A151799 -id:A151799 - cp's OEIS frontend

A373149 Fully additive with a(2) = 1 and a(p) = prevprime(p) for odd primes p.

0, 1, 2, 2, 3, 3, 5, 3, 4, 4, 7, 4, 11, 6, 5, 4, 13, 5, 17, 5, 7, 8, 19, 5, 6, 12, 6, 7, 23, 6, 29, 5, 9, 14, 8, 6, 31, 18, 13, 6, 37, 8, 41, 9, 7, 20, 43, 6, 10, 7, 15, 13, 47, 7, 10, 8, 19, 24, 53, 7, 59, 30, 9, 6, 14, 10, 61, 15, 21, 9, 67, 7, 71, 32, 8, 19, 12, 14, 73, 7, 8, 38, 79, 9, 16, 42, 25, 10, 83, 8, 16

Offset: 1

Antti Karttunen, Jun 02 2024

nonn

Cf. A064989, A151799.

Cf. also A001414, A059975, A276085.

A373149[n_] := If[n == 1, 0, Total[MapApply[#2*If[# == 2, 1, NextPrime[#, -1]] &, FactorInteger[n]]]];
Array[A373149, 100] (* Paolo Xausa, Dec 17 2024 *)

A373149(n) = { my(f = factor(n)); sum(i = 1, #f~, f[i, 2]*if(2==f[i, 1], 1, precprime(f[i, 1] - 1))); };

Additive with a(p^e) = e*A064989(p).

A378363 Greatest number <= n that is 1 or not a perfect-power.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 7, 7, 10, 11, 12, 13, 14, 15, 15, 17, 18, 19, 20, 21, 22, 23, 24, 24, 26, 26, 28, 29, 30, 31, 31, 33, 34, 35, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 63, 65, 66, 67

Offset: 1

Gus Wiseman, Nov 24 2024

Perfect-powers (A001597) are 1 and numbers with a proper integer root, complement A007916.

			In the non-perfect-powers ... 5, 6, 7, 10, 11 ... the greatest term <= 8 is 7, so a(8) = 7.

The union is A007916, complement A001597.
The version for prime numbers is A007917 or A151799, opposite A159477.
The version for prime-powers is A031218, opposite A000015.
The version for squarefree numbers is A067535, opposite A070321.
The version for perfect-powers is A081676, opposite A377468.
The version for composite numbers is A179278, opposite A113646.
Terms appearing multiple times are A375704, opposite A375703.
The run-lengths are A375706.
Terms appearing only once are A375739, opposite A375738.
The version for nonsquarefree numbers is A378033, opposite A120327.
The opposite version is A378358.
Subtracting n gives A378364, opposite A378357.
The version for non-prime-powers is A378367 (subtracted A378371), opposite A378372.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289.
A007916 lists the non-perfect-powers, differences A375706.
A069623 counts perfect-powers <= n.
A076411 counts perfect-powers < n.
A131605 lists perfect-powers that are not prime-powers.
A377432 counts perfect-powers between primes, zeros A377436.
Cf. A007918, A014210, A014234, A045542, A052410, A065514, A076412, A162966, A188951, A345531.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[NestWhile[#-1&,n,#>1&&perpowQ[#]&],{n,100}]

from sympy import mobius, integer_nthroot
def A378363(n):
    def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    a = n-f(n)
    m, k = a, f(a)+a
    while m != k: m, k = k, f(k)+a
    return m # Chai Wah Wu, Nov 26 2024

A136548 a(n) = max {k >= 1 | sigma(k) <= n}.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 5, 7, 7, 7, 7, 11, 11, 13, 13, 13, 13, 17, 17, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 29, 29, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 41, 41, 43, 43, 43, 43, 47, 47, 47, 47, 47, 47, 53, 53, 53, 53, 53, 53, 59, 59, 61, 61, 61, 61, 61, 61, 67, 67, 67, 67, 71, 71

Offset: 1

Roger L. Bagula, Mar 26 2008

Old name was "Extended 'previous prime' version 2".
This is the same as A151799 if n >= 3 and falls back to 1, if no prime smaller than n exists.
a(n+1) is the largest number k such that A007955(k) <= n, where A007955 is the product of divisors. - Jaroslav Krizek, Apr 01 2010
For every k >= 1, the equation n - a(n) = k has infinitely many solutions. - Bernard Schott, Mar 05 2019

P. Tauvel, Exercices d'Algèbre Générale et d'Arithmétique, Dunod, 2004, Exercice 18 p. 204.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

Cf. A000203 (sigma), A000142, A006530, A151799.

A136548[1]:= 1; A136548[2]:= 1; A136548[n_]:= Prime[PrimePi[n-1]]; Array[A136548,50] (* Enrique Pérez Herrero, Jul 23 2011 *)

a(n) = A006530(A000142(n-1)). - Michel Marcus, Jun 20 2014

For n > 1, a(n) < n. If p is prime, a(p+1) = p. - Bernard Schott, Mar 05 2019

Definition clarified by N. J. A. Sloane, Mar 14 2019 based on a suggestion from Jaroslav Krizek, Mar 01 2010.

A245396 Largest prime not exceeding prime(n)^(1 + 1/n).

Original entry on oeis.org

3, 5, 7, 11, 17, 19, 23, 23, 31, 37, 41, 47, 53, 53, 59, 67, 73, 73, 83, 83, 89, 89, 97, 107, 113, 113, 113, 113, 127, 131, 139, 151, 157, 157, 167, 173, 179, 181, 181, 193, 199, 199, 211, 211, 211, 223, 233, 241, 251, 251, 257, 263, 263, 277, 283, 283, 293, 293, 293, 307, 307, 317, 331, 337

Offset: 1

M. F. Hasler, Nov 03 2014

nonn

Firoozbakht's conjecture, prime(n+1) < prime(n)^(1 + 1/n), is equivalent to a(n) > prime(n). See also A182134.
Here prime(n) = A000040(n). The conjecture is also equivalent to a(n) - prime(n) >= A001223(n), the n-th gap between primes. See also A246778(n) = floor(prime(n)^(1 + 1/n)) - prime(n).
It is also conjectured that the equality a(n) - prime(n) = A001223(n) holds only for n in the set {1, 2, 3, 4, 8}, see A246782. a(n) is also largest prime less than prime(n)^(1 + 1/n), since prime(n)^(1 + 1/n) is never prime. - Farideh Firoozbakht, Nov 03 2014
a(n) = A007917(A249669(n)) = A244365(n,A182134(n)) = A006530(A245722(n)). - Reinhard Zumkeller, Nov 18 2014

A. Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015
A. Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, J. Int. Seq. 18 (2015) 15.11.2
Nilotpal Kanti Sinha, On a new property of primes that leads to a generalization of Cramer's conjecture, arXiv:1010.1399 [math.NT]
Wikipedia, Firoozbakht's conjecture

Cf. A000040, A001223, A007917, A182134, A246778, A249669.

Cf. A244365.

a245396 n = a244365 n (a182134 n)  -- Reinhard Zumkeller, Nov 16 2014

seq(prevprime(ceil(ithprime(n)^(1+1/n))),n=1..100); # Robert Israel, Nov 03 2014

Table[NextPrime[Prime[n]^(1 + 1/n), -1], {n, 64}] (* Farideh Firoozbakht, Nov 03 2014 *)

PARI
```
a(n)=precprime(prime(n)^(1+1/n))
```

a(n)=precprime(sqrtnint(prime(n)^(n+1),n)) \\ Charles R Greathouse IV, Oct 29 2018

A245396 = A007917 o A249669, i.e., a(n) = A007917(A249669(n)). Although one could say "less than" in the definition of this sequence, one cannot use A151799 in this formula because for n = 2 and n = 4, one has a(n) = A249669(n).

A285703 a(n) = A000203(A064216(n)).

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 12, 12, 14, 18, 18, 20, 13, 15, 24, 30, 24, 24, 32, 36, 38, 42, 28, 44, 31, 42, 48, 32, 54, 54, 60, 42, 48, 62, 60, 68, 72, 39, 48, 74, 31, 80, 56, 72, 84, 72, 90, 72, 90, 56, 98, 102, 72, 104, 108, 96, 110, 80, 84, 84, 57, 114, 40, 114, 126, 128, 108, 60, 132, 138, 132, 96, 96, 93, 140, 150, 98, 120, 152, 144, 120, 158, 96, 164, 133, 126

Offset: 1

Antti Karttunen, Apr 26 2017

nonn

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

Cf. A000203, A064216, A099774, A151799, A285702, A285704, A285705.

Table[DivisorSigma[1, #] &@ If[n == 1, 1, Apply[Times, FactorInteger[2 n - 1] /. {p_, e_} /; p > 2 :> NextPrime[p, -1]^e]], {n, 86}] (* Michael De Vlieger, Apr 26 2017 *)

(define (A285703 n) (A000203 (A064216 n)))

a(n) = A000203(A064216(n)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = Product_{p prime} (p^3/((p+1)*(p^2-q(p)))) = 0.8168476756..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

A013638 a(n) = prevprime(n)*nextprime(n).

Original entry on oeis.org

10, 15, 21, 35, 55, 77, 77, 77, 91, 143, 187, 221, 221, 221, 247, 323, 391, 437, 437, 437, 551, 667, 667, 667, 667, 667, 713, 899, 1073, 1147, 1147, 1147, 1147, 1147, 1271, 1517, 1517, 1517, 1591, 1763, 1927

Offset: 3

N. J. A. Sloane

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 3..10000

Cf. A151799, A151800, A030664.

a013638 n = a151799 n * a151800 n  -- Reinhard Zumkeller, May 22 2015

[ seq(prevprime(i)*nextprime(i),i=3..70) ];

a[n_] := NextPrime[n, -1] NextPrime[n];
Table[a[n], {n, 3, 50}] (* Jean-François Alcover, Aug 02 2018 *)

a(n) = A151799(n)*A151800(n). - Reinhard Zumkeller, May 22 2015

A030460 Previous prime concatenated with this prime p is a prime.

Original entry on oeis.org

3, 37, 89, 157, 163, 173, 211, 239, 257, 263, 269, 277, 337, 359, 379, 439, 479, 521, 541, 547, 607, 659, 673, 683, 733, 947, 977, 1019, 1039, 1187, 1193, 1213, 1229, 1277, 1373, 1459, 1471, 1663, 1693, 1721, 1747, 1867, 1979, 2081, 2179

Offset: 1

Patrick De Geest

Terms 157, 257, 263, 541, 1187, 1459, 2179 also belong to A030459. [Carmine Suriano, Jan 27 2011]

Harvey P. Dale, Table of n, a(n) for n = 1..10000 [extending M.F. Hasler's prior b-File beyond n=4110]

Transpose[Select[Partition[Prime[Range[400]],2,1], PrimeQ[FromDigits[ Flatten[IntegerDigits/@#]]]&]][[2]] (* Harvey P. Dale, Dec 23 2011 *)

c=0; o=2; s=1; forprime(p=3,default(primelimit), p>s & s*=10; isprime(o*s+o=p) & write("b030460.txt",c++," ",p))  \\ M. F. Hasler, Feb 06 2011

A030459(n) = A151799(a(n)). - M. F. Hasler, Feb 06 2011

A030461(n) = concat(A030459(n),a(n)) = A045533( A000720( A030459(n))). - M. F. Hasler, Feb 06 2011

Offset changed from 0 to 1 by M. F. Hasler, Feb 06 2011

A054805 Second term of strong prime quartets: prime(m+1)-prime(m) > prime(m+2)-prime(m+1) > prime(m+3)-prime(m+2).

Original entry on oeis.org

37, 67, 97, 223, 277, 307, 457, 479, 613, 631, 719, 751, 853, 877, 929, 1087, 1297, 1423, 1447, 1471, 1543, 1657, 1663, 1693, 1733, 1777, 1783, 1847, 1861, 1867, 1987, 1993, 2053, 2137, 2333, 2371, 2377, 2459, 2467, 2503, 2521, 2531, 2579, 2609, 2647

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

Second member of pairs of consecutive primes in A051634 (strong primes). - M. F. Hasler, Oct 27 2018

M. F. Hasler, Table of n, a(n) for n = 1..10000, Oct 27 2018

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quartets (= consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartet, quintet, sextet; A054819 .. A054840: members of weak prime quartet, quintet, sextet, septets.

a(n) = nextprime(A054804(n))= prevprime(A054806(n)), nextprime = A151800, prevprime = A151799. - M. F. Hasler, Oct 27 2018

Offset corrected to 1 by M. F. Hasler, Oct 27 2018

Definition clarified by N. J. A. Sloane, Aug 28 2021

A285705 a(n) = 2n - A285703(n) = 2n - A000203(A064216(n)).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 2, 4, 4, 2, 4, 4, 13, 13, 6, 2, 10, 12, 6, 4, 4, 2, 18, 4, 19, 10, 6, 24, 4, 6, 2, 22, 18, 6, 10, 4, 2, 37, 30, 6, 51, 4, 30, 16, 6, 20, 4, 24, 8, 44, 4, 2, 34, 4, 2, 16, 4, 36, 34, 36, 65, 10, 86, 14, 4, 4, 26, 76, 6, 2, 10, 48, 50, 55, 10, 2, 56, 36, 6, 16, 42, 6, 70, 4, 37, 46, 6, 98, 16, 6, 2, 4, 58, 76, 100, 10, 2, 52, 4, 2, 16, 60, 54

Offset: 1

Antti Karttunen, Apr 26 2017

Question: Are all terms positive? - Yes, they are, see A286385. (Note added Jul 24 2022).

For listening: fast tempo and percussive instrument, default "modulo 88" pitch mapping, all 10000 terms.

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

Cf. A000203, A033879, A064989, A064216, A151799, A285703, A285704, A286385.

Table[2 n - DivisorSigma[1, #] &@ If[n == 1, 1, Apply[Times, FactorInteger[2 n - 1] /. {p_, e_} /; p > 2 :> NextPrime[p, -1]^e]], {n, 103}] (* Michael De Vlieger, Apr 26 2017 *)

A064989(n) = { my(f = factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A285705(n) = (n+n - sigma(A064989(n+n-1))); \\ Antti Karttunen, Jul 24 2022

(define (A285705 n) (- (* 2 n) (A285703 n)))

a(n) = 2*n - A285703(n) = 2*n - A000203(A064216(n)).
a(n) = 1 + A286385(A064216(n)). - Antti Karttunen, Jul 24 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - Product_{p prime} (p^3/((p+1)*(p^2-q(p)))) = 0.1831523243..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

A349382 Dirichlet convolution of A064989 with A346234 (Dirichlet inverse of A003961), where A003961 and A064989 are fully multiplicative sequences that shift the prime factorization of n one step towards larger and smaller primes respectively.

Original entry on oeis.org

1, -2, -3, -2, -4, 6, -6, -2, -6, 8, -6, 6, -6, 12, 12, -2, -6, 12, -6, 8, 18, 12, -10, 6, -12, 12, -12, 12, -8, -24, -8, -2, 18, 12, 24, 12, -10, 12, 18, 8, -6, -36, -6, 12, 24, 20, -10, 6, -30, 24, 18, 12, -12, 24, 24, 12, 18, 16, -8, -24, -8, 16, 36, -2, 24, -36, -10, 12, 30, -48, -6, 12, -8, 20, 36, 12, 36, -36

Offset: 1

Antti Karttunen, Nov 17 2021

Multiplicative because both A064989 and A346234 are.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A064989, A151799, A151800, A346234, A349381 (Dirichlet inverse), A349383 (sum with it).

Cf. also A349355, A349356.

f[p_, e_] := If[p == 2, -2, NextPrime[p, -1]^e - NextPrime[p]*NextPrime[p, -1]^(e - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 17 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f = factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A346234(n) = (moebius(n)*A003961(n));
A349382(n) = sumdiv(n,d,A064989(n/d)*A346234(d));

a(n) = Sum_{d|n} A064989(n/d) * A346234(d).
a(n) = A349383(n) - A349381(n).
Multiplicative with a(p^e) = -2 if p = 2, and prevprime(p)^e - nextprime(p) * prevprime(p)^(e-1) otherwise, where prevprime function is A151799 and nextprime function is A151800. - Amiram Eldar, Nov 17 2021

cp's OEIS Frontend

A373149 Fully additive with a(2) = 1 and a(p) = prevprime(p) for odd primes p.

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A378363 Greatest number <= n that is 1 or not a perfect-power.

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A136548 a(n) = max {k >= 1 | sigma(k) <= n}.

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A245396 Largest prime not exceeding prime(n)^(1 + 1/n).

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A285703 a(n) = A000203(A064216(n)).

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A013638 a(n) = prevprime(n)*nextprime(n).

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A030460 Previous prime concatenated with this prime p is a prime.

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A054805 Second term of strong prime quartets: prime(m+1)-prime(m) > prime(m+2)-prime(m+1) > prime(m+3)-prime(m+2).

A285705 a(n) = 2n - A285703(n) = 2n - A000203(A064216(n)).