A008864 -id:A008864 - cp's OEIS frontend

A175317 a(n) = Sum_{d|n} A007955(d) where A007955(m) = product of divisors of m.

1, 3, 4, 11, 6, 42, 8, 75, 31, 108, 12, 1778, 14, 206, 234, 1099, 18, 5901, 20, 8116, 452, 498, 24, 333618, 131, 692, 760, 22166, 30, 810372, 32, 33867, 1104, 1176, 1238, 10085333, 38, 1466, 1538, 2568180, 42, 3112382, 44, 85690, 91386, 2142, 48, 255138610

Offset: 1

Jaroslav Krizek, Apr 01 2010

nonn

			For n = 4, with b(n) = A007955(n), a(4) = b(1) + b(2) + b(4) = 1 + 2 + 8 = 11.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A007429, A007955, A206032, A266265.

Subsequences: A008864, A181388 \ {0}.

a[n_] := DivisorSum[n, #^(DivisorSigma[0, #]/2) &]; Array[a, 50] (* Amiram Eldar, Oct 23 2021 *)

a(n) = sumdiv(n, d, vecprod(divisors(d))); \\ Michel Marcus, Dec 09 2014 and Oct 23 2021

from math import isqrt
from sympy import divisor_count, divisors
def A175317(n): return sum(isqrt(d)**c if (c:=divisor_count(d)) & 1 else d**(c//2) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 24 2022

From Bernard Schott, Oct 26 2021: (Start)
a(1) = 1 (the only fixed point).
a(p) = p+1 for prime p only.
a(2^k) = A181388(k+1). (End)

Corrected by Jaroslav Krizek, Apr 02 2010

Edited and more terms from Michel Marcus, Dec 09 2014

A346871 Irregular triangle read by rows in which row n lists the row A000040(n) of A237591, n >= 1.

Original entry on oeis.org

2, 2, 1, 3, 2, 4, 2, 1, 6, 3, 1, 1, 7, 3, 2, 1, 9, 4, 2, 1, 1, 10, 4, 2, 2, 1, 12, 5, 2, 2, 1, 1, 15, 6, 3, 2, 1, 1, 1, 16, 6, 3, 2, 2, 1, 1, 19, 7, 4, 2, 2, 1, 1, 1, 21, 8, 4, 2, 2, 2, 1, 1, 22, 8, 4, 3, 2, 1, 2, 1, 24, 9, 4, 3, 2, 2, 1, 1, 1, 27, 10, 5, 3, 2, 2, 1, 2, 1

Offset: 1

Omar E. Pol, Aug 06 2021

The characteristic shape of the symmetric representation of sigma(prime(n)) consists in that the diagram contains exactly two regions (or parts) and each region is a rectangle (or bar), except for the first prime number (the 2) whose symmetric representation of sigma(2) consists of only one region which contains three cells.
So knowing this characteristic shape we can know if a number is prime (or not) just by looking at the diagram, even ignoring the concept of prime number.
Therefore we can see a geometric pattern of the exact distribution of prime numbers in the stepped pyramid described in A245092.
T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(prime(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000040(n).
T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th prime into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th prime into exactly k + 1 consecutive parts.

			Triangle begins:
   2;
   2, 1;
   3, 2;
   4, 2, 1;
   6, 3, 1, 1;
   7, 3, 2, 1;
   9, 4, 2, 1, 1;
  10, 4, 2, 2, 1;
  12, 5, 2, 2, 1, 1;
  15, 6, 3, 2, 1, 1, 1;
  16, 6, 3, 2, 2, 1, 1;
  19, 7, 4, 2, 2, 1, 1, 1;
  21, 8, 4, 2, 2, 2, 1, 1;
  22, 8, 4, 3, 2, 1, 2, 1;
  24, 9, 4, 3, 2, 2, 1, 1, 1;
...
Illustration of initial terms:
Row 1:    _
        _| |
       |_ _|
         2                         Semilength = 2
.
Row 2:      _
           | |
        _ _|_|
       |_ _|1                      Semilength = 3
         2
.
Row 3:          _
               | |
               | |
              _|_|
        _ _ _|                     Semilength = 5
       |_ _ _|2
          3
.
Row 4:              _
                   | |
                   | |
                   | |
                  _|_|
                _|
        _ _ _ _| 1                 Semilength = 7
       |_ _ _ _|2
           4
.
Row 5:                         _
                              | |
                              | |
                              | |
                              | |
                              | |
                           _ _|_|
                         _|
                       _|1         Semilength = 11
                      |1
           _ _ _ _ _ _|
          |_ _ _ _ _ _|3
                6
.
The area (also the number of cells) of the successive diagrams gives A008864.

Row sums give A000040.
For the characteristic shape of sigma(A000079(n)) see A346872.
For the characteristic shape of sigma(A000217(n)) see A346873.
For the visualization of Mersenne numbers A000225 see A346874.
For the characteristic shape of sigma(A000384(n)) see A346875.
For the characteristic shape of sigma(A000396(n)) see A346876.
For the characteristic shape of sigma(A008588(n)) see A224613.
Cf. A000203, A008864, A018252, A237591, A237593, A245092, A249351, A262626.

A375738 Minimum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

Original entry on oeis.org

2, 3, 6, 7, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 29, 30, 31, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 83, 84, 85, 86, 87, 88

Offset: 1

Gus Wiseman, Sep 10 2024

nonn

Non-perfect-powers (A007916) are numbers with no proper integer roots.

An anti-run of a sequence is an interval of positions at which consecutive terms differ by more than one.

			The initial anti-runs are the following, whose minima are a(n):
  (2)
  (3,5)
  (6)
  (7,10)
  (11)
  (12)
  (13)
  (14)
  (15,17)
  (18)
  (19)
  (20)
  (21)
  (22)
  (23)
  (24,26,28)

For composite numbers we have A005381, runs A008864 (except first term).
For prime-powers we have A120430, runs A373673 (except first term).
For squarefree numbers we have A373408, runs A072284.
For nonsquarefree numbers we have A373410, runs A053806.
For non-prime-powers we have A373575, runs A373676.
For anti-runs of non-perfect-powers:
- length: A375736
- first: A375738 (this)
- last: A375739
- sum: A375737
For runs of non-perfect-powers:
- length: A375702
- first: A375703
- last: A375704
- sum: A375705
A001597 lists perfect-powers, differences A053289.
A007916 lists non-perfect-powers, differences A375706.
Cf. A007674, A045542, A046933, A061399, A216765, A251092, A373403, A373679, A375708, A375714.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Min/@Split[Select[Range[100],radQ],#1+1!=#2&]//Most

A055669 Number of prime Hurwitz quaternions of norm prime(n).

Original entry on oeis.org

24, 96, 144, 192, 288, 336, 432, 480, 576, 720, 768, 912, 1008, 1056, 1152, 1296, 1440, 1488, 1632, 1728, 1776, 1920, 2016, 2160, 2352, 2448, 2496, 2592, 2640, 2736, 3072, 3168, 3312, 3360, 3600, 3648, 3792, 3936, 4032, 4176, 4320, 4368, 4608, 4656, 4752

Offset: 1

N. J. A. Sloane, Jun 09 2000

Number of vectors of norm p in D_4 lattice (cf. A004011).

L. E. Dickson, Algebras and Their Arithmetics, Dover, 1960, Section 91.

T. D. Noe, Table of n, a(n) for n=1..1000
L. E. Dickson, Algebras and Their Arithmetics, Univ. of Chicago Press, 1923, Section 91 (page 147).

Cf. A055670, A055671, A055672.

Cf. A240068 (number of prime Lipschitz quaternions having norm prime(n)).

Join[{24},24(#+1)&/@Prime[Range[2,50]]] (* Harvey P. Dale, Mar 12 2013 *)

a(n) = 24 * (prime(n)+1) = 24 * A008864(n) for n >= 2.
a(n) = 24*A055670(n).
a(n) = A004011(prime(n)). - R. J. Mathar, Aug 01 2025

More terms from David W. Wilson, May 02 2001

A077067 Squarefree numbers of the form prime + 1.

Original entry on oeis.org

3, 6, 14, 30, 38, 42, 62, 74, 102, 110, 114, 138, 158, 174, 182, 194, 230, 258, 278, 282, 314, 318, 354, 374, 390, 398, 402, 410, 422, 434, 458, 462, 510, 542, 570, 602, 614, 618, 642, 654, 662, 674, 678, 710, 734, 758, 762, 770, 798, 822, 830, 854, 858, 878

Offset: 1

Reinhard Zumkeller, Oct 23 2002

nonn

			A005117(28) = 42 = 2*3*7 is a term as 42 = A000040(13) + 1 = 41+1.

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

Cf. A000040, A005117, A008864, A049097, A077064, A077066.

Select[Prime[Range[200]]+1,SquareFreeQ] (* Harvey P. Dale, Aug 20 2017 *)

isok(n) = issquarefree(n) && isprime(n-1); \\ Michel Marcus, Mar 22 2016

lista(nn) = forprime(p=2, nn, if (issquarefree(p+1), print1(p+1, ", "))); \\ Michel Marcus, Mar 22 2016

A077066(a(n)) = a(n).

a(n) = A049097(n)+1. - Zak Seidov, Aug 15 2006

A084922 a(n) = (prime(n)-1)*(prime(n)+1)/6.

Original entry on oeis.org

4, 8, 20, 28, 48, 60, 88, 140, 160, 228, 280, 308, 368, 468, 580, 620, 748, 840, 888, 1040, 1148, 1320, 1568, 1700, 1768, 1908, 1980, 2128, 2688, 2860, 3128, 3220, 3700, 3800, 4108, 4428, 4648, 4988, 5340, 5460, 6080, 6208, 6468, 6600, 7420

Offset: 3

Reinhard Zumkeller, Jun 11 2003

Vincenzo Librandi, Table of n, a(n) for n = 3..840

Cf. A000040, A006093, A008864, A084920, A084921.

[(p^2-1)/6: p in PrimesInInterval(4, 250)]; // Vincenzo Librandi, Apr 11 2013

Select[Range[0, 7000], PrimeQ[Sqrt[6 # + 1]]&] (* Vincenzo Librandi, Apr 11 2013 *)
(Prime[Range[3,60]]^2 -1)/6 (* G. C. Greubel, May 02 2024 *)

a(n) = (prime(n)^2-1)/6; \\ Michel Marcus, Mar 22 2016

[(n^2-1)//6 for n in prime_range(4,301)] # G. C. Greubel, May 02 2024

a(n) = A084920(n)/6.

a(n) = A084921(n)/3.

A175144 a(n) = d(p(n)-1) + d(p(n)+1), where p(n) is the n-th prime, and where d(m) is the number of divisors of m.

Original entry on oeis.org

3, 5, 7, 8, 10, 10, 11, 12, 12, 14, 14, 13, 16, 14, 14, 14, 16, 16, 14, 20, 16, 18, 16, 20, 18, 17, 16, 16, 20, 18, 20, 20, 16, 20, 18, 20, 16, 16, 20, 14, 22, 26, 22, 18, 21, 24, 22, 20, 16, 20, 20, 28, 26, 26, 17, 20, 22, 26, 16, 24, 14, 18, 24, 24, 20, 14, 22, 26, 16, 24, 20

Offset: 1

Leroy Quet, Feb 24 2010

nonn

If a(n) is a record, then the n-th prime is in sequence A090481.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000005, A006093, A008864, A090481, A090482, A175146.

taudiff := proc(n) numtheory[tau](n-1)+numtheory[tau](n+1) ; end proc: A175144 := proc(n) taudiff(ithprime(n)) ; end proc: seq(A175144(n),n=1..80) ; # R. J. Mathar, Mar 03 2010

Table[p = Prime[n]; DivisorSigma[0, p - 1] + DivisorSigma[0, p + 1], {n, 100}]
Total[DivisorSigma[0,{#-1,#+1}]]&/@Prime[Range[80]] (* Harvey P. Dale, Feb 25 2012 *)

a(n) = numdiv(prime(n)-1) + numdiv(prime(n)+1); \\ Amiram Eldar, Apr 17 2024

lista(pmax) = forprime(p = 1, pmax, print1(numdiv(p-1) + numdiv(p+1), ", ")); \\ Amiram Eldar, Apr 17 2024

a(n) = A000005(A006093(n)) + A000005(A008864(n)). - R. J. Mathar, Mar 03 2010

More terms from R. J. Mathar, Mar 03 2010

A214583 Numbers m such that for all k with gcd(m, k) = 1 and m > k^2, m - k^2 is prime.

Original entry on oeis.org

3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 48, 54, 60, 62, 68, 72, 80, 84, 90, 98, 108, 110, 132, 138, 140, 150, 180, 182, 198, 252, 318, 360, 398, 468, 570, 572, 930, 1722

Offset: 1

Hans Ruegg, Jul 21 2012

No further terms < 10^10.
This sequence is based on a remark in a paper distributed over the Internet (see the Leo Moser link) under the heading "Unsolved Problems and Conjectures" (page 84):
"Is 968 the largest number n such that for all k with (n, k) = 1 and n > k^2, n - k^2 is prime? (Erdős)"
The statement by Moser contains an error: 968 does NOT have this property (968-25*25 = 343 = 7*7*7), and the largest such number (1722) is larger than 968.
A224076(n) <= A064272(a(n)+1). - Reinhard Zumkeller, Mar 31 2013

			For example, the number 20 is part of this sequence because 20-1*1 = 19 (prime), and 20-3*3 = 11 (prime). Not considered are 20-2*2 and 20-4*4, because 2 and 4 are not relative primes to 20.

Leo Moser, An Introduction to the Theory of Numbers, The Trillia Group 2004, page 84.

Cf. A065428.
Cf. A057828, A000290, A010051.
Cf. A224075; subsequence of A008864.

a214583 n = a214583_list !! (n-1)
a214583_list = filter (p 3 1) [2..] where
   p i k2 x = x <= k2 || (gcd k2 x > 1 || a010051' (x - k2) == 1) &&
                         p (i + 2) (k2 + i) x
-- Reinhard Zumkeller, Mar 31 2013, Jul 22 2012

Reap[For[p = 2, p < 2000, p = NextPrime[p], n = p+1; q = True; k = 1; While[k*k < n, If[GCD[k, n] == 1, If[! PrimeQ[n - k^2], q = False; Break[]]]; k += 1]; If[q, Sow[n]]]] [[2, 1]] (* Jean-François Alcover, Oct 11 2013, after Joerg Arndt's Pari program *)
gQ[n_]:=AllTrue[n-#^2&/@Select[Range[Floor[Sqrt[n]]],CoprimeQ[ #, n]&], PrimeQ]; Select[Range[2000],gQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 02 2018 *)

N=10^10;
default(primelimit,N);
{ forprime (p=2, N,
    n = p + 1;
    q = 1;
    k = 1;
    while ( k*k < n,
        if ( gcd(k,n)==1,
            if ( ! isprime(n-k^2),  q=0; break() );
        );
        k += 1;
    );
    if ( q, print1(n,", ") );
); }
/* Joerg Arndt, Jul 21 2012 */

A250177 Numbers n such that Phi_21(n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

3, 6, 7, 12, 22, 27, 28, 35, 41, 59, 63, 69, 112, 127, 132, 133, 136, 140, 164, 166, 202, 215, 218, 276, 288, 307, 323, 334, 343, 377, 383, 433, 474, 479, 516, 519, 521, 532, 538, 549, 575, 586, 622, 647, 675, 680, 692, 733, 790, 815, 822, 902, 909, 911, 915, 952, 966, 1025, 1034, 1048, 1093

Offset: 1

Eric Chen, Dec 24 2014

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A008864 (1), A006093 (2), A002384 (3), A005574 (4), A049409 (5), A055494 (6), A100330 (7), A000068 (8), A153439 (9), A250392 (10), A162862 (11), A246397 (12), A217070 (13), A250174 (14), A250175 (15), A006314 (16), A217071 (17), A164989 (18), A217072 (19), A250176 (20), this sequence (21), A250178 (22), A217073 (23), A250179 (24), A250180 (25), A250181 (26), A153440 (27), A250182 (28), A217074 (29), A250183 (30), A217075 (31), A006313 (32), A250184 (33), A250185 (34), A250186 (35), A097475 (36), A217076 (37), A250187 (38), A250188 (39), A250189 (40), A217077 (41), A250190 (42), A217078 (43), A250191 (44), A250192 (45), A250193 (46), A217079 (47), A250194 (48), A250195 (49), A250196 (50), A217080 (53), A217081 (59), A217082 (61), A006315 (64), A217083 (67), A217084 (71), A217085 (73), A217086 (79), A153441 (81), A217087 (83), A217088 (89), A217089 (97), A006316 (128), A153442 (243), A056994 (256), A056995 (512), A057465 (1024), A057002 (2048), A088361 (4096), A088362 (8192), A226528 (16384), A226529 (32768), A226530 (65536), A251597 (131072), A244150 (524287), A243959 (1048576).

Cf. A085398 (Least k>1 such that Phi_n(k) is prime).

a250177[n_] := Select[Range[n], PrimeQ@Cyclotomic[21, #] &]; a250177[1100] (* Michael De Vlieger, Dec 25 2014 *)

{is(n)=isprime(polcyclo(21,n))};
for(n=1,100, if(is(n)==1, print1(n, ", "), 0)) \\ G. C. Greubel, Apr 14 2018

A304411 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*k_j).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 9, 8, 18, 12, 24, 14, 24, 24, 12, 18, 24, 20, 36, 32, 36, 24, 36, 12, 42, 12, 48, 30, 72, 32, 15, 48, 54, 48, 48, 38, 60, 56, 54, 42, 96, 44, 72, 48, 72, 48, 48, 16, 36, 72, 84, 54, 36, 72, 72, 80, 90, 60, 144, 62, 96, 64, 18, 84, 144, 68, 108, 96, 144, 72, 72

Offset: 1

Ilya Gutkovskiy, May 12 2018

			a(24) = a(2^3*3) = (2 + 1)*3 * (3 + 1)*1 = 36.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000 .
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

Cf. A000005, A000026, A000040, A000203, A003959, A001615, A003961, A005117, A005361, A007947, A008864, A045965, A048250, A064478, A081294 (numbers n such that a(n) is odd), A181797, A304407, A304408, A304409, A304412.

Cf. A072691, A330596.

a[n_] := Times @@ ((#[[1]] + 1) #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 72}]
Table[Total[Select[Divisors[n], SquareFreeQ]] DivisorSigma[0, n/Last[Select[Divisors[n], SquareFreeQ]]], {n, 72}]

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); (p+1)*e)} \\ Andrew Howroyd, Jul 24 2018

a(n) = A005361(n)*A048250(n) = A000005(n/A007947(n))*A000203(A007947(n)).
a(p^k) = (p + 1)*k where p is a prime and k > 0.
a(n) = Product_{p|n} (p + 1) if n is a squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = A072691 * A330596 = 0.6156455744... . - Amiram Eldar, Nov 30 2022

cp's OEIS Frontend

A175317 a(n) = Sum_{d|n} A007955(d) where A007955(m) = product of divisors of m.

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A346871 Irregular triangle read by rows in which row n lists the row A000040(n) of A237591, n >= 1.

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A375738 Minimum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

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Mathematica

A055669 Number of prime Hurwitz quaternions of norm prime(n).

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A077067 Squarefree numbers of the form prime + 1.

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A084922 a(n) = (prime(n)-1)*(prime(n)+1)/6.

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A175144 a(n) = d(p(n)-1) + d(p(n)+1), where p(n) is the n-th prime, and where d(m) is the number of divisors of m.

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