A120944 -id:A120944 - cp's OEIS frontend

A303606 Powers of composite squarefree numbers that are not squarefree.

36, 100, 196, 216, 225, 441, 484, 676, 900, 1000, 1089, 1156, 1225, 1296, 1444, 1521, 1764, 2116, 2601, 2744, 3025, 3249, 3364, 3375, 3844, 4225, 4356, 4761, 4900, 5476, 5929, 6084, 6724, 7225, 7396, 7569, 7776, 8281, 8649, 8836, 9025, 9261, 10000, 10404, 10648, 11025, 11236

Offset: 1

Ilya Gutkovskiy, Apr 26 2018

nonn

			196 is in the sequence because 196 = 2^2*7^2.
4900 is in the sequence because 4900 = 2^2*5^2*7^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Power.
Eric Weisstein's World of Mathematics, Squarefree.

Intersection of A024619 and A072777.
Intersection of A072774 and A126706.
Intersection of A013929 and A182853.
Cf. A000469, A001597, A005117, A120944, A286708.

Select[Range[12000], Length[Union[FactorInteger[#][[All, 2]]]] == 1 && ! SquareFreeQ[#] && ! PrimePowerQ[#] &]
seq[max_] := Module[{sp = Select[Range[Floor@Sqrt[max]], SquareFreeQ[#] && PrimeNu[#] > 1 &], s = {}}, Do[s = Join[s, sp[[k]]^Range[2, Floor@Log[sp[[k]], max]]], {k, 1, Length[sp]}]; Union@s]; seq[10^4] (* Amiram Eldar, Feb 12 2021 *)

from math import isqrt
from sympy import mobius, primepi, integer_nthroot
def A303606(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))-primepi(x))
    def f(x): return n-3+x+(y:=x.bit_length())-sum(g(integer_nthroot(x,k)[0]) for k in range(2,y))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 19 2024

Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A120944(n)-1)*A120944(n)) = Sum_{k>=2} (zeta(k)/zeta(2*k) - P(k) - 1) = 0.07547719891508850482..., where P(k) is the prime zeta function. - Amiram Eldar, Feb 12 2021

A364054 a(1) = 1; for n > 1, a(n) is the least positive integer not already in the sequence such that a(n) == a(n-1) (mod prime(n-1)).

Original entry on oeis.org

1, 3, 6, 11, 4, 15, 2, 19, 38, 61, 32, 63, 26, 67, 24, 71, 18, 77, 16, 83, 12, 85, 164, 81, 170, 73, 174, 277, 384, 57, 283, 29, 160, 23, 162, 13, 315, 158, 321, 154, 327, 148, 329, 138, 331, 134, 333, 122, 345, 118, 347, 114, 353, 112, 363, 106, 369, 100, 371, 94, 375, 92, 385

Offset: 1

Ali Sada, Oct 19 2023

5 is the smallest positive integer missing from the first 1000 terms. Also in the interval a(100) to a(1000) there are no entries less than 100. (From W. Edwin Clark via SeqFan.)
Comments from N. J. A. Sloane, Oct 22 2023 (Start)
It appears that the graph of this sequence is dominated by pairs of diverging lines, as suggested by the sketch (see link). For example, around step n = 4619, a descending line is changing to a descending line around a(4619) = 65, a companion ascending line is coming to an end near a(4594) = 44518, and a strong ascending line is starting up around a(4620) = 88899.
It would be nice to have more terms, in order to get better estimates of the times t_i where these transitions happen, and heights alpha_i, beta_i, gamma_i where line breaks are.
The only well-defined points are the (t_i, alpha_i) where the descending lines end, as can be seen from the b-file, where the end point a(4619) = 65 is well-defined. The other transitions, where an ascending line changes to a descending line, are less obvious.  It would be nice to know more.
Can the t_i and alpha_i sequences be traced back to the start of the sequence? Of course the alpha_i sequence is not monotonic, and in particular we do not know at present if some alpha_i is equal to 5.
(End)
a(28149) = 7. - Chai Wah Wu, Oct 22 2023
Comment from N. J. A. Sloane, Mar 05 2024 (Start):
At present there is no OEIS entry for the inverse sequence, since it is not known if 5 appears here.
The initial values of the inverse sequence are
n.....1..2..3..4..5..6....7.....8..9..10..11... . . .
index.1..7..2..5..?..3..28149..81..?...?...4... . . . (End)

			For n = 2, prime(2-1) = prime(1) = 2; a(1) = 1, so a(1) mod 2 = 1, so a(2) is the least positive integer == 1 (mod 2) that has not yet appeared; 1 has appeared, so a(2) = 3.
For n = 3, prime(3-1) = 3; a(2) mod 3 = 0, so a(3) is the least unused integer == 0 mod 3, which is 6, so a(3) =  6.
For n = 4, prime(4-1) = 5; a(3) mod 5 = 1, and 6 has already been used, so a(4) = 11.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Sketch showing the main features of the graph
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, 2048 X 2048 raster showing a(n) , n = 1..4194304 in rows of 2048 terms, left to right, then continued below for 2048 rows total. Color indicates terms as follows: black = empty product {1}, red = prime (A40), gold = composite prime power (A246547), bright green = primorial A2110(k), k > 1, light green = squarefree semiprime (A6881 \ {6}), dark green = squarefree composite (A120944 \ {A2110 U A6881}), blue = numbers neither squarefree nor composite (A126706 \ A286708 = A332785), purple = squareful numbers that are not prime powers (A286708).
Chai Wah Wu, Graph of first 10^8 terms

Cf. A006509, A125717.
For a(n-1) (mod prime(n-1)) see A366470.
Records: A368384, A368385.
See also A366475, A366477.

a[1] = 1; a[n_] := a[n] = Module[{p = Prime[n - 1], k = 2, s = Array[a, n - 1]}, While[! FreeQ[s, k] || ! Divisible[k - a[n - 1], p], k++]; k]; Array[a, 100] (* Amiram Eldar, Oct 20 2023 *)
nn = 2^20; c[] := False; m[] := 0; a[1] = j = 1; c[0] = c[1] = True;
  Monitor[Do[p = Prime[n - 1]; r = Mod[j, p];
    While[Set[k, p m[p] + r ]; c[k], m[p]++];
    Set[{a[n], c[k], j}, {k, True, k}], {n, 2, nn}], n];
Array[a, nn] (* Michael De Vlieger, Oct 26 2023, fast, based on congruence, avoids search *)

from itertools import count, islice
from sympy import nextprime
def A364054_gen(): # generator of terms
    a, aset, p = 1, {0,1}, 2
    while True:
        yield a
        for b in count(a%p,p):
            if b not in aset:
                aset.add(b)
                a, p = b, nextprime(p)
                break
A364054_list = list(islice(A364054_gen(),30)) # Chai Wah Wu, Oct 22 2023

A160135 Sum of non-exponential divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 5, 1, 8, 1, 10, 1, 10, 9, 9, 1, 15, 1, 12, 11, 14, 1, 30, 1, 16, 10, 14, 1, 42, 1, 29, 15, 20, 13, 19, 1, 22, 17, 40, 1, 54, 1, 18, 18, 26, 1, 58, 1, 33, 21, 20, 1, 60, 17, 50, 23, 32, 1, 78, 1, 34, 20, 49, 19, 78, 1, 24, 27, 74, 1, 75, 1, 40, 34, 26, 19, 90, 1, 76, 28

Offset: 1

Jaroslav Krizek, May 02 2009

nonn

The non-exponential divisors d|n of a number n = p(i)^e(i) are divisors d not of the form p(i)^s(i), s(i)|e(i) for all i.

			a(8) = A000203(8) - A051377(8) = 15 - 10 = 5. a(8) = a(2^3) = (2^4-1)/(2-1) - (2^1+2^3) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A051377, A000040, A006881, A120944, A000961, A053810.

lpowp := proc(n,p) local e; for e from 0 do if n mod p^(e+1) <> 0 then RETURN(e) ; fi; od: end:
expdvs := proc(n) local a,d,nfcts,b,f,iseDiv ; a := {} ; nfcts := ifactors(n)[2] ; for d in ( numtheory[divisors](n) minus {1} ) do iseDiv := true; for f in nfcts do b := lpowp(d,op(1,f) ) ; if b = 0 or op(2,f) mod b <> 0 then iseDiv := false; fi; od: if iseDiv then a := a union {d} ; fi; od: a ; end proc:
A051377 := proc(n) local k ; add( k, k = expdvs(n)) ; end: A160135 := proc(n) if n = 1 then 1; else numtheory[sigma][1](n)-A051377(n) ; fi; end: seq(A160135(n),n=1..120) ; # R. J. Mathar, May 08 2009

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; a[1] = 1; a[n_] := DivisorSigma[1, n] - esigma[n]; Array[a, 100] (* Amiram Eldar, Oct 26 2021 after Jean-François Alcover at A051377 *)

A051377(n) = { my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2], d, f[i, 1]^d)); }; \\ From A051377
A160135(n) = if(1==n,n, sigma(n) - A051377(n)); \\ Antti Karttunen, Mar 04 2018

a(n) = A000203(n) - A051377(n) for n >= 2.
a(1) = 1, a(p) = 1, a(p*q) = 1 + p + q, a(p*q*...*z) = (p + 1)*(q + 1)*...*(z + 1) - p*q*...*z, for p, q,..,z = primes (A000040), p*q = product of two distinct primes (A006881), p*q*...*z = product of k (k > 0) distinct primes (A120944).
a(p^k) = (p^(k+1)-1)/(p-1)- Sum_{d|k} p^d for p = primes (A000040), p^k = prime powers A000961(n>1), k = natural numbers (A000027)>
a(p^q) = 1+(p^1-p^1)+p^2+p^3+...+p^(q-1), for p, q = primes (A000040), p^q = prime powers of primes (A053810).

Edited by R. J. Mathar, May 08 2009

A163109 a(n) = phi(tau(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 4, 1, 2, 1, 2, 2, 2, 1, 4, 2, 2, 2, 2, 1, 4, 1, 2, 2, 2, 2, 6, 1, 2, 2, 4, 1, 4, 1, 2, 2, 2, 1, 4, 2, 2, 2, 2, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 2, 6, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 2, 2, 2, 4, 1, 4, 4, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 2, 2, 2, 2, 4, 1, 2, 2, 6, 1, 4, 1, 4, 4

Offset: 1

Jaroslav Krizek, Jul 20 2009

			a(16) = a(2^(5-1)) = 5-1 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A000010, A062821, A163377, A163378, A163379.

Table[EulerPhi[DivisorSigma[0, n]], {n, 1, 80}] (* Carl Najafi, Aug 15 2011 *)

a(n) = eulerphi(numdiv(n)); \\ Michel Marcus, Aug 22 2015

a(n) = A000010(A000005(n)). - Charles R Greathouse IV, Aug 11 2009

a(1) = 1, a(p) = 1 for p = primes (A000040), a(p*q) = 2 for p*q = product of two distinct primes (A006881), a(p*q*...*z) = 2^(k-1) for p*q*...*z = product of k (k > 2) distinct primes p, q, ..., z (A120944), a(p^(q-1)) = q - 1 for p, q = primes (A000040).

More terms from Carl Najafi, Aug 15 2011

Further extended by Antti Karttunen, Jul 23 2017

A355524 Minimal difference between adjacent prime indices of n > 1, or 0 if n is prime.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 5, 0, 0, 0, 1, 0, 0, 3, 6, 1, 0, 0, 7, 4, 0, 0, 1, 0, 0, 0, 8, 0, 0, 0, 0, 5, 0, 0, 0, 2, 0, 6, 9, 0, 0, 0, 10, 0, 0, 3, 1, 0, 0, 7, 1, 0, 0, 0, 11, 0, 0, 1, 1, 0, 0, 0, 12, 0, 0, 4, 13, 8

Offset: 2

Gus Wiseman, Jul 10 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 9842 are {1,4,8,12}, with differences (3,4,4), so a(9842) = 3.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Positions of first appearances are A077017 w/o the first term.
Positions of terms > 0 are A120944.
Positions of zeros are A130091.
Triangle A238353 counts m such that A056239(m) = n and a(m) = k.
For maximal difference we have A286470 or A355526.
Positions of terms > 1 are A325161.
If singletons (k) have minimal difference k we get A355525.
Positions of 1's are A355527.
Prepending 0 to the prime indices gives A355528.
A115720 and A115994 count partitions by their Durfee square.
A287352, A355533, A355534, A355536 list the differences of prime indices.
Cf. A000005, A066312, A238354, A238710, A286469, A351294, A352822, A355530, A355531, A355532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[PrimeQ[n],0,Min@@Differences[primeMS[n]]],{n,2,100}]

A229276 Composite squarefree numbers n such that p-tau(n) divides n+sigma(n), where p are the prime factors of n, tau(n) = A000005(n) and sigma(n) = A000203(n).

Original entry on oeis.org

6, 10, 15, 66, 145, 231, 435, 1221, 11571, 99093, 105502, 292434, 449854, 585429, 643858, 968014, 1372494, 1787091, 1939434, 4659114, 5524014, 5654334, 6250371, 6974007, 19495374, 19821714, 28488039, 34701369, 46183893, 81133734, 213352233, 230140869

Offset: 1

Paolo P. Lava, Sep 19 2013

nonn

Subsequence of A120944.

			Prime factors of 435 are 3, 5, 29 and sigma(435) = 720, tau(435) = 8.
435 + 720 = 1155 and 1155 / (3 - 8) = -231, 1155 / (5 - 8) = -385, 1155 / (29 - 8) = 55.

Kevin P. Thompson, Table of n, a(n) for n = 1..62

Cf. A000005, A000203, A228299-A228302, A229273-A229275.

with (numtheory); P:=proc(q) local a, b, c, i, ok, p, n;
for n from 2 to q do  if not isprime(n) then a:=ifactors(n)[2]; ok:=1;
for i from 1 to nops(a) do if a[i][2]>1 or a[i][1]=tau(n) then ok:=0; break;
else if not type((n+sigma(n))/(a[i][1]-tau(n)), integer) then ok:=0; break; fi; fi; od; if ok=1 then print(n); fi; fi; od; end: P(2*10^6);

a(21)-a(33) from Giovanni Resta, Sep 20 2013

First term deleted by Paolo P. Lava, Sep 23 2013

A355525 Minimal difference between adjacent prime indices of n, or k if n is the k-th prime.

Original entry on oeis.org

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

Offset: 2

Gus Wiseman, Jul 10 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 9842 are {1,4,8,12}, with differences (3,4,4), so a(9842) = 3.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Positions of first appearances are 4 followed by A000040.
Positions of 0's are A013929, see also A130091.
Triangle A238709 counts m such that A056239(m) = n and a(m) = k.
For maximal instead of minimal difference we have A286470.
Positions of terms > 1 are A325160, also A325161.
See also A355524, A355528.
Positions of 1's are A355527.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A238352 counts partitions by fixed points, rank statistic A352822.
A287352, A355533, A355534, A355536 list the differences of prime indices.
Cf. A066312, A120944, A238353, A279945, A286469, A325351, A325352, A351294, A355526, A355530, A355531, A355532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[PrimeQ[n],PrimePi[n],Min@@Differences[primeMS[n]]],{n,2,100}]

A366825 Numbers of the form p^2 * m, squarefree m > 1, prime p < lpf(m), where lpf(m) = A020639(m).

Original entry on oeis.org

12, 20, 28, 44, 45, 52, 60, 63, 68, 76, 84, 92, 99, 116, 117, 124, 132, 140, 148, 153, 156, 164, 171, 172, 175, 188, 204, 207, 212, 220, 228, 236, 244, 260, 261, 268, 275, 276, 279, 284, 292, 308, 315, 316, 325, 332, 333, 340, 348, 356, 364, 369, 372, 380, 387

Offset: 1

Michael De Vlieger, Dec 15 2023

Proper subset of A126706. Proper subset of A364996.
Prime signature of a(n) is 2 followed by at least one 1.
Numbers of the form A065642(A120944(k)) for some k.
The asymptotic density of this sequence is (6/Pi^2) * Sum_{p prime} (1/p^2) * (Product_{primes q <= p} (q/(q+1))) = 0.155068688392... . - Amiram Eldar, Dec 18 2023

			a(1) = 12 = 4*3 = p^2 * m, squarefree m > 1; sqrt(4) < lpf(3), i.e., 2 < 3.
a(5) = 45 = 9*5 = p^2 * m, squarefree m > 1; sqrt(9) < lpf(5), i.e., 3 < 5.
Prime powers p^k, k > 2, are not in the sequence since m = p^(k-2) is not squarefree and p = lpf(m).

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

Cf. A007947, A020639, A065642, A120944, A126706, A364996.

Select[Select[Range[500], PrimeOmega[#] > PrimeNu[#] > 1 &], First[#1] == 2 && Union[#2] == {1} & @@ TakeDrop[FactorInteger[#][[All, -1]], 1] &]

is(n) = {my(e = factor(n)[, 2]); #e > 1 && e[1] == 2 && vecmax(e[2..#e]) == 1;} \\ Amiram Eldar, Dec 18 2023

A158523 Moebius transform of negated primes in factorization of n.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Mar 20 2009

			a(72) = a(2^3*3^2) = (-1)^3*(2+1)*2^(3-1) * (-1)^2*(3+1)*3^(2-1) = (-12)*12 = -144.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A061019, A008683, A061020, A007427, A000012, A007428, A000005, A001615, A001222, A000040, A006881, A120944, A000961, A378434.

f[p_, e_] := (-1)^e*(p + 1)*p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 05 2023 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (-1)^f[i,2]*(f[i,1]+1)*f[i,1]^(f[i,2]-1));} \\ Amiram Eldar, Jan 05 2023

Multiplicative with a(p^e) = (-1)^e*(p+1)*p^(e-1), e>0. a(1)=1.
a(n) = mu(n) * A061019(n) = A008683(n) * A061019(n) = A061020(n) * A007427(n) = A061020(n) * A007428(n) * A000012(n) = A007427(n) * A000012(n) * A061019(n) = A007428(n) * A000005(n) * A061019(n), where operation * denotes Dirichlet convolution. Dirichlet convolution of functions b(n), c(n) is function a(n) = b(n) * c(n) = Sum_{d|n} b(d)*c(n/d).
Inverse Moebius transform gives A061019.
a(n) = (-1)^A001222(n)*A001615(n).
Apparently the Dirichlet inverse of A048250. - R. J. Mathar, Jul 15 2010
Dirichlet g.f.: zeta(2*s-2)/(zeta(s-1)*zeta(s)). - Amiram Eldar, Jan 05 2023

More terms from Antti Karttunen, Nov 26 2024

A177492 Products of squares of 2 or more distinct primes.

Original entry on oeis.org

36, 100, 196, 225, 441, 484, 676, 900, 1089, 1156, 1225, 1444, 1521, 1764, 2116, 2601, 3025, 3249, 3364, 3844, 4225, 4356, 4761, 4900, 5476, 5929, 6084, 6724, 7225, 7396, 7569, 8281, 8649, 8836, 9025, 10404, 11025, 11236, 12100, 12321, 12996, 13225, 13924

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 10 2010

nonn

			36=2^2*3^2, 100=2^2*5*2, 196=2^2*7^2,..900=2^2*3^2*5^2,..

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

Cf. A000469, A074985, A085986, A120944, A162142.

q:= n-> not isprime(n) and numtheory[issqrfree](n):
map(x-> x^2, select(q, [$4..120]))[];  # Alois P. Heinz, Aug 02 2024

f1[n_]:=Length[Last/@FactorInteger[n]]; f2[n_]:=Union[Last/@FactorInteger[n]]; lst={};Do[If[f1[n]>1&&f2[n]=={2},AppendTo[lst,n]],{n,0,8!}];lst
Reap[Do[{p, e} = Transpose[FactorInteger[n]]; If[Length[p]>1 && Union[e]=={2}, Sow[n]], {n, 13225}]][[2, 1]]
(* Second program *)
Select[Range[120], And[CompositeQ[#], SquareFreeQ[#]] &]^2 (* Michael De Vlieger, Aug 17 2023 *)

from math import isqrt
from sympy import primepi, mobius
def A177492(n):
    def f(x): return n+1+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n+1, f(n+1)
    while m != k:
        m, k = k, f(k)
    return m**2 # Chai Wah Wu, Aug 02 2024

a(n) = A120944(n)^2. - R. J. Mathar, Dec 06 2010

Definition corrected by R. J. Mathar, Dec 06 2010

cp's OEIS Frontend

A303606 Powers of composite squarefree numbers that are not squarefree.

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A364054 a(1) = 1; for n > 1, a(n) is the least positive integer not already in the sequence such that a(n) == a(n-1) (mod prime(n-1)).

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Mathematica

Python

A160135 Sum of non-exponential divisors of n.

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Maple

Mathematica

PARI

Formula

Extensions

A163109 a(n) = phi(tau(n)).

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PARI

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A355524 Minimal difference between adjacent prime indices of n > 1, or 0 if n is prime.

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Mathematica

A229276 Composite squarefree numbers n such that p-tau(n) divides n+sigma(n), where p are the prime factors of n, tau(n) = A000005(n) and sigma(n) = A000203(n).

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Maple

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A355525 Minimal difference between adjacent prime indices of n, or k if n is the k-th prime.

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Mathematica