A069675 -id:A069675 - cp's OEIS frontend

A069693 Triangular numbers with either no internal digits or all internal digits are 0.

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 300, 406, 703, 903, 3003, 4005, 8001

Offset: 1

Amarnath Murthy, Apr 06 2002

			The internal digits of 3003 are "00", which are both 0. 15 has no internal digits.

Max Alekseyev, Proof of finiteness of A069693 - A069700

Cf. A069675-A069684, A069693-A069700.

Do[ If[ Union[ Drop[ RotateLeft[ IntegerDigits[n(n + 1)/2]], -2]] == {0}, Print[n(n + 1)/2]], {n, 14, 2 10^6}]

Corrected by Sascha Kurz, Jan 02 2003

A069700 Triangular numbers with internal digits 9.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 190, 496, 595, 990, 5995

Offset: 1

Amarnath Murthy, Apr 06 2002

For proof of finiteness see A069693.

Cf. A069675 to A069684, A069693 to A069700.

Do[ If[ Union[ Drop[ RotateLeft[ IntegerDigits[n(n + 1)/2]], -2]] == {9}, Print[n(n + 1)/2]], {n, 14, 10^6}]

A069695 Triangular numbers with internal digits 3.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 136, 231, 435, 630, 333336

Offset: 1

Amarnath Murthy, Apr 06 2002

For proof of finiteness see A069693.

Cf. A069675 to A069684, A069693 to A069700.

Do[ If[ Union[ Drop[ RotateLeft[ IntegerDigits[n(n + 1)/2]], -2]] == {3}, Print[n(n + 1)/2]], {n, 14, 2 10^6}]

Extended by Robert G. Wilson v, Apr 07 2002

A273460 Numbers n such that sum of the divisors of n (except 1 and n) is equal to the product of the digits of n.

Original entry on oeis.org

98, 101, 103, 107, 109, 307, 329, 401, 409, 503, 509, 601, 607, 701, 709, 809, 907, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1201, 1301, 1303, 1307, 1409, 1601, 1607, 1609, 1709, 1801, 1901

Offset: 1

Michel Lagneau, May 23 2016

Or numbers n such that A048050(n) = A007954(n).

Most of the terms are primes which have at least one 0 among their digits (A056709). The composite numbers of the sequence are 98, 329, 3383, 4343, 5561, 6623, 12773, 17267, 21479, 57721, 129383, 136259, 142943, 172793, 246959, 256631, 292571,...

			sigma(98) - 98 - 1 = 171 - 98 - 1 = 72 and 8*9 = 72 so 98 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007954, A048050, A056709, A069675.

with(numtheory):
for n from 1 to 3000 do:
  q:=convert(n,base,10):n0:=nops(q):
  pr:=product('q[i]', 'i'=1..n0):p:=sigma(n)-n-1:
   if p=pr
    then
    printf(`%d, `,n):
    else
   fi:
od:

Do[If[DivisorSigma[1, n]-n-1==Apply[Times, IntegerDigits[n]], Print[n]], {n, 2000}]
Select[Range[2,2000],Total[Most[Rest[Divisors[#]]]]==Times@@ IntegerDigits[ #]&] (* Harvey P. Dale, Jul 20 2019 *)

A069694 Triangular numbers with internal digits 2.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 120, 325, 528, 820, 1225

Offset: 1

Amarnath Murthy, Apr 06 2002

For proof of finiteness see A069693.

Cf. A069675 to A069684, A069693 to A069700.

Do[ If[ Union[ Drop[ RotateLeft[ IntegerDigits[n(n + 1)/2]], -2]] == {2}, Print[n(n + 1)/2]], {n, 14, 2 10^6}]

A069696 Triangular numbers with internal digits 4.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 741, 946, 6441, 544446

Offset: 1

Amarnath Murthy, Apr 06 2002

For proof of finiteness see A069693.

Cf. A069675 to A069684, A069693 to A069700.

Extended by Robert G. Wilson v, Apr 07 2002

A069697 Triangular numbers with internal digits 5.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 153, 253, 351, 2556, 6555

Offset: 1

Amarnath Murthy, Apr 06 2002

For proof of finiteness see A069693.

Cf. A069675 to A069684, A069693 to A069700.

Do[ If[ Union[ Drop[ RotateLeft[ IntegerDigits[n(n + 1)/2]], -2]] == {5}, Print[n(n + 1)/2]], {n, 14, 2 10^6}]
Select[Accumulate[Range[0,300]],IntegerLength[#]<3||Union[Most[Rest[IntegerDigits[#]]]]=={5}&] (* Harvey P. Dale, Aug 13 2025 *)

Extended by Robert G. Wilson v, Apr 07 2002

A069698 Triangular numbers with internal digits 6.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 465, 561, 666, 861, 46665, 5666661

Offset: 1

Amarnath Murthy, Apr 06 2002

For proof of finiteness see A069693.

Cf. A069675 to A069684, A069693 to A069700.

Do[ If[ Union[ Drop[ RotateLeft[ IntegerDigits[n(n + 1)/2]], -2]] == {6}, Print[n(n + 1)/2]], {n, 14, 2 10^6}]

Extended by Robert G. Wilson v, Apr 07 2002

Corrected by Mark Hudson (mrmarkhudson(AT)hotmail.com), Aug 25 2004

A069699 Triangular numbers with internal digits 7.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 171, 276, 378, 1770, 2775, 5778, 8778

Offset: 1

Amarnath Murthy, Apr 06 2002

For proof of finiteness see A069693.

Cf. A069675 to A069684, A069693 to A069700.

Do[ If[ Union[ Drop[ RotateLeft[ IntegerDigits[n(n + 1)/2]], -2]] == {6}, Print[n(n + 1)/2]], {n, 14, 2 10^6}]

Corrected by Sascha Kurz, Jan 02 2003

A177998 The n-digit prime with the largest number of zero decimal digits, the largest of these if there are more than one. Zero if no such prime exists.

Original entry on oeis.org

0, 0, 907, 9007, 90007, 900007, 8000009, 40000003, 700000001, 9000000001, 40000000003, 700000000009, 8000000000009, 70000000000009, 940000000000007, 9000000000000007, 70000000000000003, 200000000000000003

Offset: 1

Lekraj Beedassy, May 17 2010

The largest number of zero digits in the n-digit prime is n-2, because leading zeros are ignored and the last digit is >0 for primes.

			3-digit primes with one zero are 101, 103, 107,.., 907. The largest of these is 907, which defines a(3).

Cf. A037053, A069675, A109902.

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A069693 Triangular numbers with either no internal digits or all internal digits are 0.

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A069700 Triangular numbers with internal digits 9.

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A069695 Triangular numbers with internal digits 3.

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A273460 Numbers n such that sum of the divisors of n (except 1 and n) is equal to the product of the digits of n.

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A069694 Triangular numbers with internal digits 2.

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A069696 Triangular numbers with internal digits 4.

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A069697 Triangular numbers with internal digits 5.

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A069698 Triangular numbers with internal digits 6.

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A069699 Triangular numbers with internal digits 7.

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