A174793 -id:A174793 - cp's OEIS frontend

A174624 Triangle read by rows: T(n,k) = prime(n) mod 2^Omega(k), where Omega() is the number of prime divisors (counted with multiplicity).

0, 0, 1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 1, 3, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 3, 1, 3, 1, 3, 0, 1, 1, 3, 1, 3, 1, 7, 3, 0, 1, 1, 1, 1, 1, 1, 5, 1, 1, 0, 1, 1, 3, 1, 3, 1, 7, 3, 3, 1, 0, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 3, 1, 3, 1, 3, 3, 3, 1, 3, 1, 3, 0, 1, 1, 3, 1, 3, 1, 7, 3, 3, 1, 7, 1, 3, 3

Offset: 1

Juri-Stepan Gerasimov, Nov 29 2010

			Triangle begins:
  0;
  0, 1;
  0, 1, 1;
  0, 1, 1, 3;
  0, 1, 1, 3, 1;
  0, 1, 1, 1, 1, 1;

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001222, A174793.

A174624 := proc(n,k) ithprime(n) mod (2^numtheory[bigomega](k)) ; end proc:
seq(seq(A174624(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Nov 30 2010

Table[Mod[Prime[n],2^PrimeOmega[k]],{n,20},{k,n}]//Flatten (* Harvey P. Dale, Jun 02 2016 *)

A279105 a(n), n>1, is the smallest number k whose symmetric representation of sigma(k) has two parts and has a larger number of legs in its two parts than a(n-1); a(1)=3.

Original entry on oeis.org

3, 10, 44, 78, 136, 348, 592, 666, 820, 1272, 1652, 1830, 2144, 2628, 3320, 3738, 4656, 5886, 6328, 7620, 8384, 9042, 10728, 13040, 14532, 15752, 16290, 18528, 21100, 21944, 24084, 25424, 28920, 32382, 32896, 35508, 39340, 42192, 46050, 48828

Offset: 1

Hartmut F. W. Hoft, Dec 06 2016

nonn

A number k with two parts in its symmetric representation of sigma(k) [ssrs(k) = 2] has the form k = q*p with q in A174973, p prime and 2*q < p. This implies that 2*q <= row(k) < p and the first 0 in the k-th row of A249223 (having row(k) = floor((sqrt(8*k+1)-1)/2) entries) occurs at position 2*q so that 2*q-1 is the number of legs in each of the two parts. Therefore, the numbers 2*q-1 with q in A174973 are the only possible leg counts when ssrs(k) = 2, and for given q in A174973 and smallest prime p(q) > 2*q the number k = q*p(q) is the smallest with a leg count of 2*q-1. Consequently, each number q*p in the column of the irregular triangle A239929 labeled by q in A174793 with p prime satisfies ssrs(q*p) = 2*q-1.
a(1) = 3 is the only odd number since 1 is the only odd number in A174973.
Every number n = 2^m * p, m >= 0, 2^(m+1) < p and p prime, in this sequence is the sum of 2^(m+1) consecutive positive integers which includes every number in A246956.

			a(3)=44 is the smallest number whose symmetric representation has 2 parts and 7 legs in each part.
a(4)=78 is the smallest number whose symmetric representation has 2 parts and 11 legs in each part.
No number k whose symmetric representation of sigma(k) has 2 parts can have 21 legs in its parts since there is no q in A174973 such that 2*q - 1 = 21.

Right border of A239929.
Supersequence of A246956 and A262259.
Cf. A007918, A174973, A237048, A237270, A237271, A237593, A246956, A249223, A279105.

a174973Q[n_] := Module[{d=Divisors[n]}, Select[Rest[d] - 2*Most[d], #>0&]=={}]
a279105[n_] := Map[# * NextPrime[2*#]&, Select[Range[n], a174973Q]]
a279105[150] (* sequence data *)

a(n) = A174973(n) * A007918(2 * A174973(n) + 1).

A279106 a(n) = number of legs in each part of the symmetric representation of sigma(A279105(n)).

Original entry on oeis.org

1, 3, 7, 11, 15, 23, 31, 35, 39, 47, 55, 59, 63, 71, 79, 83, 95, 107, 111, 119, 127, 131, 143, 159, 167, 175, 179, 191, 199, 207, 215, 223, 239, 251, 255, 263, 279, 287, 299, 311, 319, 323, 335, 351, 359, 383, 391, 395, 399, 407, 415, 419, 431, 439, 447, 455, 467, 479

Offset: 1

Hartmut F. W. Hoft, Dec 06 2016

nonn

			a(3) = 7 = 2 * A279105(3) - 1;
21 is not in the sequence since 11=(21+1)/2 is not in A174793.

Cf. A174973, A237270, A237271, A237593, A279105.

a174973Q[n_] := Module[{d=Divisors[n]}, Select[Rest[d] - 2*Most[d], #>0&]=={}]
a279106[n_]:=2*Select[Range[n], a174973Q] - 1
a279106[250] (* sequence data *)

a(n) = 2 * A279105(n) - 1.

cp's OEIS Frontend

A174624 Triangle read by rows: T(n,k) = prime(n) mod 2^Omega(k), where Omega() is the number of prime divisors (counted with multiplicity).

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A279105 a(n), n>1, is the smallest number k whose symmetric representation of sigma(k) has two parts and has a larger number of legs in its two parts than a(n-1); a(1)=3.

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A279106 a(n) = number of legs in each part of the symmetric representation of sigma(A279105(n)).

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