A045344 -id:A045344 - cp's OEIS frontend

A173494 Numbers m such that no square greater than 1 can be written as sum of distinct divisors of m.

1, 2, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 38, 41, 43, 47, 53, 58, 59, 61, 65, 67, 71, 73, 74, 77, 79, 82, 83, 85, 86, 89, 91, 97, 101, 103, 106, 107, 109, 113, 127, 131, 133, 134, 137, 139, 145, 146, 149, 151, 157, 163, 167, 173, 178, 179, 181, 185, 187, 191, 193, 197

Offset: 1

Reinhard Zumkeller, Feb 20 2010

nonn

2546 is the smallest term having more than two prime factors: 2546 = 2*19*67.

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

Cf. A173493.

A045344 is a subsequence (the primes except 3).

q[m_] := Module[{d = Divisors[m], sum, sq, x}, sum = Plus @@ d; sq = Range[2, Floor[Sqrt[sum]]]^2; Total[CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sq]]] == 0]; Select[Range[200], q] (* Amiram Eldar, Apr 16 2025 *)

A173493(a(n)) = 1.

A235480 Primes whose base-3 representation is also the base-9 representation of a prime.

Original entry on oeis.org

2, 5, 7, 11, 17, 19, 23, 31, 37, 41, 43, 53, 67, 71, 73, 83, 89, 97, 103, 149, 157, 199, 239, 251, 257, 271, 277, 293, 307, 313, 331, 337, 359, 383, 397, 421, 431, 433, 499, 541, 557, 571, 587, 599, 601, 613, 631, 653, 659, 661, 683, 691, 709, 727, 751, 769, 823, 887, 911, 983, 1009, 1021, 1031, 1049, 1051, 1063, 1129, 1163, 1217

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of a two-dimensional array of sequences, given in the LINK, based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.

Appears to be a subsequence of A015919, A045344, A052085, A064555 and A143578.

			5 = 12_3 and 12_9 = 11 are both prime, so 5 is a term.

Giovanni Resta, Table of n, a(n) for n = 1..10000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235265, A235473 - A235479, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395, A235461 - A235482. See the LINK for further cross-references.

Select[Prime@ Range@ 500, PrimeQ@ FromDigits[ IntegerDigits[#, 3], 9] &] (* Giovanni Resta, Sep 12 2019 *)

is(p,b=9,c=3)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: Code only valid for b > c.

A279508 a(n) = smallest number k such that floor(phi(k)/tau(k)) = n.

Original entry on oeis.org

2, 1, 5, 7, 27, 11, 13, 58, 17, 19, 55, 23, 65, 106, 29, 31, 85, 142, 37, 158, 41, 43, 115, 47, 119, 125, 53, 133, 145, 59, 61, 254, 262, 67, 274, 71, 73, 298, 1180, 79, 187, 83, 203, 346, 89, 209, 235, 382, 97, 394, 101, 103, 169, 107, 109, 253, 113, 458, 295

Offset: 0

Jaroslav Krizek, Dec 13 2016

nonn

a(n) = the smallest number k such that floor(A000010(k)/A000005(k)) = A279507(k) = n.
Sequences b_n of numbers k such that floor(phi(k)/tau(k)) = n for n = 0..2:
b_0: 2, 4, 6, 12;
b_1: 1, 3, 8, 10, 14, 16, 18, 20, 24, 30, 36, 42, 48, 60;
b_2: 5, 9, 15, 22, 28, 32, 40, 54, 66, 72, 84, 90, 96, 120, 180.
Sequences b_n are finite for all n >=0. See A279509 (largest number k such that floor(phi(k)/tau(k)) = n).
Supersequence of A045344 (primes excluding 3).

			For n = 2; a(2) = 5 because 5 is the smallest number with floor(phi(5) / tau(5)) = floor(4/2) = 2.

Cf. A000005, A000010, A020488, A020490, A045344, A279289, A279507, A279509.

[Min([n: n in[1..100000] | Floor(EulerPhi(n)/NumberOfDivisors(n)) eq k]): k in [0..60]]

Table[k = 1; While[Floor[EulerPhi[k]/DivisorSigma[0, k]] != n, k++]; k, {n, 0, 58}] (* Michael De Vlieger, Dec 14 2016 *)

a(n) = my(k=1); while(floor((eulerphi(k)/numdiv(k)))!=n, k++); k \\ Felix Fröhlich, Dec 14 2016

a((p-1)/2) = p for p = prime > 3.

A343859 Partial sums of the primes excluding 3.

Original entry on oeis.org

2, 7, 14, 25, 38, 55, 74, 97, 126, 157, 194, 235, 278, 325, 378, 437, 498, 565, 636, 709, 788, 871, 960, 1057, 1158, 1261, 1368, 1477, 1590, 1717, 1848, 1985, 2124, 2273, 2424, 2581, 2744, 2911, 3084, 3263, 3444, 3635, 3828, 4025, 4224, 4435, 4658, 4885, 5114, 5347

Offset: 1

Omar E. Pol, May 01 2021

Partial sums of the primes congruent to {1, 2} mod 3.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Partial sums of A045344.
Apart from the initial term this is the column 4 of A343809.
Cf. A007504.

Accumulate[Join[{2},Prime[Range[3,100]]]] (* Paolo Xausa, Sep 01 2023 *)

a(n) = A007504(n+1) - 3.

A173907 Primes of form x^y+y^x where x and y are composite numbers.

Original entry on oeis.org

43143988327398957279342419750374600193, 5052785737795758503064406447721934417290878968063369478337, 205688069665150755269371147819668813122841983204711281293004769, 3329896365316142756322307042065269797678257903507506764421250291562312417, 814539297859635326656252304265822609649892589675472598580095801187688932052096060144958129

Offset: 1

Juri-Stepan Gerasimov, Mar 02 2010

nonn

			The first 5 terms are 15^32+32^15, 33^38+38^33, 8^69+69^8, 9^76+76^9, 21^68+68^21.

Robert Israel, Table of n, a(n) for n = 1..33 (all terms < 10^999)
Robert Israel, Representations of the first 46 terms as x^y + y^x

Cf. A061119, A094133, A162488, A162490, A045344.

N:= 10^100: # for terms <= N
R:= NULL:
for x from 4 while 2*x^x < N do
  if isprime(x) then next fi;
  for y from x+1 do
    if igcd(x,y) > 1 or isprime(y) then next fi;
    q:= x^y + y^x;
    if q > N then break fi;
    if isprime(q) then R:= R,q  fi;
od od:
sort([R]); # Robert Israel, Jul 11 2025

a(3)-a(5) from Franklin T. Adams-Watters, Mar 22 2010

Definition corrected by N. J. A. Sloane, Apr 13 2010

A176551 Products of 2 primes of the form 3*k-+1.

Original entry on oeis.org

4, 10, 14, 22, 25, 26, 34, 35, 38, 46, 49, 55, 58, 62, 65, 74, 77, 82, 85, 86, 91, 94, 95, 106, 115, 118, 119, 121, 122, 133, 134, 142, 143, 145, 146, 155, 158, 161, 166, 169, 178, 185, 187, 194, 202, 203, 205, 206, 209, 214, 215, 217, 218, 221, 226, 235, 247

Offset: 1

Juri-Stepan Gerasimov, Apr 20 2010

nonn

Semiprimes without 3*primes (or triple the primes).

Numbers of the form A045344(i)*A045344(j), any i, j. [From R. J. Mathar, Apr 27 2010]

Cf. A001358, A001748, A100959.

Entries checked by R. J. Mathar, Apr 27 2010

A176569 a(n) = (-1)^n + (n-th prime of the form 3k-+1).

Original entry on oeis.org

1, 6, 6, 12, 12, 18, 18, 24, 28, 32, 36, 42, 42, 48, 52, 60, 60, 68, 70, 74, 78, 84, 88, 98, 100, 104, 106, 110, 112, 128, 130, 138, 138, 150, 150, 158, 162, 168, 172, 180, 180, 192, 192, 198, 198, 212, 222, 228, 228, 234, 238, 242, 250, 258, 262, 270, 270, 278

Offset: 1

Juri-Stepan Gerasimov, Apr 20 2010

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10001

Cf. A045344.

A045344 := proc(n) if n = 1 then 2; else ithprime(n+1) ; end if; end proc:
A176569 := proc(n) (-1)^n+A045344(n) ; end proc:
seq(A176569(n),n=1..120) ; # R. J. Mathar, Apr 27 2010

Total /@ Nest[Append[#1, Block[{p = NextPrime@ #1[[-1, -1]]}, While[Mod[p, 3] == 0, p = NextPrime@ p]; {(-1)^#2, p}]] & @@ {#, Length@ # + 1} &, {{-1, 2}}, 57] (* Michael De Vlieger, Feb 09 2019 *)

a045344(n) = if(n<2, 2, prime(n+1));
a(n) = (-1)^n + a045344(n); \\ Michel Marcus, Feb 07 2019

a(n) = (-1)^n + A045344(n). - Michel Marcus, Feb 07 2019

Corrected (double-5 replaced by double-6, 146 removed) by R. J. Mathar, Apr 27 2010

A215390 Primes congruent to {1, 2} mod 11.

Original entry on oeis.org

2, 13, 23, 67, 79, 89, 101, 167, 199, 211, 233, 277, 331, 353, 397, 409, 419, 431, 463, 541, 563, 607, 617, 661, 673, 683, 727, 739, 761, 827, 859, 881, 937, 947, 991, 1013, 1069, 1091, 1123, 1201, 1223, 1277, 1289, 1321, 1399, 1409, 1453, 1487

Offset: 1

Vincenzo Librandi, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A045344, A045372, A045391.

[p: p in PrimesUpTo(1500) | p mod 11 in [1, 2]];

Select[Prime[Range[300]],MemberQ[{1,2},Mod[#,11]]&]

A215391 Primes congruent to {1, 2} mod 13.

Original entry on oeis.org

2, 41, 53, 67, 79, 131, 157, 197, 223, 313, 353, 379, 431, 443, 457, 509, 521, 547, 587, 599, 613, 677, 691, 743, 769, 821, 859, 911, 937, 977, 1093, 1171, 1223, 1237, 1249, 1289, 1301, 1327, 1367, 1471, 1483, 1523, 1549, 1601, 1613, 1627, 1783, 1847

Offset: 1

Vincenzo Librandi, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A045344, A045372,

[p: p in PrimesUpTo(2000) | p mod 13 in [1, 2]];

Select[Prime[Range[300]],MemberQ[{1,2},Mod[#,13]]&]

A215392 Primes congruent to {1, 2} mod 17.

Original entry on oeis.org

2, 19, 53, 103, 137, 223, 239, 257, 307, 359, 409, 443, 461, 563, 613, 631, 647, 733, 919, 937, 953, 971, 1021, 1039, 1123, 1259, 1277, 1327, 1361, 1429, 1447, 1481, 1531, 1549, 1583, 1667, 1753, 1787, 1871, 1889, 1973, 2143, 2161, 2297, 2347, 2381, 2399, 2467

Offset: 1

Vincenzo Librandi, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A045344, A045372.

[p: p in PrimesUpTo(2500) | p mod 17 in [1, 2]];

Select[Prime[Range[300]],MemberQ[{1,2},Mod[#,17]]&]

cp's OEIS Frontend

A173494 Numbers m such that no square greater than 1 can be written as sum of distinct divisors of m.

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A235480 Primes whose base-3 representation is also the base-9 representation of a prime.

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A279508 a(n) = smallest number k such that floor(phi(k)/tau(k)) = n.

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Magma

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A343859 Partial sums of the primes excluding 3.

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A173907 Primes of form x^y+y^x where x and y are composite numbers.

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A176551 Products of 2 primes of the form 3*k-+1.

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A176569 a(n) = (-1)^n + (n-th prime of the form 3k-+1).

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A215390 Primes congruent to {1, 2} mod 11.

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