A376678
Position of first zero in the n-th differences of the primes, or 0 if it does not appear.
Original entry on oeis.org
0, 0, 2, 7, 69, 13, 47, 58, 9, 43, 3553, 100, 7019, 14082, 68097, 14526, 149677, 2697, 481054, 979719, 631894, 29811, 25340978, 50574254, 7510843, 210829337, 67248861, 224076286, 910615647, 931510269, 452499644, 2880203722, 396680865, 57954439970, 77572822440, 35394938648
Offset: 0
The third differences of the primes begin:
-1, 2, -4, 4, -4, 4, 0, -6, 8, ...
so a(3) = 7.
This is the position at which 0 first appears in row n of
A095195.
For composite instead of prime we have
A377037.
For squarefree instead of prime we have
A377042, nonsquarefree
A377050.
For prime-power instead of prime we have
A377055.
Cf.
A000720,
A007442,
A030016,
A065890,
A084758,
A140119,
A258025,
A258026,
A333254,
A349643,
A376681,
A376682,
A376683.
-
nn=100000;
u=Table[Differences[Select[Range[nn],PrimeQ],k],{k,2,16}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
m=Table[Position[u[[k]],0][[1,1]],{k,mnrm[Union[First/@Position[u,0]]]}]
A376855
Position of first 0 in the n-th differences of the noncomposite numbers (A008578), or 0 if it does not appear.
Original entry on oeis.org
0, 0, 1, 8, 70, 14, 48, 59, 10, 44, 3554, 101, 7020, 14083, 68098, 14527, 149678, 2698, 481055, 979720, 631895, 29812, 25340979, 50574255, 7510844, 210829338, 67248862, 224076287, 910615648, 931510270, 452499645, 2880203723, 396680866, 57954439971, 77572822441, 35394938649
Offset: 0
The third differences of the noncomposite numbers begin: 1, -1, 2, -4, 4, -4, 4, 0, -6, 8, ... so a(3) = 8.
For firsts instead of positions of zeros we have
A030016, modern
A007442.
For row-sums instead of zero-positions we have
A376683, modern
A140119.
For composite instead of noncomposite we have
A377037.
For squarefree instead of noncomposite we have
A377042, nonsquarefree
A377050.
For prime-power instead of noncomposite we have
A377055.
-
nn=10000;
u=Table[Differences[Select[Range[nn],#==1||PrimeQ[#]&],k],{k,2,16}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
m=Table[Position[u[[k]],0][[1,1]],{k,mnrm[Union[First/@Position[u,0]]]}]
A377042
Position of first zero in the n-th differences of the squarefree numbers (A005117), or 0 if it does not appear.
Original entry on oeis.org
0, 0, 1, 11, 8, 57, 14, 11, 13, 1019, 44, 1250, 43, 2721, 42, 249522, 2840, 1989839, 2839, 3373774, 4933, 142715511, 42793, 435650856, 5266, 30119361, 104063, 454172978707, 100285, 434562125244, 2755089, 2409925829164, 2485612
Offset: 0
The fourth differences begin: -3, 3, 1, -6, 7, -5, 3, 0, -2, ... so a(4) = 8
This is the first position of 0 in each row of
A377038.
For nonsquarefree numbers we have
A377050.
A073576 counts integer partitions into squarefree numbers, factorizations
A050320.
Cf.
A007674,
A053797,
A053806,
A061398,
A072284,
A076259,
A112925,
A120992,
A376311,
A376590,
A376591,
A377046.
-
nn=10000;
u=Table[Differences[Select[Range[nn],SquareFreeQ],k],{k,2,16}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
m=Table[Position[u[[k]],0][[1,1]],{k,mnrm[Union[First/@Position[u,0]]]}]
A377055
Position of first appearance of zero in the n-th differences of the prime-powers (A246655), or 0 if it does not appear.
Original entry on oeis.org
0, 0, 1, 1, 4, 48, 61, 83, 29, 57, 290, 121, 7115, 14207, 68320, 14652, 149979, 122704, 481540, 980376, 632441, 29973, 25343678, 50577935, 7512418, 210836403, 67253056, 224083553, 910629561, 931524323, 452509699, 2880227533, 396690327, 57954538325, 77572935454, 35395016473
Offset: 0
The fourth differences of A246655 begin: 1, -3, 3, 0, -2, 2, ... so a(4) = 4.
These are the positions of first zeros in each row of
A377051.
A246655 lists the prime-powers, differences
A057820 (except first term).
-
nn=10000;
u=Table[Differences[Select[Range[nn],PrimePowerQ],k],{k,2,16}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
m=Table[Position[u[[k]],0][[1,1]], {k,mnrm[Union[First/@Position[u,0]]]}]
A377285
Position of first 0 in the n-th differences of the strict partition numbers A000009, or 0 if 0 does not appear.
Original entry on oeis.org
0, 1, 1, 5, 5, 8, 20, 7, 22
Offset: 0
The 7th differences of A000009 are: 25, -16, 7, -6, 10, -9, 0, 10, ... so a(7) = 7.
For squarefree numbers we have
A377042.
For nonsquarefree numbers we have
A377050.
Position of first zero in each row of
A378622. See also:
-
A175804 is the version for partitions.
-
A293467 gives first column (up to sign).
-
A378971 gives row-sums of absolute value.
-
Table[Position[Differences[PartitionsQ/@Range[0,100],k],0][[1,1]],{k,1,8}]
-
a(n, nn=100) = my(q='q+O('q^nn), v=Vec(eta(q^2)/eta(q))); for (i=1, n, my(w=vector(#v-1, k, v[k+1]-v[k])); v = w;); my(vz=select(x->x==0, v, 1)); if (#vz, vz[1]); \\ Michel Marcus, Dec 15 2024
A377036
First term of the n-th differences of the composite numbers. Inverse zero-based binomial transform of A002808.
Original entry on oeis.org
4, 2, 0, -1, 2, -2, 0, 4, -8, 8, 0, -16, 32, -32, -1, 78, -233, 687, -2363, 8160, -25670, 72352, -184451, 430937, -933087, 1888690, -3597221, 6479696, -11086920, 18096128, -28307626, 42644791, -62031001, 86466285, -110902034, 110907489, -52325, -483682930
Offset: 0
The version for prime instead of composite is
A007442.
For noncomposite numbers we have
A030016.
This is the first column (n=1) of
A377033.
For squarefree instead of composite we have
A377041, nonsquarefree
A377049.
For prime-power instead of composite we have
A377054.
Cf:
A018252,
A065310,
A065890,
A140119,
A173390,
A333214,
A376602 (zero),
A376603 (nonzero),
A376651 (positive),
A376652 (negative),
A376680.
-
q=Select[Range[100],CompositeQ];
t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[1+k]],{k,0,j}],{j,0,Length[q]-1}]
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