A096911 -id:A096911 - cp's OEIS frontend

A379317 Positive integers with a unique even prime index.

3, 6, 7, 12, 13, 14, 15, 19, 24, 26, 28, 29, 30, 33, 35, 37, 38, 43, 48, 51, 52, 53, 56, 58, 60, 61, 65, 66, 69, 70, 71, 74, 75, 76, 77, 79, 86, 89, 93, 95, 96, 101, 102, 104, 106, 107, 112, 113, 116, 119, 120, 122, 123, 130, 131, 132, 138, 139, 140, 141, 142

Offset: 1

Gus Wiseman, Dec 29 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
   3: {2}
   6: {1,2}
   7: {4}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  19: {8}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  29: {10}
  30: {1,2,3}
  33: {2,5}
  35: {3,4}
  37: {12}
  38: {1,8}
  43: {14}
  48: {1,1,1,1,2}

Partitions of this type are counted by A038348 (strict A096911).
For all even parts we have A066207, counted by A035363 (strict A000700).
For no even parts we have A066208, counted by A000009 (strict A035457).
Positions of 1 in A257992.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A349158.
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379311 old prime, see A204389, A320629, A379312-A379315.
Cf. A000720, A038550, A087436.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[Select[prix[#],EvenQ]]==1&]

A116679 Triangle read by rows: T(n,k) is the number of partitions of n into distinct part and having exactly k even parts (n >= 0, k >= 0).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 2, 3, 1, 2, 4, 2, 2, 5, 3, 2, 6, 4, 3, 7, 4, 1, 3, 8, 6, 1, 3, 10, 8, 1, 4, 11, 10, 2, 5, 13, 11, 3, 5, 15, 14, 4, 5, 18, 18, 5, 6, 20, 21, 7, 7, 23, 24, 9, 1, 8, 26, 29, 12, 1, 8, 30, 36, 14, 1, 9, 34, 41, 18, 2, 11, 38, 47, 23, 3, 12, 43, 55, 28, 4

Offset: 0

Emeric Deutsch, Feb 22 2006

Row n contains floor((1 + sqrt(1+4*n))/2) terms.
Row sums yield A000009.
T(n,0) = A000700(n), T(n,1) = A096911(n) for n >= 1.
Sum_{k>=0} k*T(n,k) = A116680(n).

			T(9,2)=2 because we have [6,2,1] and [4,3,2].
Triangle starts:
  1;
  1;
  0, 1;
  1, 1;
  1, 1;
  1, 2;
  1, 2, 1;
  1, 3, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A000009, A000700, A096911, A116680.

g:=product((1+x^(2*j-1))*(1+t*x^(2*j)),j=1..25): gser:=simplify(series(g,x=0,38)): P[0]:=1: for n from 1 to 27 do P[n]:=sort(coeff(gser,x^n)) od: for n from 0 to 27 do seq(coeff(P[n],t,j),j=0..floor((sqrt(1+4*n)-1)/2)) od; # yields sequence in triangular form

With[{m=25}, CoefficientList[CoefficientList[Series[Product[(1+x^(2*j- 1))*(1+t*x^(2*j)), {j,1,m+2}], {x,0,m}, {t,0,m}], x], t]]//Flatten (* G. C. Greubel, Jun 07 2019 *)

G.f.: Product_{j>=1} (1+x^(2*j-1))*(1+t*x^(2*j)).

A116928 Number of 1's in all self-conjugate partitions of n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 3, 2, 4, 4, 6, 6, 8, 9, 11, 12, 15, 17, 20, 22, 26, 29, 34, 37, 43, 48, 55, 60, 69, 76, 86, 94, 106, 117, 131, 143, 160, 176, 195, 213, 236, 259, 285, 311, 342, 374, 410, 446, 488, 533, 581, 631, 688, 748, 813, 881, 957, 1038, 1125, 1216, 1317, 1425

Offset: 1

Emeric Deutsch, Feb 26 2006

nonn

a(n)=Sum(k*A116927(n,k), k>=0).

			a(12)=6 because the self-conjugate partitions of 12 are [6,2,1,1,1,1],[5,3,2,1,1] and [4,4,2,2], containing a total of six 1's.

Cf. A116927.

f:=x+sum(x^(k^2+2)/(1-x^2)/product(1-x^(2*j),j=1..k),k=1..10): fser:=series(f,x=0,70): seq(coeff(fser,x^n),n=1..67);

G.f.=x+sum(x^(k^2+2)/(1-x^2)/product(1-x^(2j), j=1..k), k=1..infinity).

a(n) = A096911(n)-(1+(-1)^n)/2, m>1. - Vladeta Jovovic, Feb 27 2006

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A379317 Positive integers with a unique even prime index.

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A116679 Triangle read by rows: T(n,k) is the number of partitions of n into distinct part and having exactly k even parts (n >= 0, k >= 0).

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A116928 Number of 1's in all self-conjugate partitions of n.

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