== 18 && $ - cp's OEIS frontend

A383938 a(n) is the least positive integer k such that b(2*j) is prime for 1 <= j <= n but not prime for j = n+1, where b(1) = k and b(m+1) = b(m) + prime(m) for m >= 1.

2, 5, 21, 129, 69, 1, 51, 23991, 171, 1371, 3, 322141431, 1431357020859

Offset: 0

Om S. M. Yadav, Aug 18 2025

Similar to A227547, primes are added in successive manner except that here the sequence breaks if an even-indexed term is not prime and considers preceding even-indexed prime as the last term of the sequence. For example, a(2) = 21 [21, 23, 26, 31, 38, 49] but since 49 is not prime, last two terms (38 and 49) are omitted leaving 31 as last term in the sequence.

a(12) is the last term, because b(j) is always divisible by 11 for some j in {2, 4, 6, 8, 10, 14, 16, 18, 22, 24, 26}. - Pontus von Brömssen, Aug 19 2025

			a(n) = k, b(m+1) = b(m) + prime(m); b(1) = k
For n = 0, a(0) = 2; b(m+1) = b(m) + prime(m): [2]
For n = 1, a(1) = 5; b(m+1) = b(m) + prime(m): [5, 7(5+2)]
For n = 2, a(2) = 21; b(m+1) = b(m) + prime(m): [21, 23(21+2), 26(23+3), 31(26+5)]
For n = 3, a(3) = 129; b(m+1) = b(m) + prime(m): [129, 131(129+2), 134(131+3), 139(134+5), 146(139+7), 157(146+11)]
For n = 4, a(4) = 69; b(m+1) = b(m) + prime(m): [69, 71(69+2), 74(71+3), 79(74+5), 86(79+7), 97(86+11), 110(97+13), 127(110+17)]
For n = 5, a(5) = 1; b(m+1) = b(m) + prime(m): [1, 3(1+2), 6(3+3), 11(6+5), 18(11+7), 29(18+11), 42(29+13), 59(42+17), 78(59+19), 101(78+23)]
For a(n), even-indexed term is prime. e.g. for a(3) = 129 [129, 131, 134, 139, 146, 157], even indexed terms 131, 139, 157 are primes.

Cf. A014284, A092163, A109723, A227547.

a(n) = my(vp=concat(2, vector(n+1, i, sum(k=1, 2*i+1, prime(k)))), v=concat(vector(n, i, 1), 0), k=1); while (apply(ispseudoprime, vector(n+1, i, vp[i]+k)) != v, k++); k; \\ Michel Marcus, Aug 19 2025

a(11) from Michel Marcus, Aug 19 2025

a(12) from Pontus von Brömssen, Aug 19 2025

A383939 Numbers k such that k and the k-th triangular number T(k) = k*(k+1)/2 have only even digits.

Original entry on oeis.org

0, 28, 40, 64, 400, 2828, 4000, 4064, 6428, 22840, 24028, 40000, 202428, 240028, 400000, 2040040, 2400028, 4000000, 6422840, 6428064, 6646624, 20044064, 20202080, 20400040, 20406080, 24000028, 40000000, 66400064, 200042428, 204000040, 228406080, 240000028

Offset: 1

Shyam Sunder Gupta, Aug 18 2025

The sequence also contains the infinite subsequence 4*10^n for n >= 1.

			64 is a term since it and T(64) = 2080 both have only even digits.

Shyam Sunder Gupta, Triangular Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 3, 82-125.

Cf. A000217, A014263, A117978, A347475.

q[k_] := And @@ (AllTrue[IntegerDigits[#], EvenQ] & /@ {k, k*(k+1)/2}); Select[Range[0, 4*10^6], q] (* Amiram Eldar, Aug 18 2025 *)

A386314 a(1) = 1 and thereafter a(n) is the smallest number k of the form 6*x+-1 not already in the sequence but where the reduced Collatz step A139391(k) is in the sequence.

Original entry on oeis.org

1, 5, 13, 17, 11, 7, 29, 19, 25, 37, 49, 53, 35, 23, 61, 65, 43, 77, 85, 101, 67, 89, 59, 113, 133, 149, 157, 173, 115, 181, 197, 131, 205, 209, 139, 185, 229, 241, 245, 163, 217, 269, 179, 119, 79, 277, 289, 301, 305, 203, 317, 211, 281, 187, 325, 341, 227, 151, 349, 373

Offset: 1

Jules Beauchamp, Jul 18 2025

These numbers are the Collatz pre-images in the form 6*x +- 1 of all previous terms not already in the sequence.
The pre-images of a term t are all p which reach t by a single odd to odd step A139391(p) = t.
These pre-images are those p = (t*2^k-1)/3 with k>=0 which are odd integers, and with here t != 0 (mod 3) there are infinitely many p != 0 (mod 3) for each t.
Multiples of 3 have no odd pre-images and are excluded here in order to have the essential part of the tree of odd to odd descents.
The trajectory of a term t reaches 1 by steps to successively earlier terms in this sequence (at various distances apart).
If the Collatz conjecture is true, then this sequence is permutation of the numbers of the form 6x +- 1 (A007310).

			a(3) = 13, since 13 (a pre-image of a(2) = 5) is the smallest unused pre-image of a(1) and a(2).
a(10) = 37 since 37 (a pre-image of a(6) = 7) is the smallest unused pre-image of all previous terms.

David A. Corneth, Table of n, a(n) for n = 1..10000 (using Michel Marcus' code)
Sean A. Irvine, Java program (github)

Cf. A007310, A178414, A178415, A139391.

lista(nn) = my(va=List(1), vs = Map(), imin=1, i=imin, nb=1); mapput(vs, 1, 1); while(#vaMichel Marcus, Aug 25 2025

A387110 Number of ways to choose a sequence of distinct integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 2, 0, 3, 2, 5, 0, 2, 3, 7, 0, 11, 5, 6, 0, 15, 2, 22, 0, 10, 7, 30, 0, 6, 11, 0, 0, 42, 6, 56, 0, 14, 15, 15, 0, 77, 22, 22, 0, 101, 10, 135, 0, 6, 30, 176, 0, 20, 6, 30, 0, 231, 0, 21, 0, 44, 42, 297, 0, 385, 56, 10, 0, 33, 14, 490, 0, 60, 15, 627, 0

Offset: 1

Gus Wiseman, Aug 18 2025

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 axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

			The prime indices of 9 are (2,2), and there are a(9) = 2 choices:
  ((2),(1,1))
  ((1,1),(2))
The prime indices of 15 are (2,3), and there are a(15) = 5 choices:
  ((2),(3))
  ((2),(2,1))
  ((2),(1,1,1))
  ((1,1),(2,1))
  ((1,1),(1,1,1))

Positions of zeros are A276078 (choosable), complement A276079 (non-choosable).
Allowing repeated partitions gives A299200, A357977, A357982, A357978.
For multiset systems see A355529, A355744, A367771, set systems A367901-A367905.
For divisors see A355731, A355733, A355734, A355735, A355737, A355739, A355740.
For prime factors instead of partitions see A355741, A355742, A387136.
The disjoint case is A383706.
For initial intervals instead of partitions we have A387111.
The case of strict partitions is A387115.
The case of constant partitions is A387120.
Taking each prime factor (instead of index) gives A387133.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A008284, A063834, A261049, A296122, A299201, A335433, A335448, A355745, A387135.

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

A387111 Number of ways to choose a sequence of distinct positive integers, one in the initial interval of each prime index of n.

Original entry on oeis.org

1, 1, 2, 0, 3, 1, 4, 0, 2, 2, 5, 0, 6, 3, 4, 0, 7, 0, 8, 0, 6, 4, 9, 0, 6, 5, 0, 0, 10, 1, 11, 0, 8, 6, 9, 0, 12, 7, 10, 0, 13, 2, 14, 0, 2, 8, 15, 0, 12, 2, 12, 0, 16, 0, 12, 0, 14, 9, 17, 0, 18, 10, 4, 0, 15, 3, 19, 0, 16, 4, 20, 0, 21, 11, 4, 0, 16, 4, 22

Offset: 1

Gus Wiseman, Aug 18 2025

The initial interval of a nonnegative integer x is the set {1,...,x}.
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 axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

			The prime indices of 75 are (2,3,3), with initial intervals ({1,2},{1,2,3},{1,2,3}), with choices (1,2,3), (1,3,2), (2,1,3), (2,3,1), so a(75) = 4.

Allowing repeated partitions gives A003963.
For constant instead of distinct we have A055396.
For multiset systems see A355529, A355744, A367771, set systems A367901-A367905.
For divisors we have A355739, zeros A355740, strict case of A355731.
For prime factors we have A355741, prime powers A355742, weakly increasing A355745.
For integer partitions we have A387110.
Positions of nonzero terms are A387112 (choosable).
Positions of 0 are A387134 (non-choosable).
A001414 adds up distinct prime divisors, counted by A001221.
A061395 gives greatest prime index.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A326841, A335433, A335448, A355733, A355737, A355747, A357980, A383706, A387120, A387133, A387135.

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

A383942 a(n) = (8*10^(2n) - 10^(n+1) + 2) / 9.

Original entry on oeis.org

78, 8778, 887778, 88877778, 8888777778, 888887777778, 88888877777778, 8888888777777778, 888888887777777778, 88888888877777777778, 8888888888777777777778, 888888888887777777777778, 88888888888877777777777778, 8888888888888777777777777778

Offset: 1

David Radcliffe, Aug 18 2025

This is one of four infinite families of triangular numbers consisting of two different digits. The other three families are A319170, A037156 (n>1), and A309597 (n>2).

Paolo Xausa, Table of n, a(n) for n = 1..495
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A000217, A037156, A062691, A073551, A256340, A309597, A319170.

Subsequence of: A119216, A119230, A119236.

A383942[n_] := (8*10^(2*n) - 10^(n+1) + 2)/9; Array[A383942, 15] (* or *)
LinearRecurrence[{111, -1110, 1000}, {78, 8778, 887778}, 15] (* Paolo Xausa, Aug 27 2025 *)

def A383942(n): return (8*10**(2*n)-10**(n+1)+2)//9

a(n) = A000217(A073551(n+1)).

G.f.: 6*x*(13 + 20*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Stefano Spezia, Aug 19 2025

A387079 Least prime factor of A386482(n).

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 2, 2, 2, 2, 7, 3, 2, 2, 2, 3, 5, 5, 2, 2, 2, 2, 2, 11, 3, 3, 2, 2, 2, 2, 19, 3, 2, 2, 2, 2, 2, 2, 2, 2, 5, 3, 3, 13, 5, 2, 2, 2, 7, 3, 3, 17, 2, 2, 2, 2, 31, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 3, 3, 23, 2, 2, 47, 3, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Michael De Vlieger, Aug 18 2025

nonn

Minimum absolute difference |s(n-1)-s(n)|, since GCD(s(n-1),s(n)) > 1, where s = A386482.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A020639, A386482, A387073, A387074, A387075.

Block[{c, j, k, p, m, nn}, nn = 120; c[] := False; m[] := 1; j = 2; c[1] = c[2] = True; {1, 2}~Join~Reap[Do[If[PrimePowerQ[j], Set[{p, k, m}, {#1, #1^(#2 - 1), #1^(#2 - 1)}] & @@ FactorInteger[j][[1]]; While[And[c[k*p], k != 0], k--];vIf[k == 0, k = m; While[c[k*p], k++]]; k *= p, k = j - 1; While[And[Or[c[k], CoprimeQ[j, k]], k != 1], k--]; If[k == 1, k += j; While[Or[c[k], CoprimeQ[j, k]], k++] ] ]; Sow[FactorInteger[k][[1, 1]] ]; Set[{c[k], j}, {True, k}], {n, 3, nn}]][[-1, 1]] ]

a(n) = A020639(A386482(n)).
a(n) <= |A386075(n-1)|.
a(m) = s(m) = A387073(i) for m = A387074(i).

A386429 Odd composites k such that A342926(k) is even and A342926(2*k) is a multiple of 3 and which satisfy Euler's condition for odd perfect numbers (A228058).

Original entry on oeis.org

45, 153, 261, 325, 369, 405, 477, 801, 909, 925, 1017, 1233, 1341, 1377, 1525, 1557, 1573, 1773, 1825, 2097, 2205, 2313, 2349, 2421, 2425, 2529, 2637, 2725, 2853, 3177, 3321, 3501, 3609, 3645, 3757, 3825, 3925, 4041, 4149, 4293, 4477, 4525, 4581, 4689, 4825, 5013, 5121, 5337, 5445, 5553, 5725, 5733, 5769, 5877, 6025

Offset: 1

Antti Karttunen, Aug 18 2025

Sequence contains also some terms of A386428: 28125, 253125, 1378125, 2278125, 3341637, 3403125, 4753125, etc.

Intersection of A228058 and A347874.
Conjectured to be also the intersection of A228058 and A349751.
Setwise difference A228058 \ A351574.
Cf. also A349755, A387162.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342926(n) = (A003415(sigma(n))-n);
isA228058(n) = if(!(n%2)||(omega(n)<2), 0, my(f=factor(n), y=0); for(i=1, #f~, if(1==(f[i, 2]%4), if((1==y)||(1!=(f[i, 1]%4)), return(0), y=1), if(f[i, 2]%2, return(0)))); (y));
isA347874(n) = ((n%2)&&!isprime(n)&&!(A342926(n)%2)&&!(A342926(2*n)%3));
isA386429(n) = (isA228058(n) && isA347874(n));

A387141 a(n) = floor((Product_{k=1..n} radical(k))^(1/n)) for n >= 1, a(0) = 1, where radical(n) is the product of distinct prime factors of n, cf. A007947.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 8, 7, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 13, 14, 14, 14, 15, 15, 15, 16, 15, 16, 16

Offset: 0

Peter Luschny, Aug 18 2025

nonn

Cf. A007947, A048803, A387140.

a := n -> if n = 0 then 1 else floor(mul(NumberTheory:-Radical(k), k = 1..n)^(1/n)) fi:

A387141[n_] := If[n == 0, 1, Floor[Power[Times @@ ResourceFunction["IntegerRadical"][Range[1, n]], 1/n]]]; Table[A387141[n], {n, 0, 74}]

a(n) = floor(A048803(n)^(1/n)) for n >= 1.

A387151 a(n) = n*n! / Product_{k=1..n} radical(k), where radical(n) is the product of distinct prime factors of n, cf. A007947.

Original entry on oeis.org

0, 1, 2, 3, 8, 10, 12, 14, 64, 216, 240, 264, 576, 624, 672, 720, 6144, 6528, 20736, 21888, 46080, 48384, 50688, 52992, 221184, 1152000, 1198080, 11197440, 23224320, 24053760, 24883200, 25712640, 424673280, 437944320, 451215360, 464486400, 2866544640

Offset: 0

David A. Corneth and Peter Luschny, Aug 18 2025

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..3167

Cf. A387140.

A387151 := n -> n*n! / mul(NumberTheory:-Radical(k), k = 1..n): seq(A387151(n), n = 0..36);

k = 1; {0}~Join~Reap[Do[k *= Times @@ FactorInteger[n][[;; , 1]]; Sow[n*n!/k], {n, 36}] ][[-1, 1]] (* Michael De Vlieger, Aug 18 2025 *)

a(n) = n! / A387140(n) for n >= 1.

cp's OEIS Frontend

User: == 18 && $

A383938 a(n) is the least positive integer k such that b(2*j) is prime for 1 <= j <= n but not prime for j = n+1, where b(1) = k and b(m+1) = b(m) + prime(m) for m >= 1.

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A383939 Numbers k such that k and the k-th triangular number T(k) = k*(k+1)/2 have only even digits.

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A386314 a(1) = 1 and thereafter a(n) is the smallest number k of the form 6*x+-1 not already in the sequence but where the reduced Collatz step A139391(k) is in the sequence.

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A387110 Number of ways to choose a sequence of distinct integer partitions, one of each prime index of n.

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A387111 Number of ways to choose a sequence of distinct positive integers, one in the initial interval of each prime index of n.

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A383942 a(n) = (8*10^(2n) - 10^(n+1) + 2) / 9.

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A387079 Least prime factor of A386482(n).

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A386429 Odd composites k such that A342926(k) is even and A342926(2*k) is a multiple of 3 and which satisfy Euler's condition for odd perfect numbers (A228058).

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