A110077 -id:A110077 - cp's OEIS frontend

A180265 a(n) = smallest k such that sigma(k) = 14^n, or zero if no such k exists.

1, 13, 0, 1164, 15132, 230484, 2823492, 36705396, 508541124, 7194470556, 100696385244, 1503091145388, 19540184890044, 273550891167372, 3811871194625676, 53378900339114532, 727176010568075796, 10177815744800162004, 142523339476298463228, 1994910930816765350844, 27918212600229725023068

Offset: 0

Walter Kehowski, Aug 22 2010

nonn

			a(0)=1 since sigma(1)=14^0=1. a(1)=13 since sigma(13)=14. a(2)=0 since no N exists such that sigma(N)=14^2. a(6)=2823492=2^2*3*7*33613 since sigma(2823492)=14^6 is the first 6th power. a(7)=36705396=2^2*3*7*13*33613 since sigma(36705396)=14^7 is the first 7th power.

Max Alekseyev, Table of n, a(n) for n = 0..100

Cf. A046528, A048251, A110077, A180460, A180461, A180462, A180463, A180464

Edited (including b-file) by N. J. A. Sloane, Oct 05 2010

Terms a(25) onward from Max Alekseyev, Mar 04 2014

A180460 a(n) is the smallest number m such that sigma(m)=22^n, or 0 if m does not exist.

Original entry on oeis.org

1, 0, 0, 10363, 136647, 3018141, 66411009, 1636922343, 31276995183, 688217286267, 15200749439001, 324029599659171, 7264291502741679, 160447401116572437, 3530475812620849113, 75514126111770824037, 1662716417771040164631, 36586320846189859358019, 804905851136700392012493, 17704604426749226872106319

Offset: 0

Walter Kehowski, Sep 06 2010

nonn

Conjecture: Given any even integer E not a power of 2 (see A078426) there exists a positive integer N such that for all n>=N the equation sigma(m)=E^n has at least one solution for m.

			a(4)=136647=3^4*7*241 since sigma(3^4*7*241)=(11^2)(2^3)(2*11^2)=2^4*11^4 and 136647 is the smallest such number.

Max Alekseyev, Table of n, a(n) for n = 0..100

Cf. A048251, A110077, A180265, A180461, A180462, A180463, A180464.

Terms a(37) onward (in b-file) from Max Alekseyev, Mar 04 2014

A180461 a(n) is the smallest number m such that sigma(m)=26^n, or 0 if m does not exist.

Original entry on oeis.org

1, 0, 0, 0, 312399, 8911029, 187131609, 5560483959, 126501010209, 3186244194273, 85514682943809, 2206011586559697, 55462952723504823, 1577373555132452973, 37744962467192492463, 974816528291192900817, 25345238283868264629273, 658976194648474007782617, 17245346702665551116114601

Offset: 0

Walter Kehowski, Sep 06 2010

nonn

			a(4)=312399=3^2*103*337 since sigma(3^2*103*337)=(13)*(2^3*13)*(2*13^2)=2^4*13^4 and 312399 is the smallest such number.

Max Alekseyev, Table of n, a(n) for n = 0..100

Cf. A048251, A110077, A180265, A180460, A180462, A180463, A180464.

Terms a(17) onward from Max Alekseyev, Mar 04 2014

A180462 a(n) is the smallest number m such that sigma(m)=34^n, or 0 if m does not exist.

Original entry on oeis.org

1, 0, 0, 38659, 1333447, 45356239, 1542131167, 52432478719, 0, 44114690056791, 1545604624685757, 52550557239372861, 1542390446782691499, 52463299805385993363, 1783774217997898256739, 60648345436543315211523, 2091915317967934350057159, 69957029735592536691912261

Offset: 0

Walter Kehowski, Sep 06 2010

nonn

			a(3)=38659=67*577 since sigma(67*577)=(2^2*17)*(2*17^2)=2^3*17^3 and 38659 is the smallest such number.

Max Alekseyev, Table of n, a(n) for n = 0..100

Cf. A048251, A110077, A180265, A180460, A180461, A180463, A180464.

Terms a(16) onward from Max Alekseyev, Mar 04 2014

A180463 a(n) is the smallest number m such that sigma(m)=38^n, or 0 if m does not exist.

Original entry on oeis.org

1, 37, 0, 0, 0, 0, 0, 85811686941, 3175032416817, 159809964979789, 4561513295803851, 155532888827892597, 5754716886632026089, 249887680992407652771, 8319571903089894983277, 307824160414326114381249, 12077787753685017948512649, 455818072892233197941369463

Offset: 0

Walter Kehowski, Sep 06 2010

nonn

			a(7)=85811686941=3*28603895647 since sigma(85811686941)= (2^2)*(2^5*19^7)=2^7*19^7 and 85811686941 is the smallest such number.

Max Alekseyev, Table of n, a(n) for n = 0..100

Cf. A048251, A110077, A180265, A180460, A180461, A180462, A180464.

Terms a(16) onward from Max Alekseyev, Mar 03 2014

A180464 a(n) is the smallest number m such that sigma(m)=46^n, or 0 if m does not exist.

Original entry on oeis.org

1, 0, 0, 0, 0, 180217597, 6217507317, 325975026477, 15034820907837, 919684211139697, 31808042757482763, 1463508270300620883, 58746435953696315769, 2709539295335742607689, 138000408890542031957601, 5718143312090439006662949, 263735727094699893912217269

Offset: 0

Walter Kehowski, Sep 06 2010

nonn

			a(5)=180217597=7*25745371 and sigma(180217597)=(2^3)*(2^2*23^5)=2^5*23^5 and 180217597 is the smallest such number.

Max Alekseyev, Table of n, a(n) for n = 0..100

Cf. A048251, A110077, A180265, A180460, A180461, A180462, A180463.

Terms a(15) onward from Max Alekseyev, Mar 03 2014

A072075 Smallest solution to phi(x) = 10^n where phi(x) = A000010(x).

Original entry on oeis.org

1, 11, 101, 1111, 10291, 100651, 1004251, 10165751, 100064101, 1000078501, 10000222501, 100062501601, 1000062516001, 10000062660001, 100002441447211, 1003922328562757, 10000390625025601, 100000002482366251, 1000000002851006251, 10000062500000160001

Offset: 0

Labos Elemer, Jun 13 2002

nonn

			n=3: a(3)=1111 because InvPhi[1000]= {1111,1255,1375,1875,2008,2222,2500,2510,2750,3012,3750}.

Ray Chandler, Table of n, a(n) for n = 0..1000
Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2

Cf. A000010, A014573, A014197, A110077, A072074, A072076.

More terms from Max Alekseyev, Apr 26 2010

a(18)-a(19) from Donovan Johnson, Feb 02 2012

A110076 a(n) is the largest number m such that sigma(m)=10^n, or if there is no such m a(n)=0.

Original entry on oeis.org

1, 0, 0, 0, 9481, 99301, 997501, 9993001, 99948001, 999795001, 9999750001, 99998670001, 999997950001, 9999986700001, 99999975000001, 999999198750001, 9999999187500001, 99999995096707501, 999999919987500001, 9999999986700000001, 99999499999999800001, 999999999907500000001, 9999999999796009687501

Offset: 0

Farideh Firoozbakht, Jul 31 2005

nonn

Conjecture: For n>3 a(n) is positive.

For 4 <= n <= 102, a(n) is the product of two distinct primes, but a(103) = a(49)*a(54) and is the product of four distinct primes: 1862645149230957031249999 * 5368709119999999999999999 * 79999999999999999999999999 * 12499999999999999999999999999. - David Wasserman, Nov 18 2008

			a(12)=999997950001 because sigma(999997950001)=sigma(799999*1249999) =800000*1250000=10^12 and 999997950001 is the largest number with this property(sigma(m)=10^12).

Max Alekseyev, Table of n, a(n) for n = 0..1000
Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2

Cf. A110077, A110078.

a[0] = 1; a[1] = a[2] = a[3] = 0; a[n_] := (For[m = 1, DivisorSigma[ 1, 10^n - m] != 10^n, m++ ];10^n - m); Do[Print[a[n]], {n, 0, 12}]

More terms from David Wasserman, Nov 18 2008

Terms a(19) onward from Max Alekseyev, Mar 06 2014

A110078 a(n) is number of solutions of the equation sigma(x)=10^n.

Original entry on oeis.org

1, 0, 0, 0, 2, 4, 7, 9, 15, 23, 36, 53, 85, 124, 202, 289, 425, 603, 864, 1209, 1699, 2397, 3386, 4665, 6440, 8801, 12101, 16338, 22078, 29565, 39557, 52615, 69823, 92338, 121622, 159435, 208513, 271775, 353436, 457759, 591191, 760763, 976412, 1250011, 1596723, 2034474, 2585159, 3277192, 4145341, 5232888, 6591553

Offset: 0

Farideh Firoozbakht, Aug 01 2005

nonn

Conjecture: For n>2, a(n+1)>a(n).

			a(4)=2 because 8743 & 9481 are all solutions of the equation sigma(x)=10^4.

Max Alekseyev, Table of n, a(n) for n = 0..1000
Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2

Cf. A110076, A110077.

{ a(d) = local(X,Y,P,L,n,f,p,m,l); X=Pol([1,0],x); Y=Pol([1,0],y); P=Set(); L=listcreate(10000); for(i=0,d, for(j=0,d, n=2^i*5^j; if(n==1,next); f=factorint(n-1)[,1]; for(k=1,length(f), p=f[k]; m=n*(p-1)+1; while(m%p==0,m\=p); if(m==1, l=setsearch(P,p); if(l==0,l=setsearch(P,p,1); P=setunion(P,[p]); listinsert(L,1,l)); L[l]+=X^i*Y^j ) ) )); R=1+O(x^(d+1))+O(y^(d+1)); for(l=1,length(L),R*=L[l]); listkill(L); vector(d+1,n,polcoeff(polcoeff(R,n-1),n-1)) } (Alekseyev)

a(n) = coefficient of x^n*y^n in Prod_p Sum_{u, v} x^u*y^v, where the product is taken over all primes p and the sum is taken over such u, v that 2^u*5^v = sigma(p^k) for some nonnegative integer k. - Max Alekseyev, Aug 08 2005

More terms from Max Alekseyev, Aug 08 2005

Terms a(44) onward from Max Alekseyev, Mar 04 2014

cp's OEIS Frontend

A180265 a(n) = smallest k such that sigma(k) = 14^n, or zero if no such k exists.

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A180460 a(n) is the smallest number m such that sigma(m)=22^n, or 0 if m does not exist.

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A180461 a(n) is the smallest number m such that sigma(m)=26^n, or 0 if m does not exist.

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A180462 a(n) is the smallest number m such that sigma(m)=34^n, or 0 if m does not exist.

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A180463 a(n) is the smallest number m such that sigma(m)=38^n, or 0 if m does not exist.

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A180464 a(n) is the smallest number m such that sigma(m)=46^n, or 0 if m does not exist.

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A072075 Smallest solution to phi(x) = 10^n where phi(x) = A000010(x).

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A110076 a(n) is the largest number m such that sigma(m)=10^n, or if there is no such m a(n)=0.

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A110078 a(n) is number of solutions of the equation sigma(x)=10^n.

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