A174965 -id:A174965 - cp's OEIS frontend

A376340 Sorted positions of first appearances in A057820, the sequence of first differences of prime-powers.

1, 4, 9, 12, 18, 24, 34, 47, 60, 79, 117, 178, 198, 206, 215, 244, 311, 402, 465, 614, 782, 1078, 1109, 1234, 1890, 1939, 1961, 2256, 2290, 3149, 3377, 3460, 3502, 3722, 3871, 4604, 4694, 6634, 8073, 8131, 8793, 12370, 12661, 14482, 14990, 15912, 17140, 19166

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     9: {2,2}
    12: {1,1,2}
    18: {1,2,2}
    24: {1,1,1,2}
    34: {1,7}
    47: {15}
    60: {1,1,2,3}
    79: {22}
   117: {2,2,6}
   178: {1,24}
   198: {1,2,2,5}
   206: {1,27}
   215: {3,14}
   244: {1,1,18}

For compression instead of sorted firsts we have A376308.
For run-lengths instead of sorted firsts we have A376309.
For run-sums instead of sorted firsts we have A376310.
The version for squarefree numbers is the unsorted version of A376311.
The unsorted version is A376341.
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.
Cf. A001597, A006549, A007916, A025475, A037201, A053289, A078147, A110969, A120430, A174965, A373948, A375706.

q=Differences[Select[Range[100],PrimePowerQ]];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A377054 First term of the n-th differences of the powers of primes. Inverse zero-based binomial transform of A000961.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, -5, 15, -34, 63, -97, 115, -54, -251, 1184, -3536, 8736, -18993, 37009, -64545, 98442, -121393, 82008, 147432, -860818, 2710023, -7110594, 17077281, -38873146, 85085287, -179965647, 367885014, -725051280, 1372311999, -2481473550, 4257624252

Offset: 0

Gus Wiseman, Oct 22 2024

sign

			The sixth differences of A000961 begin: -5, 10, -9, 1, 6, -10, 16, -18, ..., so a(6) = -5.

The version for primes is A007442, noncomposites A030016, composites A377036.
For squarefree numbers we have A377041, nonsquarefree A377049.
This is the first column of the array A377051.
For antidiagonal-sums we have A377052, absolute A377053.
For positions of first zeros we have A377055.
A000040 lists the primes, differences A001223, seconds A036263.
A000961 lists the powers of primes, differences A057820.
A001597 lists perfect-powers, complement A007916.
A008578 lists the noncomposites, differences A075526.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
Cf. A025475, A053707, A093555, A174965, A175804, A246655, A361102, A376340, A376596, A377056.

q=Select[Range[100],#==1||PrimePowerQ[#]&];
Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[1+k]],{k,0,j}],{j,0,Length[q]/2}]

The inverse zero-based binomial transform of a sequence (q(0), q(1), q(2), ...) is the sequence p given by:

p(j) = sum_{k=0..j} (-1)^(j-k)*binomial(j,k)*q(k)

A375740 Numbers k such that A007916(k+1) - A007916(k) = 1. In other words, the k-th non-perfect-power is 1 less than the next.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Gus Wiseman, Sep 10 2024

nonn

Positions in A007916 of numbers k such that k+1 is also a member.
Positions of 1's in A375706 (first differences of A007916).
Non-perfect-powers (A007916) are numbers with no proper integer roots.

			The non-perfect-powers are 2, 3, 5, 6, 7, 10, 11, 12, 13, ... which increase by one after positions 1, 3, 4, 6, ...

The version for non-prime-powers is A375713, differences A373672.
The complement is A375714, differences A375702.
The version for prime-powers is A375734, differences A373671.
The complement for non-prime-powers is A375928, differences A110969.
A000040 lists the prime numbers, differences A001223.
A000961 lists prime-powers (inclusive), differences A057820.
A001597 lists perfect-powers, differences A053289.
A002808 lists the composite numbers, differences A073783.
A018252 lists the nonprime numbers, differences A065310.
Non-perfect-powers:
- terms: A007916
- differences: A375706
- runs: sum A375705, A375703, A375704, A375702
- anti-runs: A375737, A375738, A375739, A375736.
Non-prime-powers (exclusive):
- terms: A361102
- differences: A375708
- runs: A373678, A373676, A373677, A110969
- anti-runs: A373679, A373575, A255346, A373672
Cf. A006549, A046933, A093555, A174965, A246655, A251092.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Join@@Position[Differences[Select[Range[100],radQ]],1]

from itertools import count, islice
from sympy import perfect_power
def A375740_gen(): # generator of terms
    a, b = -1, 0
    for n in count(2):
        c = not perfect_power(n)
        if c:
            a += 1
        if b&c:
            yield a
    b = c
A375740_list = list(islice(A375740_gen(), 52)) # Chai Wah Wu, Sep 11 2024

A375713 Indices of consecutive non-prime-powers (A361102) differing by 1. Numbers k such that the k-th and (k+1)-th non-prime-powers differ by just one.

Original entry on oeis.org

5, 8, 9, 15, 16, 17, 19, 20, 23, 24, 27, 28, 30, 31, 32, 33, 36, 38, 40, 41, 44, 45, 46, 47, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 63, 64, 67, 68, 71, 72, 74, 75, 76, 77, 78, 79, 81, 82, 85, 87, 88, 89, 90, 93, 94, 95, 96, 97, 98, 99, 100, 103, 104, 105, 106

Offset: 1

Gus Wiseman, Sep 02 2024

nonn

			The initial non-prime-powers are 1, 6, 10, 12, 14, 15, 18, 20, 21, which first increase by one after the fifth and eighth terms.

The inclusive version is a(n) - 1.
For prime-powers inclusive (A000961) we have A375734, differences A373671.
For nonprime numbers (A002808) we have A375926, differences A373403.
For prime-powers exclusive (A246655) we have A375734(n+1) + 1.
First differences are A373672.
Positions of 1's in A375708.
For non-perfect-powers we have A375740.
Prime-powers inclusive:
- terms: A000961
- differences: A057820
- runs: A373675, A373673, A373674, A174965
- antiruns: A373576, A120430, A006549, A373671
Non-prime-powers inclusive:
- terms: A361102
- differences: A375708
- runs: A373678, A373676, A373677, A110969
- antiruns: A373679, A373575, A255346, A373672
A000040 lists all of the primes, differences A001223.
A007916 lists non-perfect-powers, differences A375706.
Cf. A046933, A053289, A073783, A093555, A176246, A251092, A375714.

Join@@Position[Differences[Select[Range[100],!PrimePowerQ[#]&]],1]

A361102(k+1) - A361102(k) = 1.

A376308 Run-compression of the sequence of first differences of prime-powers.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 20 2024

nonn

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

			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, ...
The run-compression is A376308 (this sequence).

For primes instead of prime-powers we have A037201, halved A373947.
For squarefree numbers instead of prime-powers we have A376305.
For run-lengths instead of compression we have A376309.
For run-sums instead of compression 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, differences A057820.
A003242 counts compressed compositions, ranks A333489.
A024619 and A361102 list non-prime-powers, differences A375708.
A116861 counts partitions by compressed sum, by compressed length A116608.
A373948 encodes compression using compositions in standard order.
Cf. A001597, A006549, A007916, A025475, A034296, A053289, A076259, A110969, A120430, A124767, A174965, A374251.

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

A373669 Least k such that the k-th maximal run of non-prime-powers has length n. Position of first appearance of n in A110969, and the sequence ends if there is none.

Original entry on oeis.org

1, 5, 7, 12, 18, 190, 28, 109, 40, 28195574, 53

Offset: 1

Gus Wiseman, Jun 14 2024

A run of a sequence (in this case A361102) is an interval of positions at which consecutive terms differ by one.
Are there only 9 terms?
From David A. Corneth, Jun 14 2024: (Start)
No. a(10) exists.
Between the prime 144115188075855859 and 144115188075855872 = 2^57 there are 12 non-prime-powers so a(12) exists. (End)

			The maximal runs of non-prime-powers begin:
   1
   6
  10
  12
  14  15
  18
  20  21  22
  24
  26
  28
  30
  33  34  35  36
  38  39  40
  42
  44  45  46
  48
  50  51  52
  54  55  56  57  58
  60

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

For composite runs we have A073051, sorted A373400, firsts of A176246.
For squarefree runs we have firsts of A120992.
For prime-powers runs we have firsts of A174965.
For prime runs we have firsts of A251092 or A175632.
For squarefree antiruns we have A373128, firsts of A373127.
For nonsquarefree runs we have A373199, firsts of A053797.
The sorted version is A373670.
For antiruns we have firsts of A373672.
For runs of non-prime-powers:
- length A110969
- min A373676
- max A373677
- sum A373678
A000961 lists the powers of primes (including 1).
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A361102 lists the non-prime-powers, without 1 A024619.
Cf. A007053, A008864, A014963, A027833, A038664, A054265, A067774, A356068, A373401, A373403, A373671.

q=Length/@Split[Select[Range[10000],!PrimePowerQ[#]&],#1+1==#2&]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[y,Range[#1]]&];
Table[Position[q,k][[1,1]],{k,spna[q]}]

A376598 Points of nonzero curvature in the sequence of prime-powers inclusive (A000961).

Original entry on oeis.org

4, 5, 7, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A376596) are nonzero.

Inclusive means 1 is a prime-power. For the exclusive version, subtract 1 from all terms.

			The prime-powers inclusive (A000961) are:
  1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...
with first differences (A057820):
  1, 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, ...
with first differences (A376596):
  0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...
with nonzeros at (A376598):
  4, 5, 7, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, ...

Gus Wiseman, Points of nonzero curvature in the prime-powers inclusive.

The first differences were A057820, see also A376340.
First differences are A376309.
These are the nonzeros of A376596 (sorted firsts A376653, exclusive A376654).
The complement is A376597.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
`A064113 lists positions of adjacent equal prime gaps.
For prime-powers inclusive: A057820 (first differences), A376597 (second differences), A376597 (inflections and undulations), A376653 (sorted firsts in second differences).
For points of nonzero curvature: A333214 (prime), A376603 (composite), A376589 (non-perfect-power), A376592 (squarefree), A376595 (nonsquarefree), A376601 (non-prime-power).
Cf. A025475, A053707, A069623, A093555, A174965, A251092, A333254, A336416, A361102.

Join@@Position[Sign[Differences[Select[Range[1000], #==1||PrimePowerQ[#]&],2]],1|-1]

A373670 Numbers k such that the k-th run-length A110969(k) of the sequence of non-prime-powers (A024619) is different from all prior run-lengths.

Original entry on oeis.org

1, 5, 7, 12, 18, 28, 40, 53, 71, 109, 170, 190, 198, 207, 236, 303, 394, 457, 606, 774, 1069, 1100, 1225, 1881, 1930, 1952, 2247, 2281, 3140, 3368, 3451, 3493, 3713, 3862, 4595, 4685, 6625, 8063, 8121, 8783, 12359, 12650, 14471, 14979, 15901, 17129, 19155

Offset: 1

Gus Wiseman, Jun 15 2024

nonn

The unsorted version is A373669.

			The maximal runs of non-prime-powers begin:
   1
   6
  10
  12
  14  15
  18
  20  21  22
  24
  26
  28
  30
  33  34  35  36
  38  39  40
  42
  44  45  46
  48
  50  51  52
  54  55  56  57  58
  60
So the a(n)-th runs begin:
   1
  14  15
  20  21  22
  33  34  35  36
  54  55  56  57  58

For nonsquarefree runs we have A373199 (if increasing), firsts of A053797.
For squarefree antiruns see A373200, unsorted A373128, firsts of A373127.
For composite runs we have A373400, unsorted A073051, firsts of A176246.
For prime antiruns we have A373402.
For runs of non-prime-powers:
- length A110969, firsts A373669, sorted A373670 (this sequence):
- min A373676
- max A373677
- sum A373678
For runs of prime-powers:
- length A174965
- min A373673
- max A373674
- sum A373675
A000961 lists the powers of primes (including 1).
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists the non-prime-powers, without 1 A024619.
Cf. A005381, A008864, A014963, A027833, A038664, A356068, A373401, A373574.

t=Length/@Split[Select[Range[10000],!PrimePowerQ[#]&],#1+1==#2&];
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

A376680 Run-lengths of first differences of composite numbers.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 1, 4, 2, 4, 1, 2, 2, 2, 1, 4, 1, 4, 2, 4, 1, 2, 2, 4, 1, 2, 1, 4, 1, 6, 1, 2, 2, 2, 2, 2, 1, 12, 1, 2, 1, 4, 2, 8, 2, 4, 1, 4, 1, 2, 1, 4, 1, 4, 2, 8, 2, 2, 2, 10, 1, 10, 1, 2, 2, 2, 1, 4, 2, 8, 1, 4, 1, 4, 1, 4, 2, 4, 1, 2, 2, 8, 1, 12, 1, 2

Offset: 1

Gus Wiseman, Oct 10 2024

nonn

Also first differences of A376603 (points of nonzero curvature in the composite numbers).

			The composite numbers (A002808) are:
  4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
  2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, ...
with runs:
  (2,2), (1,1), (2,2), (1,1), (2,2), (1,1), (2), (1,1,1,1), (2,2), (1,1,1,1), ...
with lengths (A376680):
  2, 2, 2, 2, 2, 2, 1, 4, 2, 4, 1, 2, 2, 2, 1, 4, 1, 4, 2, 4, 1, 2, 2, 4, 1, 2, ...

These are the run-lengths of A073783, ones A375929.
For prime instead of composite we have A333254, first appearances A335406.
These are the first differences of A376603.
A000040 lists the prime numbers, first differences A001223, second differences A036263.
A002808 lists the composite numbers, differences A073783.
A064113 lists positions of adjacent equal prime gaps.
A073445 gives second differences of composite numbers, zeros A376602.
Cf. A054546, A076259, A174965, A251092, A376651, A376652.

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

A377052 Antidiagonal-sums of the array A377051(n,k) = n-th term of k-th differences of powers of primes.

Original entry on oeis.org

1, 3, 4, 5, 6, 13, -6, 45, -50, 113, -98, 73, 274, -1159, 3563, -8707, 19024, -36977, 64582, -98401, 121436, -81961, -147383, 860871, -2709964, 7110655, -17077217, 38873213, -85085216, 179965720, -367884935, 725051361, -1372311916, 2481473639, -4257624155

Offset: 0

Gus Wiseman, Oct 22 2024

sign

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

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

The version for primes is A140119, noncomposites A376683, composites A377034.
For squarefree numbers we have A377039, nonsquarefree A377047.
These are the antidiagonal-sums of A377051.
The unsigned version is A377053.
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.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
Cf. A000961, A025475, A053707, A057820, A093555, A174965, A246655, A361102, A376340, A376596, A376598.

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

cp's OEIS Frontend

A376340 Sorted positions of first appearances in A057820, the sequence of first differences of prime-powers.

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A377054 First term of the n-th differences of the powers of primes. Inverse zero-based binomial transform of A000961.

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A375740 Numbers k such that A007916(k+1) - A007916(k) = 1. In other words, the k-th non-perfect-power is 1 less than the next.

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A375713 Indices of consecutive non-prime-powers (A361102) differing by 1. Numbers k such that the k-th and (k+1)-th non-prime-powers differ by just one.

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A376308 Run-compression of the sequence of first differences of prime-powers.

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A373669 Least k such that the k-th maximal run of non-prime-powers has length n. Position of first appearance of n in A110969, and the sequence ends if there is none.

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A376598 Points of nonzero curvature in the sequence of prime-powers inclusive (A000961).

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A373670 Numbers k such that the k-th run-length A110969(k) of the sequence of non-prime-powers (A024619) is different from all prior run-lengths.

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A376680 Run-lengths of first differences of composite numbers.

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