A061345 -id:A061345 - cp's OEIS frontend

A115232 Primes p which can be written in the form 2^i + q^j where i >= 0, j >= 1, q = odd prime.

5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281

Offset: 1

Reinhard Zumkeller, Jan 17 2006

nonn

a(n)=A000040(n+2) for n <= 32, but A000040(35)=149 is a term of A115231;
A115233 is a subsequence; the union with A115231 gives all primes (A000040);
A006512 and A053703 are subsequences.

Cf. A000079, A061345.

Cf. A006512, A053703, A115231, A115233.

maxp = 281;
Union[Sort[Reap[Do[p = 2^i + q^j; If[p <= maxp && PrimeQ[p], Sow[p]], {i, 0, Log[2, maxp]//Ceiling}, {j, 1, Log[3, maxp]//Ceiling}, {q, Prime[Range[2, PrimePi[maxp]]]}]][[2, 1]]]] (* Jean-François Alcover, Aug 03 2018 *)

Recomputed (based on recomputation of A115230) by R. J. Mathar and Reinhard Zumkeller, Apr 29 2010
Edited by N. J. A. Sloane, Apr 30 2010
Terms a(1)=2 and a(2)=3 removed from Data by Jean-François Alcover, Aug 03 2018

A235868 Union of 2 and powers of odd primes.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229

Offset: 1

José María Grau Ribas, Feb 23 2014

nonn

Numbers n such that the group G_n:={x+yi: x^2+y^2==1 (mod n); 0<=x,yA060968(n) = A235863(n).

Jose María Grau, A. M. Oller-Marcen, Manuel Rodriguez and D. Sadornil, Fermat test with Gaussian base and Gaussian pseudoprimes, arXiv:1401.4708 [math.NT], 2014.

Cf. A060968, A061345, A235863.

Select[ Range[230], # == 2 || Mod[#, 2] == 1 && PrimeNu[#] < 2 &] (* and modified by Robert G. Wilson v, Dec 29 2016 *)

{2} UNION A061345. - R. J. Mathar, Jul 19 2024

A280200 Expansion of 1/(1 - Sum_{k>=2} floor(1/omega(2k-1))x^(2*k-1)), where omega() is the number of distinct prime factors (A001221).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 11, 16, 21, 26, 37, 47, 61, 84, 108, 143, 191, 249, 331, 437, 575, 763, 1004, 1326, 1754, 2311, 3055, 4036, 5323, 7033, 9288, 12257, 16193, 21379, 28223, 37278, 49212, 64984, 85815, 113297, 149614, 197551, 260839, 344439, 454795, 600517, 792958, 1047023, 1382519, 1825533, 2410456, 3182845

Offset: 0

Ilya Gutkovskiy, Dec 28 2016

nonn

Number of compositions (ordered partitions) into odd prime powers (1 excluded).

			a(10) = 3 because we have [7, 3], [5, 5] and [3, 7].

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Prime Power
Index entries for sequences related to compositions

Cf. A001221, A023359, A061345, A246655, A280151, A280195.

nmax = 60; CoefficientList[Series[1/(1 - Sum[Floor[1/PrimeNu[2 k - 1]] x^(2 k - 1), {k, 2, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=2} floor(1/omega(2*k-1))*x^(2*k-1)).

A361924 Numbers whose infinitary divisors have distinct values of the infinitary totient function iphi (A091732).

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 25, 27, 28, 29, 31, 33, 35, 36, 37, 39, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 85, 87, 89, 91, 92, 93, 95, 97, 99, 100, 101

Offset: 1

Amiram Eldar, Mar 30 2023

nonn

First differs from A003159 at n = 57.
Numbers k such that A361923(k) = A037445(k).
Since Sum_{d infinitary divisor of k} iphi(d) = k, these are numbers k such that the multiset {iphi(d) | d infinitary divisor of k} is a partition of k into distinct parts.
Includes all the odd prime powers (A061345) and all the powers of 4 (A000302).
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 6, 66, 651, 6497, 64894, 648641, 6485605, 64851632, 648506213, 6485025363, ... . Apparently, this sequence has an asymptotic density 0.6485...

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

Cf. A000302, A003159, A037445, A061345, A077609, A091732, A361923.

Similar sequences: A326835, A348004.

f[p_, e_] := p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], 1]));
iphi[1] = 1; iphi[n_] := Times @@ (Flatten@ (f @@@ FactorInteger[n]) - 1);
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]; idivs[1] = {1};
q[n_] := Length @ Union[iphi /@ (d = idivs[n])] == Length[d]; Select[Range[100], q]

iphi(n) = {my(f=factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k)) - 1, 1)))}
isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(d = divisors(n), f = factor(n), idiv = []); for (k=1, #d, if(isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
is(k) = {my(d = idivs(k)); #Set(apply(x->iphi(x), d)) == #d;}

A110268 Consider the sequence A110566: lcm{1,2,...,n}/denominator of harmonic number H(n). a(n) is the factor that is changed going from A110566(n) to A110566(n+1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 3, 1, 1, 3, 5, 1, 3, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 11, 1, 1, 1, 1, 7, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 11, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 5

Offset: 1

Robert G. Wilson v, Sep 17 2005

nonn

a(n) is always an odd prime power, A061345.

			A110566(4) through A110566(10) are {1,1,3,3,3,1,1}, therefore the factors are 1,3,1,1,3,1.

Cf. A110566, A112811.

f[n_] := LCM @@ Range[n]/Denominator[HarmonicNumber[n]]; Table[ LCM[f[n], f[n + 1]]/GCD[f[n], f[n + 1]], {n, 104}]

f(n) = lcm(vector(n, k, k))/denominator(sum(k=1, n, 1/k));
a(n) = my(x = f(n+1)/f(n)); if (x > 1, x, 1/x); \\ Michel Marcus, Mar 07 2018

A117119 Number of partitions of 2*n into two odd prime powers.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 5, 6, 6, 6, 7, 8, 6, 9, 7, 6, 8, 7, 6, 8, 7, 7, 9, 8, 7, 9, 8, 7, 11, 9, 7, 12, 8, 7, 9, 9, 8, 10, 8, 9, 12, 11, 9, 12, 9, 8, 13, 9, 8, 13, 10, 11, 14, 11, 8, 13, 12, 10, 13, 9, 9, 16, 10, 11, 14, 10, 10, 15, 10, 9, 16, 12, 9, 16, 12, 11, 18

Offset: 1

Reinhard Zumkeller, Apr 15 2006

nonn

Conjecture: For all n, a(n) > 0; a(n) > A002375(n).

			a(1) = #{1+1} = 1; a(2) = #{1+3} = 1; a(3) = #{1+5, 3+3} = 2;
a(20) = #{3+37, 3^2+31, 11+29, 13+3^3, 17+23} = 5;
a(21) = #{1+41, 5+37, 11+31, 13+29, 17+5^2, 19+23} = 6.

Eric Weisstein's World of Mathematics, Goldbach Conjecture
Wikipedia, Goldbach's conjecture
Wikipedia, Waring-Goldbach problem
Index entries for sequences related to Goldbach conjecture

Cf. A061345.

isA061345 := proc(n)
    if n = 1 then
        true;
    elif type(n,'even') then
        false;
    elif nops(numtheory[factorset](n)) = 1 then
        true;
    else
        false;
    end if;
end proc:
A117119 := proc(n)
    local a,j,i;
    a := 0 ;
    for i from 1 do
        j := 2*n-i ;
        if j < i then
            break;
        end if;
        if isA061345(i) and isA061345(j) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A117119(n),n=1..60) ; # R. J. Mathar, Jul 09 2016

oppQ[n_] := n == 1 || OddQ[n] && PrimeNu[n] == 1; a[n_] := (k = 0; For[i = 1, True, i++, j = 2n - i; If[j < i, Break[]]; If[oppQ[i] && oppQ[j], k++] ]; k); Array[a, 100] (* Jean-François Alcover, Feb 13 2018 *)

A182946 Array of odd prime powers p^j, where j>=1, by antidiagonals.

Original entry on oeis.org

3, 9, 5, 27, 25, 7, 81, 125, 49, 11, 243, 625, 343, 121, 13, 729, 3125, 2401, 1331, 169, 17, 2187, 15625, 16807, 14641, 2197, 289, 19, 6561, 78125, 117649, 161051, 28561, 4913, 361, 23, 19683, 390625, 823543, 1771561, 371293, 83521, 6859, 529, 29

Offset: 1

Clark Kimberling, Dec 14 2010

The monotonic ordering of A182946, with 1 prefixed, is A061345. The joint-rank array of A182946 is A182870.

Cf. A061345, A182445.

 width=9;Transpose[Table[Table[Prime[n+1]^j,{n,1,width},{j,1,width}]]]; Flatten[Table[Table[%[[z-k+1]][[k]],{k,1,z}],{z,1,width}]]

A228485 Odd prime powers p^k such that p is congruent to 2 or 5 mod 9.

Original entry on oeis.org

5, 11, 23, 25, 29, 41, 47, 59, 83, 101, 113, 121, 125, 131, 137, 149, 167, 173, 191, 227, 239, 257, 263, 281, 293, 311, 317, 347, 353, 383, 389, 401, 419, 443, 461, 479, 491, 509, 529, 563, 569, 587, 599, 617, 625, 641, 653, 659, 677, 743, 761, 797, 821, 839

Offset: 1

Arkadiusz Wesolowski, Aug 23 2013

nonn

For any n, the equation x^3 + y^3 = a(n)*z^3 is not solvable in nonzero integers. Therefore, these numbers do not occur in A020898.

Henri Cohen, Number Theory. Volume I: Tools and Diophantine Equations, Graduate Texts in Mathematics 239, Springer, 2007, pp. 374-375.

Wikipedia, Prime power

Cf. A020898, A025473. Subsequence of A061345.

forstep(n=3, 839, 2, p=isprimepower(n); if(p>0, m=Mod(round(n^(1/p)), 9); if(m==2||m==5, print1(n, ", "))));

A273938 Sum of the divisors of the n-th odd prime power.

Original entry on oeis.org

1, 4, 6, 8, 13, 12, 14, 18, 20, 24, 31, 40, 30, 32, 38, 42, 44, 48, 57, 54, 60, 62, 68, 72, 74, 80, 121, 84, 90, 98, 102, 104, 108, 110, 114, 133, 156, 128, 132, 138, 140, 150, 152, 158, 164, 168, 183, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230

Offset: 1

R. J. Mathar, Jun 04 2016

nonn

Cf. A061345, A086455, A247837.

A273938 := proc(n)
        numtheory[sigma](A061345(n-1)) ;
end proc:

a(n) = A000203(A061345(n-1)).

A274915 Powers of odd non-Fermat primes.

Original entry on oeis.org

1, 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 263, 269, 271, 277, 281, 283, 293, 307, 311

Offset: 1

Juri-Stepan Gerasimov, Nov 11 2016

n is in the sequence if n = p^m where p is in A138889 and m >= 0. - Robert Israel, Sep 15 2017
The difference between two divisors of n is never a power of 2. The first number with this property that is not in the sequence is 91. - Robert Israel, Sep 15 2017
Subsequence of A061345.

			49 is in this sequence because 49 = 7^2 and 7 is not a Fermat prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A019434, A061345, A092506, A138889, A277994.

N:= 500: # to get all terms <= N
P:= select(isprime, {seq(i,i=7..N,2)}) minus {seq(2^i+1, i=1..ilog2(N))}:
sort(convert(map(p -> seq(p^k,k=0..floor(log[p](N))), P), list)); # Robert Israel, Sep 15 2017

A277994(a(n)) = 0.

Edited, new name, and corrected by Robert Israel, Sep 15 2017

cp's OEIS Frontend

A115232 Primes p which can be written in the form 2^i + q^j where i >= 0, j >= 1, q = odd prime.

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A235868 Union of 2 and powers of odd primes.

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A280200 Expansion of 1/(1 - Sum_{k>=2} floor(1/omega(2*k-1))*x^(2*k-1)), where omega() is the number of distinct prime factors (A001221).

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A361924 Numbers whose infinitary divisors have distinct values of the infinitary totient function iphi (A091732).

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A110268 Consider the sequence A110566: lcm{1,2,...,n}/denominator of harmonic number H(n). a(n) is the factor that is changed going from A110566(n) to A110566(n+1).

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Mathematica

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A117119 Number of partitions of 2*n into two odd prime powers.

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A182946 Array of odd prime powers p^j, where j>=1, by antidiagonals.

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Mathematica

A228485 Odd prime powers p^k such that p is congruent to 2 or 5 mod 9.

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PARI

A280200 Expansion of 1/(1 - Sum_{k>=2} floor(1/omega(2k-1))x^(2*k-1)), where omega() is the number of distinct prime factors (A001221).