A084758 -id:A084758 - cp's OEIS frontend

A084759 Composite numbers in ascending order such that the difference of successive terms is unique. a(m) - a(m-1) = a(k) - a(k-1) iff m = k.

4, 6, 9, 10, 14, 20, 25, 32, 40, 49, 60, 70, 82, 95, 110, 124, 140, 158, 175, 194, 214, 235, 258, 280, 304, 329, 355, 382, 410, 440, 469, 500, 532, 565, 600, 634, 670, 707, 745, 784, 824, 865, 908, 950, 994, 1040, 1085, 1132, 1180, 1230, 1281, 1330, 1382, 1435

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 17 2003

nonn

The sequence of first differences is 2, 3, 1, 4, 6, 5, 7, 8, 9, 11, 10, 12, 13, 15, 14, 16, 18, 17, 19, 20, 21, 23, 22, 24, 25, 26, 27, 28, ... Conjecture: every number is a term of this sequence. For every number r there exists some k such that a(k) - a(k-1) = r.

			The term after 14 is 20 and not 18 or 16 as 6-4 = 16-14 = 2, 18-14 = 14-10 = 4.

Cf. A002808, A084758.

More terms from David Wasserman, Jan 05 2005

A375182 Slowest increasing sequence of semiprimes such that the differences of successive terms are distinct.

Original entry on oeis.org

4, 6, 9, 10, 14, 21, 26, 34, 46, 55, 65, 82, 93, 106, 121, 141, 155, 161, 177, 201, 219, 247, 274, 295, 314, 339, 361, 391, 417, 446, 469, 501, 535, 566, 611, 649, 685, 718, 753, 793, 835, 878, 917, 958, 995, 1041, 1094, 1138, 1186, 1238, 1285, 1339, 1389, 1438, 1497, 1555, 1618, 1673, 1735, 1795, 1851

Offset: 1

Zak Seidov and Robert Israel, Sep 23 2024

nonn

For n >= 2, a(n) is the least semiprime k > a(n-1) such that k - a(n-1) <> a(j) - a(j-1) for j < n.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358, A084758.

R:= 4: x:= 4: S:= {}: count:= 1:
for i from 6 while count < 100 do
  if not member(i-x, S) and numtheory:-bigomega(i) = 2 then
    R:= R,i; S:= S union {i-x}; x:= i; count:= count+1
  fi
od:
R;

s = {4, 6};  df  = {2}; Do[k = 1; While [MemberQ[df, k] ||
2 != PrimeOmega[a = s[[-1]] + k], k++]; AppendTo[s, a]; AppendTo[df, k], {98}];
s

A379542 Second term of the n-th differences of the prime numbers.

Original entry on oeis.org

3, 2, 0, 2, -6, 14, -30, 62, -122, 220, -344, 412, -176, -944, 4112, -11414, 26254, -53724, 100710, -175034, 281660, -410896, 506846, -391550, -401486, 2962260, -9621128, 24977308, -57407998, 120867310, -236098336, 428880422, -719991244, 1096219280

Offset: 0

Gus Wiseman, Jan 12 2025

sign

Also the inverse zero-based binomial transform of the odd prime numbers.

For all primes (not just odd) we have A007442.
Including 1 in the primes gives A030016.
Column n=2 of A095195.
The version for partitions is A320590 (first column A281425), see A175804, A053445.
For nonprime instead of prime we have A377036, see A377034-A377037.
Arrays of differences: A095195, A376682, A377033, A377038, A377046, A377051.
A000040 lists the primes, differences A001223, A036263.
A002808 lists the composite numbers, differences A073783, A073445.
A008578 lists the noncomposite numbers, differences A075526.
Cf. A064113, A065890, A084758, A140119, A173390, A258025, A258026, A293467, A333214, A333254, A377041.

nn=40;Table[Differences[Prime[Range[nn+2]],n][[2]],{n,0,nn}]

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * prime(k+2)); \\ Michel Marcus, Jan 12 2025

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * prime(k+2).

A084760 Squarefree numbers in ascending order such that the difference of successive terms is unique. a(m) - a(m-1) = a(k) - a(k-1) iff m = k.

Original entry on oeis.org

2, 3, 5, 10, 13, 17, 23, 30, 38, 47, 57, 69, 82, 93, 107, 122, 138, 155, 173, 193, 214, 233, 255, 278, 302, 327, 353, 381, 410, 437, 467, 498, 530, 563, 597, 633, 670, 705, 743, 782, 822, 863, 905, 949, 994, 1037, 1085, 1131, 1178, 1227, 1277, 1329, 1382, 1433

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 17 2003

nonn

The sequence of first differences is 1, 2, 5, 3, 4, 6, 7, 8, 9, 10, 12, 13, 11, 14, 15, 16, 17, 18, 20, 21, 19, ... Conjecture: (1) every number is a term of this sequence. For every number r there exists some k such that a(k) - a(k-1) = r. Question: What is the longest string of consecutive integers in this sequence (of successive differences)?

Answer: 5, as exemplified by the 6 values 17 to 57. Any longer series with differences consecutive integers must include a multiple of 4, as can be seen by enumerating all possibilities modulo 4. - Franklin T. Adams-Watters, Jul 14 2006

			After 5 the next term is 10 and not 6 or 7, as 6-5 = 3-2 =1 and 7-5 = 5-3 = 2.

Cf. A005117, A084758, A084759.

More terms from Franklin T. Adams-Watters, Jul 14 2006

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A084759 Composite numbers in ascending order such that the difference of successive terms is unique. a(m) - a(m-1) = a(k) - a(k-1) iff m = k.

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A375182 Slowest increasing sequence of semiprimes such that the differences of successive terms are distinct.

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A379542 Second term of the n-th differences of the prime numbers.

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A084760 Squarefree numbers in ascending order such that the difference of successive terms is unique. a(m) - a(m-1) = a(k) - a(k-1) iff m = k.

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