A036689 -id:A036689 - cp's OEIS frontend

A280187 Numbers n such that 2 * (1^n + 2^n + 3^n + ... + n^n) is not 0 (mod n), but 2 * (1^d + 2^d + 3^d + ... + d^d) is 0 (mod d) for each other d | n.

6, 20, 110, 272, 506, 812, 2162, 2756, 3422, 4970, 6806, 7832, 11342, 12656, 17030, 18632, 22052, 27722, 29756, 31862, 36290, 38612, 51302, 54056, 56882, 62750, 65792, 68906, 72092, 85556, 96410, 100172, 120062, 124256, 128522

Offset: 1

Charles R Greathouse IV, Dec 28 2016

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..4000

Primitive elements of A228870.

Subsequence of A002943. Also a subsequence of A028689, A036689, A053198, A068377, A079143, A128672, A220211 and other sequences ...- Paolo P. Lava, Jan 10 2017

has(n)=my(f=factor(n)[,1]); for(i=1,#f, if(n%(f[i]-1)==0 && f[i]>2, return(1))); 0
is(n)=if(n%2, return(0)); if(n%3==0, return(n==6)); if(n%20==0, return(n==20)); if(!has(n), return(0)); my(f=factor(n)[,1]); for(i=1,#f, if(has(n/f[i]), return(0))); 1 \\ Charles R Greathouse IV, Dec 28 2016

A341109 a(n) = denominator(p(n, x)) / (n!denominator(bernoulli(n, x))), where p(n, x) = Sum_{k=0..n} E2(n, k)binomial(x + k, 2*n) / Product_{j=1..n} (j - x) and E2(n, k) are the second-order Eulerian numbers A201637.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 96, 192, 1152, 768, 1536, 3072, 18432, 36864, 221184, 147456, 884736, 1769472, 10616832, 21233664, 637009920, 424673280, 2548039680, 5096079360, 152882380800, 61152952320, 366917713920, 81537269760, 163074539520, 326149079040, 1956894474240

Offset: 0

Peter Luschny, Feb 06 2021

nonn

The challenge is to characterize the sequence purely arithmetically, i.e., without reference to the Eulerian numbers or the Bernoulli polynomials.

András Bazsó and István Mező, On the coefficients of power sums of arithmetic progressions, J. Number Th., 153 (2015), 117-123.
J.-L. Chabert and P.-J. Cahen, Old problems and new questions around integer-valued polynomials and factorial sequences In: J. W. Brewer, S. Glaz, W. J. Heinzer, B. M. Olberding (eds), Multiplicative Ideal Theory in Commutative Algebra. Springer, Boston, MA., 2006.
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, arXiv:1705.03857 [math.NT] 2017, Amer. Math. Monthly.

Cf. A100655, A053657 (Minkowski), A341107, A341108, A318256, A144845, A163176, A201637 (Eulerian2), A036689, A324370, A007947, A324369, A195441.

Epoly := proc(n, x) add(combinat:-eulerian2(n, k)*binomial(x+k, 2*n), k = 0..n) / mul(j-x, j = 1..n): simplify(expand(%)) end:
seq(denom(Epoly(n, x)) / (n!*denom(bernoulli(n, x))), n = 0..30);

A053657[n_] := Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k,0,n}],{p, Prime[Range[n]]}];
A144845[n_] := Denominator[Together[BernoulliB[n, x]]];
A163176[n_] := A053657[n] / n!;
Table[(n + 1) A163176[n + 1] / A144845[n], {n, 0, 30}]

def A341109(n): # uses[A341108, A318256]
    return A341108(n)//A318256(n)
print([A341109(n) for n in (0..30)])

a(n) = A053657(n+1)/(n!*A144845(n)).
a(n) = (n+1)*A163176(n+1)/A144845(n).
a(n) = A341108(n)/A318256(n).
a(n) = A341107(n)*A324369(n+1).
a(n) = A341108(n)/A324370(n+1).
a(n) = A341108(n)*A007947(n+1)/A144845(n).
a(n) = A341108(n)*A324369(n+1)/A195441(n).
prime(n) divides a(k) for k >= A036689(n).
2^(n-1) divides exactly a(n) for n >= 2.

A072006 Number of terms in InversePhi set of prime(n)*(prime(n)-1) = phi(p(n)^2), where prime(n) is the n-th prime and phi=A000010.

Original entry on oeis.org

3, 4, 5, 4, 2, 7, 5, 2, 2, 2, 2, 6, 10, 2, 2, 2, 2, 7, 4, 2, 16, 4, 2, 8, 19, 5, 2, 2, 2, 13, 2, 2, 2, 4, 5, 4, 2, 4, 2, 5, 2, 14, 2, 21, 2, 2, 2, 2, 2, 5, 5, 2, 28, 2, 2, 2, 2, 2, 8, 8, 2, 2, 2, 2, 4, 5, 2, 14, 2, 7, 5, 2, 2, 5, 4, 2, 2, 11, 7, 17, 2, 11, 2, 26, 2, 2, 12, 4, 5, 2, 2, 2, 2, 2, 2, 2, 5, 5

Offset: 1

Labos Elemer, Jun 04 2002

nonn

p^2 and 2p^2 are always in inverse set, so a(n) >= 2.

			For n = 5: prime(5) = 11, a(5) = 2 because InvPhi(110) = {121, 242}.
For n = 6: prime(6) = 13, a(6) = 7 because InvPhi(13*12) = InvPhi(156) = {157, 169, 237, 314, 316, 338, 474}.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A036689, A000010.

[seq(nops(invphi(ithprime(j)*(-1+ithprime(j)))),j=1..128)];

a(n) = my(p=prime(n)); #invphi(p*(p-1)); \\ Michel Marcus, Mar 25 2020

a(n) = Card[InvPhi(p(n)*(p(n)-1))] = Card[InvPhi(A036689(n))].

A079143 Numbers divisible by prime ceilings of their square roots + 1.

Original entry on oeis.org

2, 4, 6, 9, 20, 25, 42, 49, 110, 121, 156, 169, 272, 289, 342, 361, 506, 529, 812, 841, 930, 961, 1332, 1369, 1640, 1681, 1806, 1849, 2162, 2209, 2756, 2809, 3422, 3481, 3660, 3721, 4422, 4489, 4970, 5041, 5256, 5329, 6162, 6241, 6806, 6889, 7832, 7921

Offset: 1

Cino Hilliard, Dec 26 2002

n is in the sequence if r=ceiling(sqrt(n)) is prime and r divides n.
Union of the 2 sequences A001248={p^2} and A036689={p(p-1)} for p prime.
Sum of the reciprocals = 1.225...

			930 is in the sequence because ceiling(sqrt(930)) = 31 and 930/31 = 30.

Flatten[ #(#-{1, 0})&/@Prime/@Range[30]]
a[n_] := (p=Prime[Ceiling[n/2]])(p-Mod[n, 2])

ipsqrt(n) = { sr= 0; for(x=1,n, v = ceil(sqrt(x)); if(isprime(v) && x%v == 0, print1(x" "); sr+=1.0/x; ); ); print(); print(sr); } \\ numbers divisible by the prime ceilings of their square roots.

a(n) = prime(ceiling(n/2))*(prime(ceiling(n/2)) - (n mod 2))

A079477 First prime after phi(prime(n)^2).

Original entry on oeis.org

3, 7, 23, 43, 113, 157, 277, 347, 509, 821, 937, 1361, 1657, 1811, 2179, 2767, 3433, 3671, 4423, 4973, 5261, 6163, 6823, 7841, 9319, 10103, 10513, 11351, 11777, 12659, 16007, 17033, 18637, 19183, 22063, 22651, 24499, 26407, 27733, 29759, 31873, 32587, 36293

Offset: 1

Jon Perry, Jan 15 2003

nonn

The sequence always increases.

			p_3=5, phi(5^2)=phi(25)=20, therefore a(3)=23

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

Cf. A036689, A151800.

a:= n-> nextprime(2*binomial(ithprime(n),2)):
seq(a(n), n=1..44);  # Alois P. Heinz, Mar 15 2023

Prime[1+PrimePi[EulerPhi[Prime[Range[41]]^2]]] (* T. D. Noe, Nov 15 2006 *)

forprime (p=2,200, print1(nextprime(eulerphi(p^2))","))

a(n) = A151800(A036689(n)). - Michel Marcus, Mar 15 2023

Corrected by T. D. Noe, Nov 15 2006

A088659 a(n) = n*(p-1) where p is the largest prime factor of n.

Original entry on oeis.org

2, 6, 4, 20, 12, 42, 8, 18, 40, 110, 24, 156, 84, 60, 16, 272, 36, 342, 80, 126, 220, 506, 48, 100, 312, 54, 168, 812, 120, 930, 32, 330, 544, 210, 72, 1332, 684, 468, 160, 1640, 252, 1806, 440, 180, 1012, 2162, 96, 294, 200, 816, 624, 2756, 108, 550, 336, 1026

Offset: 2

Benoit Cloitre, Nov 21 2003

nonn

It is conjectured that sequence gives period length of the periodic sequence {A088957(k) mod n}_{k>n}.

The records of this sequence are given by A036689 (product of a prime and the previous number). - Michel Marcus, May 19 2015

Ivan Neretin, Table of n, a(n) for n = 2..1000

seq(n*(max(numtheory:-factorset(n))-1), n=2..100); # Robert Israel, May 19 2015

Table[n*(FactorInteger[n][[-1, 1]] - 1), {n, 2, 57}] (* Ivan Neretin, May 19 2015 *)

PARI
```
a(n)=n*(component(factor(n),1)-1)
```

For p the k-th prime, a(p) = A036689(k). - Michel Marcus, May 19 2015

a(n) = n*A070777(n). - Michel Marcus, May 19 2015

A137148 a(n) = k*phi(k), where k is the n-th nonprime number.

Original entry on oeis.org

1, 8, 12, 32, 54, 40, 48, 84, 120, 128, 108, 160, 252, 220, 192, 500, 312, 486, 336, 240, 512, 660, 544, 840, 432, 684, 936, 640, 504, 880, 1080, 1012, 768, 2058, 1000, 1632, 1248, 972, 2200, 1344, 2052, 1624, 960, 1860, 2268, 2048, 3120, 1320, 2176, 3036

Offset: 1

Artur Jasinski, Jan 23 2008

Numbers that occur in A002618 but not in A036689.

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

Cf. A002618, A018252, A036689, A065484, A136141.

a = {}; Do[If[!PrimeQ[n], AppendTo[a, n EulerPhi[n]]], {n, 1, 100}]; a

a(n) = A002618(A018252(n)). - R. J. Mathar, Jan 18 2021

Sum_{n>=1} 1/a(n) = A065484 - A136141 = 1.430699927388... . - Amiram Eldar, Oct 26 2024

A228529 a(n) = prime(n*prime(n)).

Original entry on oeis.org

3, 13, 47, 107, 257, 397, 653, 881, 1279, 1889, 2293, 3119, 3847, 4423, 5323, 6563, 7937, 8819, 10391, 11833, 12889, 14831, 16477, 18713, 21599, 23603, 25189, 27409, 29063, 31511, 37159, 39869, 43321, 45589, 50923, 53281, 57271, 61561, 65173, 69821, 74383

Offset: 1

Omar E. Pol, Oct 20 2013

nonn

			For n = 2, prime(2*prime(2)) = prime(2*3) = prime(6) = 13, so a(2) = 13.

Cf. A000040, A006450, A033286, A036689, A080697, A217622, A230098, A230325, A230285, A230460.

Table[Prime[n*Prime[n]], {n, 100}] (* T. D. Noe, Oct 22 2013 *)

a(n) = prime(n*prime(n)); \\ Michel Marcus, Oct 22 2013

a(n) = A000040(A033286(n)).

A308394 Numbers which can be written in the form m^k - m with m prime and k a positive integer.

Original entry on oeis.org

0, 2, 6, 14, 20, 24, 30, 42, 62, 78, 110, 120, 126, 156, 240, 254, 272, 336, 342, 506, 510, 620, 726, 812, 930, 1022, 1320, 1332, 1640, 1806, 2046, 2162, 2184, 2394, 2756, 3120, 3422, 3660, 4094, 4422, 4896, 4970, 5256, 6162, 6558, 6806, 6840, 7832, 8190, 9312

Offset: 1

Craig J. Beisel, May 24 2019

The only known terms which have two representations where m is prime are 6 and 2184. It is conjectured by Bennett these are the only terms with this property.

			a(9) = 2^6 - 2 = 62.
For the two terms known to have two representations we have a(3) = 6 = 2^3 - 2 = 3^2 - 3 and a(33)= 2184 = 3^7 - 3 = 13^3 - 13.

Robert Israel, Table of n, a(n) for n = 1..10000
Michael Bennett, On some exponential equations of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922.
Dana Mackenzie, 2184: An Absurd (and Adsurd) Tale, Integers (Electronic Journal of Combinatorial Number Theory), 18 (2018), A33.

Cf. A057895, A057896, A246068, A308324.

Subsequences: A000918 (2^n - 2), A036689 (p^2 - p), A058809 (3^n - 3), A178671 (5^n - 5).

N:= 10^6; # to get all terms <= N
P:= select(isprime,[2,seq(i,i=3..floor((1+sqrt(1+4*N))/2),2)]):
S:= {0,seq(seq(m^k-m,k=2..floor(log[m](N+m))),m=P)}:
sort(convert(S,list)); # Robert Israel, Aug 11 2019

x=List([]); lim=10000; forprime(m=2, lim, for(k=1, 100, y=(m^k-m); if(y>lim, break, i=setsearch(x, y, 1); if(i>0, listinsert(x, y, i))))); for(i=1, #x, print(x[i]));

isok(n) = {forprime(p=2, oo, my(keepk = 0); for (k=1, oo, if ((x=p^k - p) == n, return(1)); if (x > n, keepk = k; break);); if (keepk == 2, break););} \\ Michel Marcus, Aug 06 2019

A119958 Numerator of determinant of n X n matrix with elements M[i,j] = (p^2 - p + 1)/(p*(p-1)) if i=j and 1 otherwise, where p=Prime[i].

Original entry on oeis.org

3, 7, 147, 301, 33411, 1748509, 36718689, 4198170109, 709490748421, 82402282638039, 1345903949754637, 1564158644309443, 855594778437265321, 5136411178193150947, 3703352459477261832787, 261798531558431048025481

Offset: 1

Alexander Adamchuk, Aug 02 2006

nonn

All square prime divisors of a(n) {7,13,43,139,19,31,61,37,607,523,67,79,1201,241,1171,157,109,...} belong to A002476[n] Primes of form 6n + 1.

Cf. A002476, A036689, A002061.

Numerator[ Table[ Det[ DiagonalMatrix[ Table[1/(Prime[i]*(Prime[i]-1)), {i, 1, n} ] + 1 ]], {n, 1, 150}]]

cp's OEIS Frontend

A280187 Numbers n such that 2 * (1^n + 2^n + 3^n + ... + n^n) is not 0 (mod n), but 2 * (1^d + 2^d + 3^d + ... + d^d) is 0 (mod d) for each other d | n.

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A341109 a(n) = denominator(p(n, x)) / (n!*denominator(bernoulli(n, x))), where p(n, x) = Sum_{k=0..n} E2(n, k)*binomial(x + k, 2*n) / Product_{j=1..n} (j - x) and E2(n, k) are the second-order Eulerian numbers A201637.

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A072006 Number of terms in InversePhi set of prime(n)*(prime(n)-1) = phi(p(n)^2), where prime(n) is the n-th prime and phi=A000010.

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A079143 Numbers divisible by prime ceilings of their square roots + 1.

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PARI

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A079477 First prime after phi(prime(n)^2).

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A088659 a(n) = n*(p-1) where p is the largest prime factor of n.

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Maple

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PARI

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A137148 a(n) = k*phi(k), where k is the n-th nonprime number.

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Mathematica

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A228529 a(n) = prime(n*prime(n)).

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A341109 a(n) = denominator(p(n, x)) / (n!denominator(bernoulli(n, x))), where p(n, x) = Sum_{k=0..n} E2(n, k)binomial(x + k, 2*n) / Product_{j=1..n} (j - x) and E2(n, k) are the second-order Eulerian numbers A201637.