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A102185 Iccanobirt semiprime indices (15 of 15): Indices of semiprime numbers in A102125.

5, 15, 16, 19, 20, 21, 22, 24, 28, 29, 35, 38, 44, 54, 58, 59, 72, 84, 106, 108, 137, 145, 174, 227, 238, 253, 258, 362, 363, 371, 388

Offset: 1

Jonathan Vos Post and Ray Chandler, Dec 31 2004

Cf. A001358, A102125, A102171-A102184, A102205.

A102125(a(n)) = A102205(n).

a(24)-a(27) from Robert Price, Nov 11 2018

Offset changed to 1 and a(28)-a(31) from Jinyuan Wang, Aug 15 2021

A102191 Iccanobirt semiprimes (1 of 15): Semiprime numbers in A102111.

Original entry on oeis.org

4, 185, 1135955, 550783729, 10755767351826313, 885150880428474601, 145045760838001005739, 276700469046311728441, 17906534239981723909956235510218343, 23104799226903739899579090259021365554, 504198286800075916303995704410366363448429, 2363732899718211729861685511671459865604449872085443

Offset: 1

Jonathan Vos Post and Ray Chandler, Dec 31 2004

Cf. A001358, A102111, A102192-A102205.

a(n) = A102111(A102171(n)).

Offset changed to 1 and more terms from Jinyuan Wang, Jul 31 2021

A102205 Iccanobirt semiprimes (15 of 15): Semiprime numbers in A102125.

Original entry on oeis.org

4, 5071, 6313, 31591, 9853, 11733, 31865, 736481, 9834802, 5123383, 906334841, 312395329, 73044385753, 39216355244851, 353123779923181, 944016528715333, 901870160743125919, 3394064622591216338731, 798539095539570459224764519, 5844680137439021618014007903

Offset: 1

Jonathan Vos Post and Ray Chandler, Dec 31 2004

Cf. A001358, A102125, A102185, A102191-A102204.

nxt[{a_,b_,c_}]:=With[{ir=IntegerReverse},{b,c,ir[ir[a]+ir[b]+ir[c]]}]; Select[NestList[nxt,{0,0,1},200][[;;,1]],PrimeOmega[#]==2&] (* Harvey P. Dale, Jul 22 2025 *)

a(n) = A102125(A102185(n)).

Offset changed to 1 and a(20) from Jinyuan Wang, Aug 14 2021

A109556 prime(k) for those k where floor((2(prime(k+1)-prime(k))PrimePi(k) mod (8*k))/k) = m with m = 2.

Original entry on oeis.org

3, 5, 11, 17, 79, 97, 103, 109, 127, 163, 193, 223, 229, 277, 307, 313, 349, 379, 397, 439, 457, 463, 487, 499, 613, 643, 677, 727, 733, 751, 941, 947, 971, 977, 991, 1013, 1033, 1063, 1097, 1103, 1117, 1123, 1181, 1187, 1217, 1223, 1231, 1283, 1291, 1321

Offset: 1

Roger L. Bagula, Jun 27 2005

Cf. A000040, A000720, A001359.

Cf. A109556-A109569 for m = 2..15.

a = Flatten[Table[If[Floor[2*Mod[(Prime[n + 1] - Prime[n])*PrimePi[n]/n, 8]] == 2, Prime[n], {}], {n, 1, 400}]]

Name edited and offset corrected by Amiram Eldar, Jun 02 2025

A109569 prime(k) for those k where floor((2(prime(k+1)-prime(k))PrimePi(k) mod (8*k))/k) = m with m = 15.

Original entry on oeis.org

19609, 25471, 35617, 44293, 58831, 89689, 107377, 162143, 218287, 253159, 290249, 302329, 360091, 404597, 507217, 514967, 517639, 531383, 570253, 619397, 621143, 639757, 642281, 673669, 716173, 736279, 794249, 795349, 815729, 873787, 912649, 933073, 937253, 1013063

Offset: 1

Roger L. Bagula, Jun 27 2005

Cf. A000040, A000720, A001359.

Cf. A109556-A109569 for m = 2..15.

a = Flatten[Table[If[Floor[2*Mod[(Prime[n + 1] - Prime[n])*PrimePi[n]/n, 8]] == 15, Prime[n], {}], {n, 1, 10000}]]

Name edited, offset corrected and more terms added by Amiram Eldar, Jun 02 2025

A128395 Numbers k such that k^2 divides 15^k-1.

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 16, 28, 56, 112, 136, 272, 452, 812, 904, 952, 1624, 1808, 1904, 3164, 3248, 6328, 11912, 12656, 15368, 18632, 23824, 27608, 30736, 37264, 47908, 55216, 60248, 83384, 91756, 95816, 102604, 107576, 113936, 120496, 130424, 166768

Offset: 1

Alexander Adamchuk, Mar 01 2007

Amiram Eldar, Table of n, a(n) for n = 1..640

Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.

a={1};For[n=1,n<200000,n++,If[PowerMod[15,n,n^2]==1, AppendTo[a, n]]]; a (* Stefan Steinerberger, Jun 10 2007 *)
Join[{1},Select[Range[167000],PowerMod[15,#,#^2]==1&]] (* Harvey P. Dale, Sep 14 2020 *)

is(k) = Mod(15, k^2)^k == 1; \\ Amiram Eldar, May 21 2024

More terms from Stefan Steinerberger, Jun 10 2007

A128402 Numbers k such that k^2 divides 22^k-1.

Original entry on oeis.org

1, 3, 7, 21, 39, 273, 507, 3081, 3549, 21567, 40053, 78117, 280371, 343239, 546819, 1015521, 2056899, 2402673, 5998317, 6171243, 7108647, 8740173, 12338859, 14398293, 18988203, 27115881, 41988219, 43198701, 47727771, 55431363

Offset: 1

Alexander Adamchuk, Mar 01 2007

Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128403, A128404.

select(t -> 22 &^ t - 1 mod t^2 = 0, [seq(2*k+1,k=0..10^6)]); # Robert Israel, Jan 23 2015

a={}; Do[r=(22^n-1)/n^2; If[r==IntegerPart[r], AppendTo[a, n]], {n, 1, 10^3}]; a (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)

{ forstep(m=11,10^8,2, if( Mod(22,m^2)^m==1, print(m) ) ) } \\ Max Alekseyev, Oct 18 2008

a(14)-a(30) from Max Alekseyev, Oct 18 2008

A128404 Numbers k such that k^2 divides 24^k-1.

Original entry on oeis.org

1, 23, 1081, 2870377, 7009273, 15954479, 134907719, 329435831, 537539141, 15001987199, 874750261127, 1991103024721, 4272172921319, 4862143429729, 7933540182019, 12816504745411, 41113262272969, 67084347257659

Offset: 1

Alexander Adamchuk, Mar 01 2007

23 divides all terms except the first.

Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403.

a={}; Do[r=(24^n-1)/n^2; If[r==IntegerPart[r], AppendTo[a, n]], {n, 1, 10^3}]; a (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)

a(5)-a(6) from Farideh Firoozbakht, Mar 05 2007
a(7)-a(10) from Ryan Propper, Feb 23 2008
Terms a(11) onward from Max Alekseyev, May 06 2010

A135577 Numbers that have only the digit "1" as first, central and final digit. For numbers with 5 or more digits the rest of digits are "0".

Original entry on oeis.org

1, 111, 10101, 1001001, 100010001, 10000100001, 1000001000001, 100000010000001, 10000000100000001, 1000000001000000001, 100000000010000000001, 10000000000100000000001, 1000000000001000000000001, 100000000000010000000000001, 10000000000000100000000000001

Offset: 1

Omar E. Pol, Feb 24 2008

Also, equal to A135576(n), written in base 2.
Essentially the same as A066138. - R. J. Mathar Apr 29 2008
a(n) has 2n-1 digits.

			----------------------------
n ............ a(n)
----------------------------
1 ............. 1
2 ............ 111
3 ........... 10101
4 .......... 1001001
5 ......... 100010001
6 ........ 10000100001
7 ....... 1000001000001
8 ...... 100000010000001
9 ..... 10000000100000001
10 ... 1000000001000000001

G. C. Greubel, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A001576, A066138, A135576, A138118, A138119, A138120, A138147, A138826.

Join[{1}, LinearRecurrence[{111, -1110, 1000}, {111, 10101, 1001001}, 25]] (* G. C. Greubel, Oct 19 2016 *)
Join[{1},Table[FromDigits[Join[{1},PadRight[{},n,0],{1},PadRight[{},n,0],{1}]],{n,0,10}]] (* Harvey P. Dale, Aug 15 2022 *)

Vec(-x*(2000*x^3-1110*x^2+1)/((x-1)*(10*x-1)*(100*x-1))  + O(x^100)) \\ Colin Barker, Sep 16 2013

a(n) = A135576(n), written in base 2.
Also, a(1)=1, for n>1; a(n)=(concatenation of 1, n-2 digits 0, 1, n-2 digits 0 and 1).
From Colin Barker, Sep 16 2013: (Start)
a(n) = 1 + 10^(n-1) + 100^(n-1) for n>1.
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>4.
G.f.: x*(2000*x^3 - 1110*x^2 + 1)/((1-x)*(10*x-1)*(100*x-1)). (End)
E.g.f.: (-111 - 200*x + 100*exp(x) + 10*exp(10*x) + exp(100*x))/100. - Elmo R. Oliveira, Jun 13 2025

A141269 (0, 1, 2, 3, 2^2, 5, 23, 7, 2^3, 3^2, 25, 11, 2^2*3, ...) transform in (0+1, 2+3, 2+2, 5+2, 3+7, 2+3, 3+2, 2+5, 11+2, 2+3, ...).

Original entry on oeis.org

1, 5, 4, 7, 10, 5, 5, 7, 13, 5, 15, 10, 7, 21, 5, 21, 4, 8, 9, 34, 5, 8, 4, 16, 5, 9, 31, 8, 33, 8, 13, 22, 9, 5, 39, 21, 16, 5, 46, 5, 50, 4, 14, 7, 25, 49, 7, 9, 7, 5, 19, 15, 55, 6, 16, 5, 10, 21, 88, 4, 8, 63, 34, 9, 8, 18, 5, 78, 4, 20, 25, 12, 73, 6, 75, 39, 8, 4, 21, 18, 5, 92, 6, 8, 6

Offset: 1

Juri-Stepan Gerasimov, Aug 08 2008

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

Cf. A136734, A141270.

Corrected at three or more places by R. J. Mathar, Feb 21 2009

cp's OEIS Frontend

A102185 Iccanobirt semiprime indices (15 of 15): Indices of semiprime numbers in A102125.

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A102191 Iccanobirt semiprimes (1 of 15): Semiprime numbers in A102111.

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A102205 Iccanobirt semiprimes (15 of 15): Semiprime numbers in A102125.

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A109556 prime(k) for those k where floor((2*(prime(k+1)-prime(k))*PrimePi(k) mod (8*k))/k) = m with m = 2.

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A109569 prime(k) for those k where floor((2*(prime(k+1)-prime(k))*PrimePi(k) mod (8*k))/k) = m with m = 15.

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A128395 Numbers k such that k^2 divides 15^k-1.

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A128402 Numbers k such that k^2 divides 22^k-1.

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A128404 Numbers k such that k^2 divides 24^k-1.

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A135577 Numbers that have only the digit "1" as first, central and final digit. For numbers with 5 or more digits the rest of digits are "0".

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A109556 prime(k) for those k where floor((2(prime(k+1)-prime(k))PrimePi(k) mod (8*k))/k) = m with m = 2.

A109569 prime(k) for those k where floor((2(prime(k+1)-prime(k))PrimePi(k) mod (8*k))/k) = m with m = 15.

A141269 (0, 1, 2, 3, 2^2, 5, 23, 7, 2^3, 3^2, 25, 11, 2^2*3, ...) transform in (0+1, 2+3, 2+2, 5+2, 3+7, 2+3, 3+2, 2+5, 11+2, 2+3, ...).