. A002144, A002145, A049084, A108546, A108549 (fixed points), A332808 (inverse permutation).
Cf. also A332815, A332817 (this permutation applied to Doudna tree and its mirror image), also A332818, A332819.
Cf. also A267099, A332212 and A348746 for other similar mappings.

terms = 72;
A111745 = Module[{prs = Prime[Range[2 terms]], m3, m1, min},
     m3 = Select[prs, Mod[#, 4] == 3&];
     m1 = Select[prs, Mod[#, 4] == 1&];
     min = Min[Length[m1], Length[m3]];
     Riffle[Take[m3, min], Take[m1, min]]];
A108546[n_] := If[n == 1, 2, A111745[[n - 1]]];
A049084[n_] := PrimePi[n]*Boole[PrimeQ[n]];
a[n_] := If[n == 1, 1, Module[{p, e}, Product[{p, e} = pe; A108546[A049084[p]]^e, {pe, FactorInteger[n]}]]];
Array[a, terms] (* Jean-François Alcover, Nov 19 2021, using Harvey P. Dale's code for A111745 *)

up_to = 26927; \\ One of the prime fixed points.
A108546list(up_to) = { my(v=vector(up_to), p,q); v[1] = 2; v[2] = 3; v[3] = 5; for(n=4,up_to, p = v[n-2]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[n] = q); (v); };
v108546 = A108546list(up_to);
A108546(n) = v108546[n];
A108548(n) = { my(f=factor(n)); f[,1] = apply(A108546,apply(primepi,f[,1])); factorback(f); }; \\ Antti Karttunen, Apr 25 2022

Name edited by Antti Karttunen, Apr 25 2022

cp's OEIS Frontend

A039703 a(n) = n-th prime modulo 5.

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A066246 a(n) = 0 unless n is a composite number A002808(k) then a(n) = k.

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A241917 If n is a prime with index i, p_i, a(n) = i, (with a(1)=0), otherwise difference (i-j) of the indices of the two largest primes p_i, p_j, i >= j in the prime factorization of n: a(n) = A061395(n) - A061395(A052126(n)).

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A065381 Primes not of the form p + 2^k, p prime and k >= 0.

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A251239 Indices of prime numbers in A098550.

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A276079 Numbers n such that prime(k)^(k+1) divides n for some k.

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A028416 Primes p such that the decimal expansion of 1/p has a periodic part of even length.

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A108548 Fully multiplicative with a(prime(j)) = A108546(j), where A108546 is the lexicographically earliest permutation of primes such that after 2 the forms 4k+1 and 4k+3 alternate, and prime(j) is the j-th prime in A000040.