A072055 -id:A072055 - cp's OEIS frontend

A165286 Primes which are the sum of two or more consecutive terms of A072055.

23, 61, 83, 101, 139, 157, 197, 263, 283, 293, 331, 349, 367, 383, 401, 449, 491, 509, 541, 569, 571, 577, 641, 659, 661, 701, 751, 773, 809, 811, 853, 859, 881, 887, 967, 977, 997, 1009, 1039, 1069, 1117, 1123, 1181, 1193, 1217, 1237, 1279, 1283, 1361

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 13 2009

nonn

The terms in A072055 are odd, so the count of terms in the sum must be odd to yield a prime.

			a(1)= 5+7+11 = sum_{i=1..3} A072055(i) = 23.
a(2) = sum_{i=1..5} A072055(i) = 5+7+11+15+23 = 61. a(3) = 7+11+15+23+27 = 83.

Cf. A165285

lst={};Do[s=2*Prime[m]+1;Do[p=Prime[n];s+=(2*p+1);If[PrimeQ[s], If[s<=6949, AppendTo[lst,s]]],{n,m+1,6!}],{m,1,6!}];lst=Take[Union@lst,200]
Module[{nn=100,terms},terms=2*Prime[Range[nn]]+1;Take[Union[Select[ Flatten[ Table[Total/@Partition[terms,n,1],{n,3,nn,2}]],PrimeQ]],nn/2]] (* Harvey P. Dale, Jun 11 2015 *)

Redefined in terms of A072055; cross-references to unrelated sequences deleted - R. J. Mathar, Sep 16 2009

A005385 Safe primes p: (p-1)/2 is also prime.

Original entry on oeis.org

5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2579, 2819, 2879, 2903, 2963

Offset: 1

N. J. A. Sloane

Then (p-1)/2 is called a Sophie Germain prime: see A005384.
Or, primes of the form 2p+1 where p is prime.
Primes p such that denominator(Bernoulli(p-1) + 1/p) = 6. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 10 2004
Primes p such that p-1 is a semiprime. - Zak Seidov, Jul 01 2005
A156659(a(n)) = 1; A156875 gives numbers of safe primes <= n. - Reinhard Zumkeller, Feb 18 2009
From Daniel Forgues, Jul 31 2009: (Start)
A safe prime p is 7 or of the form 6k-1, k >= 1, i.e., p == 5 (mod 6).
A prime p of the form 6k+1, k >= 2, i.e., p = 1 (mod 6), cannot be a safe prime since (p-1)/2 is composite and divisible by 3. (End)
If k is the product of the n-th safe prime p and its corresponding Sophie Germain prime (p-1)/2, then a(n) = 2(k-phi(k))/3 + 1, where phi is Euler's totient function. - Wesley Ivan Hurt, Oct 03 2013
From Bob Selcoe, Apr 14 2014: (Start)
When the n-th prime is divided by all primes up to the (n-1)-th prime, safe primes (p) have remainders of 1 when divided by 2 and (p-1)/2 and no other primes. That is, p(mod j)=1 iff j={2,(p-1)/2}; p>j, {p,j}=>prime.  Explanation: Generally, x(mod y)=1 iff x=y'+1, where y' is the set of divisors of y, y'>1. Since safe primes (p) are of the form p(mod j)=1 iff p and j are prime, then j={j'}. That is, since j is prime, there are no divisors of j (greater than 1) other than j.  Therefore, no primes other than j exist which satisfy the equation p(mod j)=1.
Except primes of the form 2^n+1 (n>=0), all non-safe primes (p') will have at least one prime (p") greater than 2 and less than (p-1)/2 such that p'(mod p")=1. Explanation: Non-safe primes (p') are of the form p'(mod k)=1 where k is composite. This means prime divisors of k exist, and p" is the set of prime divisors of k (example p'=89: k=44; p"={2,11}).  The exception applies because p"={2} iff p'=2^n+1.
Refer to the rows in triangle A207409 for illustration and further explanation. (End)
Conjecture: there is a strengthening of the Bertrand postulate for n >= 24: the interval (n, 2*n) contains a safe prime. It has been tested by Peter J. C. Moses up to n = 10^7. - Vladimir Shevelev, Jul 06 2015
The six known safe primes p such that (p-1)/2 is a Fibonacci prime are in A263880. - Jonathan Sondow, Nov 04 2015
The only term in common with A005383 is 5. - Zak Seidov, Dec 31 2015
From the fourth entry onward, do these correspond to Smarandache's problem 34 (see A007931 link), specifically values which cannot be used (do not meet conditions) to confirm the conjecture? - Bill McEachen, Sep 29 2016
Primes p with the property that there is a prime q such that p+q^2 is a square. - Zak Seidov, Feb 16 2017
It is conjectured that there are infinitely many safe primes, and their estimated asymptotic density ~ 2C/(log n)^2 (where C = 0.66... is the twin prime constant A005597) converges to the actual value as far as we know. - M. F. Hasler, Jun 14 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
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
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
A. R. Ashrafi and F. Koorepazan-Moftakhar, Towards the Classification of Finite Simple Groups with exactly Three or Four Supercharacter Theories, arXiv preprint arXiv:1605.08971 [math.GR], 2016.
R. P. Boas & N. J. A. Sloane, Correspondence, 1974.
Siji Chen and Sheng Chen, Connectedness of digraphs from quadratic polynomials, Involve (2020) Vol. 13, No. 2, 357-360.
Benoit Cloitre, On the fractal behavior of primes, 2011.
L. H. Gallardo and O. Rahavandrainy, There are finitely many even perfect polynomials over F_p with p+1 irreducible divisors, Acta Mathematica Universitatis Comenianae, Vol. 83, No. 2, 2016, 261-275.
David Naccache, Double-Speed Safe Prime Generation, IACR, Report 2003/175, 2003.
Planetmath, Safe prime.
Michael J. Wiener, Safe Prime Generation with a Combined Sieve, IACR, Report 2003/186, 2003.
Wikipedia, Safe prime.

Cf. A007700, A023272, A023302, A023330, A057331, A005602, A207409, A263880.
Except for the initial term, this is identical to A079148.
Cf. A005384, A005383.
Subsequence of A088707.
Primes in A072055.

a005385 n = a005385_list !! (n-1)
a005385_list = filter ((== 1) . a010051 . (`div` 2)) a000040_list
-- Reinhard Zumkeller, Sep 18 2011

[p: p in PrimesUpTo(3000) | IsPrime((p-1) div 2)]; // Vincenzo Librandi, Jul 06 2015

with(numtheory); [ seq(safeprime(i),i=1..3000) ]: convert(%,set); convert(%,list); sort(%);
A005385_list := n->select(i->isprime(iquo(i,2)),select(i->isprime(i),[$1..n])): # Peter Luschny, Nov 08 2010

Select[Prime[Range[1000]],PrimeQ[(#-1)/2]&] (* Zak Seidov, Jan 26 2011 *)

g(n) = forprime(x=2,n,y=x+x+1;if(isprime(y),print1(y","))) \\ Cino Hilliard, Sep 12 2004

[x|x<-primes(10^3), bigomega(x-1)==2] \\ Altug Alkan, Nov 04 2015

from sympy import isprime, primerange
def aupto(limit):
  alst = []
  for p in primerange(1, limit+1):
    if isprime((p-1)//2): alst.append(p)
  return alst
print(aupto(2963)) # Michael S. Branicky, May 07 2021

a(n) = 2 * A005384(n) + 1.

More terms from Larry Reeves (larryr(AT)acm.org), Feb 15 2001

A286623 Square array A(n,k) = A276943(n,k)/A002110(n-1), read by descending antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

1, 3, 1, 4, 4, 1, 5, 6, 6, 1, 7, 7, 10, 8, 1, 9, 16, 11, 14, 12, 1, 10, 19, 36, 15, 22, 14, 1, 11, 21, 41, 78, 23, 26, 18, 1, 13, 22, 45, 85, 144, 27, 34, 20, 1, 15, 31, 46, 91, 155, 222, 35, 38, 24, 1, 16, 34, 71, 92, 165, 235, 324, 39, 46, 30, 1, 17, 36, 76, 155, 166, 247, 341, 438, 47, 58, 32, 1, 18, 37, 80, 162, 287, 248, 357, 457, 668, 59, 62, 38, 1

Offset: 1

Antti Karttunen, Jun 28 2017

			The top left 12 X 12 corner of the array:
  1,  3,  4,  5,    7,    9,   10,   11,   13,   15,   16,   17
  1,  4,  6,  7,   16,   19,   21,   22,   31,   34,   36,   37
  1,  6, 10, 11,   36,   41,   45,   46,   71,   76,   80,   81
  1,  8, 14, 15,   78,   85,   91,   92,  155,  162,  168,  169
  1, 12, 22, 23,  144,  155,  165,  166,  287,  298,  308,  309
  1, 14, 26, 27,  222,  235,  247,  248,  443,  456,  468,  469
  1, 18, 34, 35,  324,  341,  357,  358,  647,  664,  680,  681
  1, 20, 38, 39,  438,  457,  475,  476,  875,  894,  912,  913
  1, 24, 46, 47,  668,  691,  713,  714, 1335, 1358, 1380, 1381
  1, 30, 58, 59,  900,  929,  957,  958, 1799, 1828, 1856, 1857
  1, 32, 62, 63, 1148, 1179, 1209, 1210, 2295, 2326, 2356, 2357
  1, 38, 74, 75, 1518, 1555, 1591, 1592, 3035, 3072, 3108, 3109

Transpose: A286625.
Cf. A002110, A276943.
Row 1: A276155.
Column 1: A000012, Column 2: A008864, Column 3: A100484, Column 4: A072055, Column 5: A023523 (from its second term onward), Column 6: A286624 (= 1 + A123134), Column 11: 2*A123134, Column 13: 3*A006094.
Cf. A276616 (analogous array).

(define (A286623 n) (A286623bi (A002260 n) (A004736 n)))
(define (A286623bi row col) (/ (A276943bi row col) (A002110 (- row 1))))

A(n,k) = A276943(n, k) / A002110(n-1).

A286625 Square array A(n,k) = A276945(n,k)/A002110(k-1), read by descending antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

1, 1, 3, 1, 4, 4, 1, 6, 6, 5, 1, 8, 10, 7, 7, 1, 12, 14, 11, 16, 9, 1, 14, 22, 15, 36, 19, 10, 1, 18, 26, 23, 78, 41, 21, 11, 1, 20, 34, 27, 144, 85, 45, 22, 13, 1, 24, 38, 35, 222, 155, 91, 46, 31, 15, 1, 30, 46, 39, 324, 235, 165, 92, 71, 34, 16, 1, 32, 58, 47, 438, 341, 247, 166, 155, 76, 36, 17, 1, 38, 62, 59, 668, 457, 357, 248, 287, 162, 80, 37, 18

Offset: 1

Antti Karttunen, Jun 28 2017

			The top left 12 X 12 corner of the array:
   1,  1,  1,   1,   1,   1,   1,   1,    1,    1,    1,    1
   3,  4,  6,   8,  12,  14,  18,  20,   24,   30,   32,   38
   4,  6, 10,  14,  22,  26,  34,  38,   46,   58,   62,   74
   5,  7, 11,  15,  23,  27,  35,  39,   47,   59,   63,   75
   7, 16, 36,  78, 144, 222, 324, 438,  668,  900, 1148, 1518
   9, 19, 41,  85, 155, 235, 341, 457,  691,  929, 1179, 1555
  10, 21, 45,  91, 165, 247, 357, 475,  713,  957, 1209, 1591
  11, 22, 46,  92, 166, 248, 358, 476,  714,  958, 1210, 1592
  13, 31, 71, 155, 287, 443, 647, 875, 1335, 1799, 2295, 3035
  15, 34, 76, 162, 298, 456, 664, 894, 1358, 1828, 2326, 3072
  16, 36, 80, 168, 308, 468, 680, 912, 1380, 1856, 2356, 3108
  17, 37, 81, 169, 309, 469, 681, 913, 1381, 1857, 2357, 3109

Transpose: A286623.
Cf. A002110, A276945.
Column 1: A276155.
Row 1: A000012, Row 2: A008864, Row 3: A100484, Row 4: A072055, Row 5: A023523 (from its second term onward), Row 6: A286624.
Cf. A276617 (analogous array).

(define (A286625 n) (A286625bi (A002260 n) (A004736 n)))
(define (A286625bi row col) (/ (A276945bi row col) (A002110 (- col 1))))

A(n,k) = A276945(n, k) / A002110(k-1).

A089241 Even numbers k such that k/2 - 1 is prime.

Original entry on oeis.org

6, 8, 12, 16, 24, 28, 36, 40, 48, 60, 64, 76, 84, 88, 96, 108, 120, 124, 136, 144, 148, 160, 168, 180, 196, 204, 208, 216, 220, 228, 256, 264, 276, 280, 300, 304, 316, 328, 336, 348, 360, 364, 384, 388, 396, 400, 424, 448, 456, 460, 468, 480, 484

Offset: 1

Giovanni Teofilatto, Dec 22 2003

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

a(n) = A072055(n) + 1.

a(n) = A029933(A000040(n)).

[2*NthPrime(n)+2: n in [1..60]]; // Vincenzo Librandi, Jan 09 2018

2Prime[Range[100]] + 2 (* Alonso del Arte, Mar 06 2017 *)

a(n)=2*prime(n) + 2 \\ Charles R Greathouse IV, Mar 06 2017

a(n) = 2*prime(n) + 2. - Alonso del Arte, Mar 02 2017

Corrected (62 replaced by 64, 202 replaced by 204, 314 by 316, 386 by 388) by R. J. Mathar, Apr 22 2010

Offset corrected by Arkadiusz Wesolowski, Aug 09 2011

A278230 Least number with the prime signature of 2*prime(n) + 1.

Original entry on oeis.org

2, 2, 2, 6, 2, 8, 6, 6, 2, 2, 12, 12, 2, 6, 6, 2, 6, 6, 24, 6, 12, 6, 2, 2, 30, 6, 12, 6, 6, 2, 30, 2, 12, 12, 6, 6, 60, 6, 6, 2, 2, 12, 2, 12, 6, 30, 12, 6, 30, 24, 2, 2, 30, 2, 6, 6, 12, 6, 30, 2, 48, 2, 30, 6, 30, 6, 30, 72, 6, 6, 6, 2, 60, 12, 30, 6, 6, 30, 6, 60, 2, 6, 2, 12, 6, 2, 6, 30, 6, 12, 30, 6, 60, 2, 24, 6, 2, 6, 6, 12, 30, 6, 12, 6, 12, 210, 12

Offset: 1

Antti Karttunen, Nov 19 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000040, A046523, A072055.
Cf. A072192 (positions of 2's), A005384.
Cf. also A278227, A278228, A278229.

Table[Times @@ MapIndexed[(Prime@ First@ #2)^#1 &, #] &@ If[Length@ # == 1 && #[[1, 1]] == 1, {0}, Reverse@ Sort@ #[[All, -1]]] &@ FactorInteger[ 2 Prime@ n + 1], {n, 120}] (* Michael De Vlieger, Nov 21 2016 *)

(define (A278230 n) (A046523 (+ 1 (* 2 (A000040 n)))))

a(n) = A046523(A072055(n)) = A046523(1 + 2*A000040(n)).

A023589 a(n) is the number of distinct prime divisors of 2*prime(n)+1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 1, 3, 2, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 3, 2, 2, 1, 1, 2, 1, 2, 2, 3, 2, 2, 3, 2, 1, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 3, 2, 3, 2, 3, 2, 2, 2, 2, 1, 3, 2, 3, 2, 2, 3, 2, 3, 1, 2, 1, 2, 2, 1, 2, 3, 2, 2, 3, 2, 3, 1, 2, 2, 1, 2, 2

Offset: 1

Clark Kimberling

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001221, A072055.

Array[PrimeNu[2 Prime[#] + 1] &, 99] (* Michael De Vlieger, Jul 09 2022 *)

a(n) = omega(2*prime(n)+1) \\ Hugo Pfoertner, Jul 09 2022

A023590 Greatest prime divisor of 2*prime(n)+1.

Original entry on oeis.org

5, 7, 11, 5, 23, 3, 7, 13, 47, 59, 7, 5, 83, 29, 19, 107, 17, 41, 5, 13, 7, 53, 167, 179, 13, 29, 23, 43, 73, 227, 17, 263, 11, 31, 23, 101, 7, 109, 67, 347, 359, 11, 383, 43, 79, 19, 47, 149, 13, 17, 467, 479, 23, 503, 103, 31, 11, 181, 37, 563, 7, 587, 41, 89

Offset: 1

Clark Kimberling

nonn

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

Cf. A006530, A072055.

Table[FactorInteger[2Prime[n]+1][[-1,1]],{n,70}] (* Harvey P. Dale, Jun 22 2013 *)

a(n) = vecmax(factor(2*prime(n)+1)[,1]); \\ Michel Marcus, Aug 08 2020

a(n) = A006530(A072055(n)). - Amiram Eldar, Aug 08 2020

A072057 Sum of divisors of 2*prime(n)+1.

Original entry on oeis.org

6, 8, 12, 24, 24, 40, 48, 56, 48, 60, 104, 124, 84, 120, 120, 108, 144, 168, 240, 168, 228, 216, 168, 180, 336, 240, 312, 264, 296, 228, 432, 264, 372, 416, 336, 408, 624, 440, 408, 348, 360, 532, 384, 572, 480, 640, 624, 600, 672, 720, 468

Offset: 1

Reinhard Zumkeller, Jun 11 2002

nonn

			Divisors of A072055(8) = 2*A000040(8)+1 = 2*19+1 = 39: {1,3,13,39}, therefore a(8) = 1+3+13+39 = 56.

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

Cf. A000040, A000203, A072055, A072056.

a072057 = a000203 . a072055  -- Reinhard Zumkeller, Oct 10 2013

DivisorSigma[1,#]&/@(2Prime[Range[60]]+1) (* Harvey P. Dale, Sep 17 2013 *)

a(n) = sigma(2*prime(n)+1); \\ Amiram Eldar, Apr 26 2024

lista(pmax) = forprime(p = 2, pmax, print1(sigma(2*p+1), ", ")); \\ Amiram Eldar, Apr 26 2024

a(n) = A000203(A072055(n)).

A072060 Smallest prime p such that 2*p+1 has n prime factors (with multiplicity).

Original entry on oeis.org

2, 7, 13, 67, 283, 607, 1093, 12757, 22963, 49207, 482233, 1623847, 797161, 14614627, 19929037, 35872267, 150663523, 2044719247, 1355971711, 9879222469, 37773497677, 200490103057, 172595827849, 235357947067, 1176789735337, 5507375961379, 10591107618037

Offset: 1

Reinhard Zumkeller, Jun 11 2002

nonn

			a(4)=67=A000040(19): 2*67+1 = 135 = 5*3^3, 4 factors.

Daniel Suteu, Table of n, a(n) for n = 1..500

Cf. A001222, A023595, A072055, A072059.

generate(A, B, n) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p,ceil(A/m)), B\m, my(t=m*q); if(isprime((t-1)/2), listput(list, (t-1)/2))), forprime(q = p, sqrtnint(B\m, n), my(t=m*q); list=concat(list, f(t, q, n-1)))); list); vecsort(Vec(f(1, 3, n)));
a(n) = my(x=2^n, y=2*x); while(1, my(v=generate(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Feb 05 2023

More terms from Don Reble, Apr 15 2003

cp's OEIS Frontend

A165286 Primes which are the sum of two or more consecutive terms of A072055.

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A005385 Safe primes p: (p-1)/2 is also prime.

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A286623 Square array A(n,k) = A276943(n,k)/A002110(n-1), read by descending antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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A286625 Square array A(n,k) = A276945(n,k)/A002110(k-1), read by descending antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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A089241 Even numbers k such that k/2 - 1 is prime.

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Magma

Mathematica

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A278230 Least number with the prime signature of 2*prime(n) + 1.

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A023589 a(n) is the number of distinct prime divisors of 2*prime(n)+1.

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