A376340 -id:A376340 - cp's OEIS frontend

A377053 Antidiagonal-sums of the absolute value of the array A377051(n,k) = n-th term of k-th differences of powers of primes.

1, 3, 4, 5, 6, 13, 24, 45, 80, 123, 174, 229, 382, 1219, 3591, 8849, 19288, 37899, 67442, 108323, 156054, 206733, 311525, 860955, 2710374, 7111657, 17080759, 38884849, 85124764, 180097856, 368321633, 726482493, 1377039690, 2496856437, 4306569569, 7016267449

Offset: 0

Gus Wiseman, Oct 22 2024

nonn

These are the row-sums of the absolute value of the triangle-version of A377051.

			The sixth antidiagonal of A377051 is (8, 1, -1, -2, -3, -4, -5), so a(6) = 24.

The version for primes is A376681, noncomposites A376684, composites A377035.
For squarefree numbers we have A377040, nonsquarefree A377048.
This is the antidiagonal-sums of the absolute value of A377051.
The signed version is A377052.
For leaders we have A377054, for primes A007442 or A030016.
For first zero-positions we have A377055.
A version for partitions is A377056, cf. A175804, A053445, A281425, A320590.
A000040 lists the primes, differences A001223, seconds A036263.
A008578 lists the noncomposites, differences A075526.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
Cf. A000961, A025475, A053707, A057820, A069623, A093555, A174965, A246655, A361102, A376340, A376596.

nn=20;
t=Table[Differences[NestList[NestWhile[#+1&, #+1,!PrimePowerQ[#]&]&,1,2*nn],k],{k,0,nn}];
Total/@Abs[Table[t[[j,i-j+1]],{i,nn},{j,i}]]

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

Gus Wiseman, Oct 22 2024

nonn

			The fourth differences of A246655 begin: 1, -3, 3, 0, -2, 2, ... so a(4) = 4.

Lucas A. Brown, Python program.

The version for primes is A376678, noncomposites A376855, composites A377037.
For squarefree numbers we have A377042, nonsquarefree A377050.
These are the positions of first zeros in each row of A377051.
For antidiagonal-sums we have A377052, absolute A377053.
For leaders we have A377054, for primes A007442 or A030016.
A000040 lists the primes, differences A001223, seconds A036263.
A000961 lists the powers of primes, differences A057820.
A008578 lists the noncomposites, differences A075526.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
A246655 lists the prime-powers, differences A057820 (except first term).
Cf. A025475, A053707, A093555, A174965, A175804, A361102, A376340, A376596, A377056.

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]]]}]

a(12)-a(27) from Pontus von Brömssen, Oct 22 2024
a(28)-a(30) from Chai Wah Wu, Oct 23 2024
a(31)-a(35) from Lucas A. Brown, Nov 03 2024

A376654 Sorted positions of first appearances in the second differences of consecutive prime-powers exclusive (A246655).

Original entry on oeis.org

3, 4, 9, 11, 17, 24, 44, 46, 47, 59, 67, 68, 70, 79, 117, 120, 177, 178, 198, 205, 206, 215, 243, 244, 303, 324, 326, 401, 465, 483, 604, 800, 879, 938, 1032, 1054, 1076, 1233, 1280, 1720, 1889, 1890, 1905, 1939, 1959, 1961, 2256, 2289, 2409, 2879, 3149

Offset: 1

Gus Wiseman, Oct 06 2024

nonn

			The prime-powers exclusive (A246655) are:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, ...
with first differences (A057820 except first term) :
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, 3, ...
with first differences (A376596 except first term):
  0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, -4, ...
with first appearances (A376654):
  1, 3, 4, 9, 11, 17, 24, 44, 46, 47, 59, 67, 68, 70, 79, 117, 120, 177, 178, 198, ...

For first differences we have A376340.
These are the sorted positions of first appearances in A376596 except first term.
The inclusive version is a(n) + 1 = A376653(n), except first term.
For squarefree instead of prime-power we have A376655.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
For prime-powers inclusive: A057820 (first differences), A376597 (inflections and undulations), A376598 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
Cf. A025475, A053707, A057820, A064113, A069623, A093555, A174965, A333254, A336416, A361102.

q=Differences[Select[Range[1000],PrimePowerQ[#]&],2];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A376309 Run-lengths of the sequence of first differences of prime-powers.

Original entry on oeis.org

3, 1, 2, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

			The sequence of prime-powers (A246655) is:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, ...
The sequence of first differences (A057820) of prime-powers is:
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, ...
with runs:
  (1,1,1),(2),(1,1),(2,2),(3),(1),(2),(4),(2,2,2,2),(1),(5),(4),(2),(4), ...
with lengths A376309 (this sequence).

Antti Karttunen, Table of n, a(n) for n = 1..20000

For runs of prime-powers increasing by one we have A174965.
For primes instead of prime-powers we have A333254.
For squarefree numbers instead of prime-powers we have A376306.
For compression instead of run-lengths we have A376308.
For run-sums instead of run-lengths we have A376310.
For positions of first appearances we have A376341, sorted A376340.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A024619 and A361102 list non-prime-powers, first differences A375708.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs in standard compositions, anti-runs A333381.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A373948 encodes compression using compositions in standard order.
Cf. A001597, A006549, A025475, A037201, A053289, A110969, A120430, A251092, A373822, A375706.

Length/@Split[Differences[Select[Range[100],PrimePowerQ]]]

up_to = 20000;
A376309list(up_to) = { my(v=vector(up_to), ppp=2, pd=1, d, rl=0, k=2, i=0); while(i<#v, k++; if(isprimepower(k), d = k-ppp; ppp = k; if(d == pd, rl++, i++; v[i] = rl; rl = 1; pd = d))); (v); };
v376309 = A376309list(up_to);
A376309(n) = v376309[n]; \\ Antti Karttunen, Jan 18 2025

More terms from Antti Karttunen, Jan 18 2025

A376310 Run-sums of the sequence of first differences of prime-powers.

Original entry on oeis.org

3, 2, 2, 4, 3, 1, 2, 4, 8, 1, 5, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 8, 4, 2, 1, 3, 6, 2, 10, 2, 12, 4, 2, 4, 6, 2, 10, 2, 4, 2, 24, 4, 2, 4, 6, 4, 8, 5, 1, 12, 2, 6, 4, 2, 6, 4, 14, 4, 2, 4, 14, 12, 4, 2, 4, 6, 2, 18, 4, 6, 8, 4, 8, 10, 2

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

			The sequence of prime-powers (A246655) is:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, ...
The sequence of first differences (A057820) of prime-powers is:
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, ...
with runs:
  (1,1,1),(2),(1,1),(2,2),(3),(1),(2),(4),(2,2,2,2),(1),(5),(4),(2),(4), ...
with sums A376310 (this sequence).

For primes instead of prime-powers we have A373822, halved A373823.
For squarefree numbers instead of prime-powers we have A376307.
For compression instead of run-sums we have A376308.
For run-lengths instead of run-sums we have A376309.
For positions of first appearances we have A376341, sorted A376340.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A024619 and A361102 list non-prime-powers, first differences A375708.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs in standard compositions, anti-runs A333381.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A373948 encodes compression using compositions in standard order.
Cf. A001597, A006549, A007916, A025475, A037201, A053289, A054265, A110969, A120430, A174965, A251092, A375706.

Total/@Split[Differences[Select[Range[100],PrimePowerQ]]]

A376341 Position of first appearance of n in A057820, the sequence of first differences of prime-powers, or 0 if n does not appear.

Original entry on oeis.org

1, 5, 10, 13, 19, 25, 199, 35, 118, 48, 28195587, 61, 3745011205066703, 80, 6635, 312, 1079, 207, 3249254387600868788, 179, 43580, 216, 21151968922, 615, 762951923, 403, 1962, 466, 12371, 245, 1480223716, 783, 494890212533313, 1110, 2064590, 1235, 375744164943287809536

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

For odd n either a(n) or a(n)+1 is in A024622 (unless a(n) = 0), corresponding to cases where the smaller or the larger term in the pair of consecutive prime powers, respectively, is a power of 2. - Pontus von Brömssen, Sep 27 2024

			a(4) = 13, because the first occurrence of 4 in A057820 is at index 13. The corresponding first pair of consecutive prime powers with difference 4 is (19, 23), and a(4) = A025528(23) = 13.
a(61) = A024622(96), because the first pair of consecutive prime powers with difference 61 is (2^96, 2^96+61), and A025528(2^96+61) = A024622(96).

Pontus von Brömssen, Table of n, a(n) for n = 1..60

For compression instead of first appearances we have A376308.
For run-lengths instead of first appearances we have A376309.
For run-sums instead of first appearances we have A376310.
For squarefree numbers instead of prime-powers we have A376311.
The sorted version is A376340.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A024619 and A361102 list non-prime-powers, first differences A375708.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A116861 counts partitions by compressed sum, by compressed length A116608.
Cf. A001597, A006549, A007916, A024622, A025475, A025528, A037201, A038664, A053289, A110969, A120430, A174965, A373400, A375706.

mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=Differences[Select[Range[100],#==1||PrimePowerQ[#]&]];
Table[Position[q,k][[1,1]],{k,mnrm[q]}]

A057820(a(n)) = n whenever a(n) > 0. - Pontus von Brömssen, Sep 24 2024

Definition modified by Pontus von Brömssen, Sep 26 2024

More terms from Pontus von Brömssen, Sep 27 2024

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A377053 Antidiagonal-sums of the absolute value of the array A377051(n,k) = n-th term of k-th differences of powers of primes.

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A377055 Position of first appearance of zero in the n-th differences of the prime-powers (A246655), or 0 if it does not appear.

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A376654 Sorted positions of first appearances in the second differences of consecutive prime-powers exclusive (A246655).

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A376309 Run-lengths of the sequence of first differences of prime-powers.

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A376310 Run-sums of the sequence of first differences of prime-powers.

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A376341 Position of first appearance of n in A057820, the sequence of first differences of prime-powers, or 0 if n does not appear.

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